High-dimensional time series regressions are often regularized to produce sparse coefficients. We show that temporal aggregation provides a powerful alternative to reduce dimensionality in high-order autoregressions and mixed-frequency regressions. To this end, we propose StarTime (Sparse Tree-based Aggregation for Time Series), a convex penalization method that uses a temporal tree to arrange lags hierarchically from high to low frequency. StarTime then flexibly selects coefficients to be aggregated at possibly varying frequencies, sparse or a combination thereof. We provide new error bounds for StarTime, demonstrate improved estimation accuracy and recovery of aggregation and sparsity in simulations relative to benchmarks, and illustrate StarTime's relevance for financial and macroeconomic applications.

Hedonic price models are widely used to assess how environmental amenities affect property values, yet methodological guidance for estimating direct price effects remains sparse. We conduct an empirical Monte Carlo simulation to evaluate the performance of traditional and causal machine learning approaches for estimating the direct unmediated price effect of spatially delineated amenities on treated properties (DUET), a conservative lower-bound approximation for welfare changes with direct applications to benefit-cost analysis. Where previous simulations rely on parametric assumptions, we retain the actual data-generating process underlying over 1 million property transactions from upstate New York (1990--2024). By randomly assigning "treatment locations" across iterations we establish a "ground truth" that allows us to precisely measure estimation error. Our results demonstrate that generalized difference-in-differences (DID) regression consistently outperforms baseline DID and two-way fixed effects models across all scenarios. Causal Machine Learning (CML) methods, particularly causal forest DID, achieve comparable performance to generalized DID in most scenarios. In larger samples (above 3,000 treated) increasingly common in contemporary hedonic studies, CML approaches offer substantial advantages when properly specified. Based on empirical simulation results, we provide a set of method-specific best practice recommendations for both traditional regression and causal machine learning approaches.

Aggregated relational data (ARD) consists of survey responses to questions of the form ``how many people do you know who~$X$?'' and is widely used in survey statistics for indirect inference about populations and social networks. The dominant ARD inference target is hidden-population size estimation via the Network Scale-Up Method (NSUM), but ARD is also used for personal-network-size estimation, mixing-pattern recovery, and inference about latent network structure. Bayesian inference for ARD almost universally assumes that, conditional on a respondent's degree, the counts reported for different subpopulations are independent. There are, however, reasons to question this assumption, as homophily, latent-space clustering, and imperfect recall may all induce cross-population dependence. We develop a simulation-based neural estimation framework for ARD which requires only a simulator, so it can be applied to generative models whose likelihood cannot be written down or efficiently evaluated. The framework trains a permutation-invariant neural Bayes estimator that returns, for each marginal parameter, a posterior median and a 95% credible interval, by minimising a multi-quantile pinball loss with a cumulative-gap construction that rules out quantile crossing by design. We demonstrate the framework on three structurally distinct intractable extensions of NSUM-style ARD inference: a stochastic block model, a latent-space model, and a recall-subset model. We apply the framework to ARD Household Survey collected in Rwanda. The framework provides inference on any new survey drawn from the training distribution, and extends the reach of ARD modelling to network-structure and cognitive-process assumptions beyond those currently accessible to likelihood-based inference.

The fraction of variance explained (FVE) in a linear model quantifies the extent to which predictors account for outcome variability. In high-dimensional settings, where traditional FVE estimators do not apply, modern FVE estimators such as GWASH or linear mix-effect model estimated through the restricted maximum likelihood (LMM-REML) struggle with strong correlation among predictors, often found, for example, in brain imaging data. We propose a decomposition framework that partitions the FVE into two components: a low-dimensional component capturing the strong correlation, estimable by low dimensional methods, and a high-dimensional component with remaining weak correlation, estimable by high dimensional methods. Simulations demonstrate that decomposing dominant principal components (PCs) and estimating the high-dimensional FVE using GWASH or LMM-REML leads to improved bias reduction compared to directly applying standard approaches such as GWASH and LMM-REML. Our method shows consistent performance asymptotically as both the number of predictors and the number of samples increase. We illustrate the method in an analysis of the Adolescent Brain Cognitive Development (ABCD) brain imaging dataset, capturing nuanced heritability signals in the FVE of cognitive measures predicted by high-resolution brain imaging data.

Advertising platforms use randomized lift tests to measure incrementality, but privacy-preserving reporting systems degrade the observed signal through match-rate loss, linkability loss, attribution-window loss, aggregation-threshold suppression, randomized reporting noise, and segment-heterogeneous signal loss. This paper formulates privacy-constrained advertising measurement as a robust causal decision problem under the mentioned signal losses. Given a randomized experiment and an ambiguity set for privacy-induced degradation, the framework projects the observation-compatible fiber of clean/unfiltered experimental worlds onto the incrementality functional and returns certified, rejected, and unresolved decisions. The main result gives a sharp decision frontier. Reports outside the frontier support uniformly valid certification or rejection, whereas reports inside it contain too little information for any method to uniformly distinguish above-threshold incrementality from non-incrementality. Supporting results give finite-sample certification, sample-complexity guarantees, a minimax lower bound showing that signal loss reduces effective information, and a reporting-granularity tradeoff. On 2.0M Criteo Uplift rows and the 64K-row Hillstrom email experiment, clean conversion lift is positive in both datasets, with lifts 0.00112 and 0.00495, respectively. Population certification survives mild degradation in Criteo and severe degradation in Hillstrom, while all considered finite-sample stress settings in both datasets remain unresolved after simultaneous uncertainty and reporting noise are included. Overall, the research contributes a decision-theoretic layer for privacy-aware incrementality measurement whose output is the strongest causal-claim justified by degraded ads signals.

Realized volatility has become a standard tool for measuring latent variation in financial assets, and its forecasting is crucial for a wide range of financial applications. We propose a network-based model for forecasting a vector of realized variance processes through the heterogeneous autoregressive (HAR) approach. The generalised network HAR (GNHAR) model incorporates cross-sectional spillovers through a directed graph inferred from Granger-causality tests or connectedness indices, yielding a parsimonious multivariate time series model specification. In an application to ten equities over tranquil and crisis regimes, the proposed GNHAR model improves upon common HAR model benchmarks under both short- and long-term forecasting. We also compare the network-based specification when the jump-continuous decomposition or node-specific option-implied variances are considered. Finally, unlike overparameterised models, our approach yields a concise set of parameters that track the strengthening or weakening of cross-market dependencies, providing a time-varying quantitative assessment of market stability.

We develop a quantitative approximation framework for diffusion distillation, viewing few-step sampling as error propagation under compositions of learned flow maps. Focusing on trajectory distillation for the probability-flow ODE, we show that local approximation errors can be strongly amplified in low-noise multimodal regimes, where the underlying dynamics become stiff. In an analytically tractable Gaussian-mixture Ornstein--Uhlenbeck setting, we separate two core difficulties: approximating the time-dependent score field and controlling the dynamical amplification governed by the time-integrated Jacobian bound of the probability-flow ODE. On the approximation side, we prove constructive L^p(p_t) guarantees showing that ReLU--ReQU networks approximate the Gaussian-mixture score uniformly over time, with depth and width scaling polylogarithmically in the target accuracy and explicitly with the mixture geometry. On the stability side, we derive an explicit bound L(t) for the spatial Lipschitz constant of the probability-flow velocity and convert it into a flow map stability estimate governed by \int_s^t L(u)\,du, making late-time amplification in stiff regimes computable. Building on these estimates, we prove that deep residual compositions efficiently approximate the long-horizon transport, with global error controlled by the stability amplification factor, and identify a Lipschitz-mismatch regime in which one-step distillation is structurally unfavorable. The resulting theory yields a stability-balanced non-uniform time grid obtained by uniform partitioning in the cumulative stability coordinate. Experiments support the prediction and reduce end-to-end relative MSE by up to 51.9\% with 8 segments compared with uniform grids.

Paired comparison models are useful for estimating latent abilities or preferences from binary outcomes, but maximum likelihood estimation can be unstable or fail when the comparison graph is disconnected or nearly separated. Ridge regularization addresses these difficulties by shrinking ability parameters toward a common center, but it can obscure the simple likelihood interpretation that makes Bradley-Terry and Thurstone-Mosteller models attractive to practitioners. This paper describes two data-augmentation perspectives on regularization. The first adds fractional pseudo-games between every pair of competitors. The second adds a fixed-strength phantom player and gives each real competitor a weighted pseudo-win and pseudo-loss against that player. Both approaches yield finite, shrunken estimates; the phantom-player construction also resolves the usual location nonidentifiability without an explicit linear constraint. For the Bradley-Terry model, the two augmentations lead to transparent penalty functions that can be compared directly with ridge penalties. An application to the 2025 Major League Baseball regular season illustrates that tuned pseudo-game and phantom-player regularization can closely reproduce ridge-regularized strength estimates while retaining an intuitive augmented-data representation.

Outliers are known to significantly distort the results of many commonly used clustering methods, often leading to unreliable partitions. To address this issue, several robust clustering approaches have been developed that not only reduce their influence but also facilitate the detection of meaningful outliers. This presentation focuses on robust clustering methods based on trimming, especially TCLUST, which extends the type of trimming used by MCD in one-population problems to the more general case of multiple and unknown clusters. While TCLUST performs well on low-dimensional data, it struggles with high-dimensional datasets due to the complexity of estimating a large number of parameters. The Robust Linear Grouping (RLG) method offers an alternative by assuming clusters lie near lower-dimensional subspaces, thereby combining clustering with dimensionality reduction. However, RLG has limitations when subspaces intersect and assumes overly simplistic isotropic orthogonal errors. A robust clustering method extending TCLUST will be presented, building on the High Dimensional Data Clustering (HDDC) approach by incorporating trimming and eigenvalue constraints. This new approach, called tHHDC, combines TCLUST and RLG, requiring careful modification and integration of both methodologies within that HDDC framework. A study of the theoretical properties of this approach, together with a feasible algorithm for its implementation, will be presented. The interest of the proposed methodology, along with the issue of selecting input parameters, will be illustrated through a simulation study and a real-data example.

Resource-constrained pricing controllers can make fixed-price inference impossible: the controller's resource state may remove the target price neighborhood from the feasible set, even when every realized action has a known positive density. We formalize this support-exclusion failure through a local non-identification result and a realized information clock. We then design a target-aware pricing controller that certifies feasible target bands and logs continuous local densities. Localized debiasing gives studentized intervals whose width is governed by this clock. The resulting regret--information accounting, stated up to pilot re-solving error, shows that cheap exploration can be insufficient for inference: polynomial target mass gives polynomial rates, while a pure $1/t$ target branch does not yield shrinking fixed-target intervals without additional local movement. Experiments show calibration in certified bands and diagnostic abstention when the resource state collapses target support.

TThis paper develops a Dynamic Mini-Max (DMM) framework for repeated surveys comprising a Dynamic Mini-Max Design and a Sequential Hierarchical Bayes Update (SHBU). The DMM jointly optimizes sample size and wave overlap subject to simultaneous precision constraints for levels and movements, a respondent burden limit, and a fieldwork budget. The methods are illustrated using 2021 Australian Census data (t = 1) and simulated waves t = 2, 3, 4. Both the DMM and the classical design start from the same 5% proportional allocation of n_A = 42,018 units. The DMM reduces this to n* = 40,251 while meeting all precision constraints, achieving a cost saving of approximately 6.3%. Level coverage is comparable between the two designs (maximum absolute relative error (MARE) ratio 0.844--1.263). Movement coverage diverges markedly: the DMM achieves 100% across all 27 domain-variable cells, while the classical design achieves only 82%--96% (87.5%--95.0% nationally). The classical confidence interval understates movement uncertainty because it addresses sampling variance only and does not account for the model variance component V_mod_hat. Additional benefits of the DMM framework -- including coherent joint inference for levels and movements, sequential updating without ad hoc composite-estimator chaining, and small area estimation -- are outlined in the paper.

We study projection-based diagnostics for distinguishing directional asymmetry from tail-ratio departure in multivariate data. The procedure reduces the problem to one-dimensional projections and computes two quantile-based summaries: a directional skewness measure evaluated over several quantile levels, and an interquantile tail-ratio evaluated relative to a chosen benchmark. The two summaries lead to a four-regime classification: symmetric benchmark-tail, symmetric tail-departed, skewed benchmark-tail, and skewed tail-departed. The quantile formulation avoids relying on third and fourth moments, which can be unstable in heavy-tailed settings. We establish population properties under central symmetry and ellipticity, uniform finite-sample bounds over the searched directions, and consistency of the threshold classifier under separated regimes. A sparse rank-one calculation is also used to show why coordinate directions can complement random directions in high dimensions. The resulting diagnostic is meant to guide subsequent modelling choices, for example whether a symmetric, skewed, tail-departed, or combined multivariate model is appropriate.

Standard conformal prediction (CP) procedures are typically formulated in terms of p-values, but reliance on p-values alone limits flexibility, for example, when combining dependent evidence across models or data splits. Recent work has explored e-value formulations for conformal inference, yet a direct connection between p- and e-value formulations in CP has been missing, especially regarding their statistical efficiency. We first identify limitations of classical p-to-e calibrators in the CP setting, showing that they are not set-preserving and can lead to overly conservative prediction sets. To address this, we propose a novel P2E calibrator that converts conformal p-values into e-values without altering the prediction set induced by the original conformal p-value. We establish both theoretically and empirically that our calibrator can yield significant efficiency gains over existing p-to-e calibrators. This e-value formulation enables principled use of recent advances in e-value merging and randomization, where we demonstrate its impact in two applications: cross-conformal prediction (CCP), whose variants typically provide only approximate $1-2α$ coverage, and conformal aggregation (CA). In both cases, our e-value-based methods satisfy the desired $1-α$ coverage guarantee while improving efficiency over standard baselines. More broadly, our approach expands the flexibility of CP and opens new directions for efficient, distribution-free uncertainty quantification.

This work proposes a novel solution to predict pulsar timing residuals with limited data, addressing the critical challenge of data scarcity across spin-frequency subgroups of millisecond pulsars in PTA datasets. The proposed solution applies a Long Short-Term Memory (LSTM) network optimized using the model-agnostic meta-learning algorithm, enabling rapid adaptation to new frequency domain by fine-tuning the LSTM network with only a few-shot of ground truth timing residuals. Particle swarm optimization algorithm is also used for automatic hyperparameter optimization, leading to improved prediction accuracy. Our solution, evaluated on the second data release of the International Pulsar Timing Array (IPTA), demonstrates robust generalization with accurate predictions in three metrics across high-frequency test frequency domains, while requiring only 10% of the timing residuals from these domains for model fine-tuning. Furthermore, our lightweight structure only costs 16.86 MB CPU memory and 18 milliseconds for single-step residual prediction. All these characteristics make our solution highly suitable for real-world applications, where effective and real-time predictions of pulsar timing residuals are essential-particularly in resource-constrained environments with limited computational power, memory, or energy availability.

While principal component analysis (PCA) is a fundamental tool for dimensionality reduction, its dense representations make it ill-suited for high-dimensional data. Existing methods address this by promoting sparsity through explicit $\ell_1$-penalties, but these are not obvious to tune due to the unsupervised nature of the task. In contrast, we propose Adversarial PCA (AdvPCA), which leverages robust optimization to achieve sparsity by optimizing the reconstruction objective against bounded, worst-case latent space perturbations. We show that this formulation admits a closed-form reduction, leading to a practical iterative algorithm that alternates between adversarial linear regression-style updates for the sparse encoder and orthogonal updates for the decoder. By theoretically characterizing the solution, we derive a data-adaptive parameterization that allows the algorithm to perform effectively out of the box. We validate these claims through numerical experiments on synthetic and real-world genomics data.

In many risk prediction problems, covariates and a response surrogate are routinely available for a large target population, whereas the true response is costly to ascertain and is observed only for a limited subset. This creates a design problem: one must decide which observations should receive response measurement in order to build a prediction model under a fixed measurement budget. We propose a surrogate-assisted optimal sampling framework for risk prediction under measurement constraints. In the target setting, the surrogate identifies confirmed positive cases, while responses for surrogate-negative observations remain unobserved and can be selectively measured, and thus the sampling design determines how the response measurement budget is allocated. Our framework constructs an optimal sampling design minimizing the leading term of the expected out-of-sample cross-entropy loss and incorporates the resulting design into an inverse-probability-weighted cross-entropy estimator. The proposed design depends only on covariates, the surrogate, and a preliminary estimator, and therefore does not require responses from unlabeled observations at the design stage. We establish consistency, asymptotic normality, and leading-order prediction optimality of the resulting estimator. Extensive simulation studies and two real data applications demonstrate that the proposed design improves prediction performance and exhibits robustness under surrogate misspecification and rare outcome settings.

Modeling high-dimensional data is challenging, yet essential to understanding many complex systems. Maximum entropy models such as Ising and Potts models have been used extensively to capture pairwise interactions from correlation patterns in data, allowing to infer graphical representations of complex systems from observations (e.g., from protein sequences or neural population activity). Recently, there has been growing interest in modeling higher-order correlation patterns involving simultaneously three or more variables. While progress has been made in binary data with high-order Ising models, we extend this framework to the more general case of discrete data. We introduce q-state spin models, a complete family of maximum entropy models that generalize the vector Potts model to include long-range and arbitrary high-order interactions. In the pairwise case, our models allow for more diverse interaction types compared to the standard vector Potts model. We discuss their statistical interpretation with examples and relate them to discrete Fourier analysis. Using a loop expansion of the partition function, we show that the statistical properties of spin models are fully captured by the algebraic structure of their interactions. We define gauge transformations under which this structure, and thus the partition function, remains invariant. Models equivalent under gauge transformations can be seen as different representations of the same abstract statistical model, despite generally having interactions of different orders, extending results from the binary case. For practical application to data analysis, we focus on a subset of models known in the binary case as Minimally Complex Models, generalizing them to discrete data. We obtain a closed-form expression for the marginal likelihood of these models, enabling fast model selection. We illustrate their use with simple real-world examples.

Many systems used in real-world environments require adding new categories and incorporating new information without forgetting what was previously learnt by the classification model. This is known as class-incremental continual learning, and in the case of multivariate time-series, is further complicated by the temporal structure of the data. In this paper, we present a novel approach for performing class incremental continual learning for the classification of multivariate time series data based upon the construction of a dual-stream feature extraction pipeline (using both deep temporal embedding features generated via a pre-trained frozen foundation model and application of statistical features). Evaluated on five benchmark datasets, the proposed system achieves competitive average accuracy across all datasets while maintaining low forgetting rates across all experimental configurations.

Concepts of calibration formalize the compatibility between probabilistic predictions and the respective outcomes. In a nutshell, the outcomes ought to be indistinguishable from random draws from the predictive distributions. In this paper, we review, extend, and bridge notions of calibration that have been proposed for classification and regression tasks. Particular emphasis is given to hierarchical relations between the various notions, as they apply to general real-valued data, continuous outcomes, count data, nominal classes, and binary outcomes. To highlight a number of contributions, we introduce the notion of modal calibration for nominal outcomes, we distinguish full, partial, and average calibration in this setting, and we show that double probability integral transform (PIT) calibration is logically independent of previously proposed concepts of calibration for discrete outcomes. Furthermore, we generalize extant results on concepts of calibration that are expressed in terms of properties or functionals of the predictive distributions, such as means, quantiles, or event probabilities. Throughout the paper, we illustrate the concepts and their hierarchical relations in worked examples, and we provide algorithmic tools that support the construction of instructive examples and counterexamples.

This paper presents a post-Bayesian approach to online filtering in nonlinear state-space models, capable of avoiding over-confident inferences in settings where either the dynamical model, the measurement model, or both, could be misspecified. This is addressed using predictively oriented (PrO) posteriors, an emerging paradigm in which learning (i.e., posterior concentration) occurs if and only if the overall model is well-specified, without strict adherence to Bayes' theorem. As the characterisation of PrO posteriors is challenging, our main technical contribution is a fast approximate linear-Gaussian update procedure, analogous to an (iterated) extended Kalman filter. The methodology, which we call EKF-PrO, has no tunable hyper-parameters and has a computational cost comparable to that of existing filtering methods. Performance is empirically assessed on a range of linear and non-linear applications, in which the state-space model is systematically misspecified.

Chain-of-thought (CoT) reasoning has become a widely used mechanism for eliciting multi-step reasoning in large language models by generating intermediate reasoning steps at inference time. Yet the scaling behavior of generalization with CoT depth remains poorly understood. To address this question, we study a theoretically solvable model of CoT for in-context weight prediction in linear regression, where test-time reasoning is represented as an iterative refinement of the weight-parameter estimate. Using tools from random matrix theory under high-dimensional asymptotics, we derive an exact formula for the generalization error as a function of reasoning depth, pretraining data amount, and context length. Our analysis reveals a sharp phase transition separating exponential and polynomial improvement, saturation, and overthinking, and characterizes how the optimal reasoning depth scales. We further show that deeper reasoning is most effective with sufficiently rich pretraining and in-context information, whereas limited pretraining or context makes longer reasoning prone to error amplification or saturation. We also validate these predictions through experiments on fully learned linear attention and softmax attention models. Our results provide a unified theoretical account of how test-time CoT depth affects generalization.

Active Statistical Inference is a new framework to make precise claims about population parameters with provable statistical guarantees. It uses a predictive "black-box" machine learning (ML) model to strategically decide which data points to label, roughly prioritizing samples for which the ML model is unsure about their label values. A major issue is that the framework can be brittle when uncertainty estimates are noisy. This paper introduces OPAL (Optimized Policy for Allocation of Labels), which learns a labeling strategy within a tractable class of smooth policies to yield estimators with the lowest variance. In effect, OPAL is an end-to-end pipeline that turns a black-box model's uncertainty scores into a data-adaptive labeling strategy and then performs inference on the collected samples. We evaluate OPAL on real datasets spanning medical imaging data, computational social science, and proteomics. As a concrete example, we consider predicting breast cancer subtype from histopathology images and using OPAL to form valid confidence intervals for odds ratios for different demographic groups. We show that OPAL achieves nominal coverage in finite samples and has the accuracy one expects from methods which have far more labeled samples.

Tail index regression studies how covariates affect tail heaviness in heavy-tailed data. In many applications, data are distributed across heterogeneous sources, where direct pooling is infeasible due to privacy or regulatory constraints. Existing methods mainly focus on single-dataset analysis and do not address heterogeneous federated settings. We develop a personalized federated framework for high-dimensional tail index regression that accommodates client heterogeneity while exploiting latent similarities across clients. The proposed estimator combines sparsity regularization with nonconcave fusion penalties to perform coefficient estimation, variable selection, and group recovery. We establish non-asymptotic convergence rates and show that the estimator enjoys an oracle property by consistently recovering the underlying grouping structure. For computation, we develop an ADMM-based federated algorithm with adaptive gradient updates and establish its convergence guarantees. We further propose a debiased federated inference procedure based on adaptive weighted aggregation across related clients, yielding valid confidence intervals and hypothesis tests with improved efficiency over target-only inference. Simulation studies and real-data analysis demonstrate the effectiveness of the proposed methods.

With the increasing scale and number of wind farms, wind turbines' daily operation and maintenance costs are increasing. To reduce operation and maintenance costs and enhance the reliability of wind turbine and system operation data before reaching catastrophic failures, monitoring the operating status of the equipment and detecting failures at an early stage is crucial. It is of great practical significance to utilize the working condition data for abnormal assessment of the operating status of wind turbines to realize abnormal monitoring of the operating status of wind turbines. However, the existing anomaly detection methods can neither perform effective relational modeling in data filled with a large amount of redundant information nor reasonably utilize the valuable anomaly data. For this reason, this paper proposes an anomaly detection model that fuses a Transformer and a generative adversarial network. Firstly, it reduces the leakage detection rate of minor deviation anomalies by amplifying the reconstruction error. Secondly, it uses autoregressive inference to extract multimodal features to enhance the stability and generalization ability of training. Finally, the temporal feature extraction module is constructed to promote the interactive learning between features of different time scales and effectively reduce the time redundancy. The results of multiple sets of experiments conducted on real WTG datasets show that TransGAN-WT achieves an average F1 score of 96.10% across multiple wind turbine datasets, which is 5.84% and 2.89% higher than several other state-of-the-art baseline methods. It also realizes a false positive rate (FPR) of 0.06%, and is verified by the Wilcoxon signed-rank test to have achieved a statistically significant performance enhancement compared to the state-of-the-art baseline methods, effectively ensuring the stable operation of wind turbines.

Count data with excess zeros arise frequently in health economics and epidemiology. The standard Poisson Hurdle Model (PHM) parametrises the underlying Poisson rate directly, so its count-component coefficients are log-rate ratios rather than log-ratios of the marginal mean. Consequently, the incidence density ratio (IDR) from the PHM is neither exact nor constant across covariate profiles, complicating applied reporting. We propose the Marginalised Poisson Hurdle Model (MPHM), which reparametrises the count component so that the coefficient vector beta directly governs the marginal mean E[Y]. A nonlinear connector equation links the structural Poisson rate to this parametrised mean. We prove existence and uniqueness of the connector solution, develop a vectorised Brent's-method solver, derive the score equations and block-diagonal Fisher information, establish asymptotic normality, and prove that exp(beta) is exactly constant across all covariate values. A simulation study with n in {100, 250, 500, 1000}, zero proportion pi in {0.2, 0.4, 0.6, 0.8}, and R = 200 replications confirms consistency, near-zero bias, and 95% Wald coverage of 0.905-0.975 across all 16 scenarios. Applied to the NMES1988 physician visit data (n = 4,406), the MPHM yields IDR = 1.163 (95% CI: 1.150-1.177) per additional chronic condition - an exact, population-wide effect not derivable from the PHM. The MPHM resolves the non-constant IDR problem by directly parametrising E[Y]. The resulting IDR holds for every individual and the whole population without further marginalisation, substantially simplifying the reporting of covariate effects in health utilisation research.

Modeling interactions among multimodal, high-dimensional data is intrinsically challenging due to ultra-high dimensionality and complex dependence structure with high level noise. Screening methods are effective for reducing dimensionality, but most existing approaches shrink only the predictor space while retaining all outcomes. In cross-modal analyses, different outcomes often select different predictor subsets, so the union remains large and the response dimension is unchanged, limiting the practical benefit of screening. This gives rise to heavy computational burdens and poor interpretability. To address these limitations, we propose a new screening framework, Graph Independence Dual Screening (GIDS), which simultaneously reduces the dimensionality of response variables and predictors. We design computationally efficient algorithms that facilitate downstream selection procedures, improving accuracy and scalability, and establish supporting theoretical results. Extensive simulation studies demonstrate that GIDS outperforms existing methods that screen only predictors. To illustrate its utility, we applied GIDS to the Alzheimer's Disease Neuroimaging Initiative (ADNI) dataset, analyzing interactions between genome-wide 865,353 DNA methylation and 49,386 transcriptomic variables. GIDS reduced the feature space to approximately 9,000 CpGs and 2,000 transcripts, uncovering blockwise interaction structures: clusters of CpG sites and gene transcripts with strong associations. These findings not only improve computational tractability but also yield interpretable biological insights, highlighting coordinated regulatory mechanisms underlying Alzheimer's disease.

Switchback experiments -- in which treatment is assigned at the level of a cluster crossed with a time period -- are widely used in marketplace and platform settings, yet no closed-form power formula exists for them. We fill this gap by deriving a closed-form, multi-level asymptotic variance approximation for the individual-level OLS estimator, facilitating power budgeting. Using this formula, we reveal a structural floor on statistical power: while idiosyncratic noise vanishes with observation density, macro-level shocks are multiplicatively penalized by cluster size imbalance. We confirm through analytical derivations and Monte Carlo simulations that the formula is exact across typical parameters and serves as a mathematically conservative upper bound in extreme boundary regimes. We study three methodological applications. First, we prove that advanced assignment designs like stratification only partially eliminate the penalty of cluster size imbalance on power. Second, we demonstrate that variance reduction techniques targeting macro-level shocks yield disproportionately greater efficiency gains than those targeting residual noise. Third, we formalize the finite-sample power trade-offs between individual-level and cell-level estimators.

The Galton--Watson process (GWP) is a discrete-time branching process model that provides a powerful tool for analyzing epidemic data and estimating key epidemiological parameters such as the basic reproduction number. When used with surveillance-based cluster size data, the GWP can also elicit information about the extent of transmission heterogeneity, even when each transmission process is not directly observable. When cluster size distribution data are available, the parameters that govern the transmission can be statistically inferred by using the probability mass function that corresponds to the observed cluster size data. For multi-type GWPs, however, real-world applications remain limited, possibly because of the absence of conceptually and practically straightforward approaches for deriving the closed-form solution of the final size distribution. In the present study, we propose a framework for computing the final size distribution of multi-type GWPs, using a method for the choice of the Cauchy integral contour. We provide examples of how our framework can be applied to both simulated data and real-world data of Middle East respiratory syndrome, and discuss potential pitfalls surrounding the identifiability of parameters for statistical inference when using likelihoods that are not conditioned on extinction.

Gradient observations can substantially improve Gaussian process (GP) surrogates, particularly in high-dimensional settings where function evaluations are expensive. However, exact inference with $n$ function values and $n$ full gradients in $d$ dimensions scales cubically in the joint state size, imposing an intractable $\mathcal{O}(n^3 d^3)$ computational bottleneck. We introduce TERA, a highly scalable derivative GP method based on target-specific exact gradient reduction. We prove that for stationary kernels, the gradient components orthogonal to the directions connecting the target and conditioning points are conditionally independent of the target function value; consequently, the exact conditional density is fully characterized by at most $m^2$ directional derivatives once a conditioning set of size $m$ is specified. By using these reduced, dimension-free conditionals as local factors in a Vecchia approximation, TERA effectively decouples $n$ and $d$ from the dense matrix inversion. This reduces the per-target evaluation cost to $\mathcal{O}(dm^2 + m^6)$ time and $\mathcal{O}(dm^2 + m^4)$ memory, leaving the underlying derivative GP model mathematically unchanged. Empirical evaluations demonstrate that TERA achieves state-of-the-art predictive accuracy while operating orders of magnitude faster than standard derivative GPs. Crucially, both computation time and peak GPU memory remain essentially flat with respect to $d$, enabling highly scalable inference in high-dimensional spaces.

Neural posterior estimation (NPE) is a simulation-based approach to Bayesian inference that trains a neural network to approximate the posterior distribution from simulated parameter - data pairs, bypassing likelihood evaluation. We apply NPE -- to our knowledge for the first time -- to stochastic susceptible-infectious-removed (SIR) epidemic models observed through final outcome data, considering both homogeneously mixing and household-structured populations. Such data arise naturally in retrospective outbreak investigations and household transmission studies, yet inference is computationally challenging: data-augmentation Markov chain Monte Carlo (MCMC) can be slow to mix in large populations and difficult to implement, while Approximate Bayesian Computation (ABC) suffers from low acceptance rates, particularly for large populations or unlikely outcomes. The discrete, low-dimensional nature of such observations makes this setting particularly well suited to NPE. We show that a logNormal posterior approximation, parameterised by a feed-forward neural network, accurately recovers reference posteriors across a range of population sizes and transmission regimes, and extends naturally to joint inference on global and local transmission rates in the household model. Once trained, the network produces approximate posterior distributions in seconds and generalises reliably to population sizes and structures not seen during training. Performance on both synthetic and real outbreak datasets is consistently strong, with results in close agreement with published analyses.

Mediation analysis plays an essential role in uncovering the mechanisms by which an exposure influences an outcome through intermediate pathways. While methodological advances for single-mediator settings are well established, rigorous tools for handling multiple, sequentially ordered mediators remain underdeveloped. Such settings are common in applications like longitudinal cohort studies, where exposures operate through complex chains of mediators over time. In this paper, we establish a general framework for sequentially ordered mediators that enables the identification and formal decomposition of the total effect into component path-specific effects. We also develop estimation procedures for mediation estimands with both continuous and categorical outcomes. Furthermore, we introduce a new testing strategy to conduct inference using a studentized statistic combined with data-splitting. This approach achieves valid Type I error control under the composite null across diverse data-generating mechanisms. Through extensive simulations and applications to two large-scale empirical studies, we demonstrate that the proposed methodology provides reliable estimation, valid inference, and improved power for discovering novel mediation pathways.

Computer experiments with both quantitative and qualitative inputs have become common across various areas. However, constructing accurate and computationally efficient emulators for such experiments at large scales remains a significant challenge. We propose a novel, scalable framework for emulating computer experiments with mixed inputs. Our approach is based on a new covariance function integrating additive Gaussian Processes (GPs) to handle the mixed inputs, with Vecchia approximation for scalability. We demonstrate that methods for large-scale computer experiments can be effectively extended when paired with our proposed modeling framework.

Gradient boosted decision trees require a stopping rule to avoid overfitting. The standard rule monitors a validation loss and stops if the loss fails to improve for a fixed patience period. However, the patience parameter has no interpretable scale and validation losses can be noisy or implicitly defined by a user-specified gradient. We propose ScoreStop, a gradient-based early-stopping rule that casts the stopping decision at each iteration as a test of the null hypothesis that the current predictor is the population risk minimizer. We use a functional score test, computed on validation data, with a statistic that is scale-invariant in the update direction, with a known asymptotic distribution under the null. Because our test uses gradients rather than loss values, the same construction applies to implicit losses such as LambdaRank, and data-dependent losses such as Cox regression via influence functions. In synthetic experiments and real-data benchmarks, we show that ScoreStop is competitive with loss-based methods.

Visual and quantitative goodness-of-fit diagnostics are an important tool in the practitioner's toolbox. The need for convincing and reliable diagnostics is particularly clear when fitting extreme value regression models, which are used for extrapolation far beyond the observable range of the response variable, and often evaluated at unobserved covariate values. Despite this, few diagnostics have been developed for extreme value regression models, and those available often suffer in terms of interpretability or scalability on low-dimensional or non-Euclidean covariate domains, often encountered in modern applications. Moreover, existing methods tend to offer a global perspective on model fit; that is, they quantify goodness-of-fit across the entire dataset, without offering insight into regions of the covariate space where the model fit may be poor. We propose two novel visual diagnostics for extreme value regression models: the standardised tail plot and the normalised residual plot. By considering the asymptotic distribution of normalised exceedance probabilities, we show that uncertainty bounds for our plots are approximately independent of the sample size used in their construction. This allows us to propose visual diagnostics which can efficiently and consistently compare goodness-of-fit at both a global and regional level, despite varying sample sizes over regions of the covariate domain. Following a discussion of summary statistics for global and regional goodness-of-fit, we provide two applications of extreme value regression models that illustrate how our diagnostics can be used to perform model comparison (across thousands of candidate models) and provide actionable findings that support model design.

Latent state-space models are widely used to study partially observed dynamical systems, yet most formulations assume that process variability is independent of latent-state position. In many biological, behavioral, and physiological systems, however, variability may depend systematically on the underlying dynamical state, producing structured stochasticity that is not captured by constant-variance models. We introduce a state-coupled stochastic volatility framework in which latent process variance depends on displacement from a latent equilibrium. To estimate this relationship under partial observation, we develop a particle expectation-maximization procedure combining bootstrap particle filtering and backward trajectory smoothing. The model includes a coupling parameter, $γ$, that quantifies the strength of association between latent-state position and process variability. A large-scale simulation benchmark evaluated recovery and detection performance across varying coupling strengths, observation noise levels, trajectory lengths, and persistence regimes. The proposed framework consistently reduced recovery bias relative to an observed-state heteroskedastic proxy, with the largest improvements occurring under strong coupling. Recovery performance improved with increasing latent persistence, while detection performance remained competitive across a broad range of conditions and became increasingly advantageous as observation noise increased. Taken together, the results demonstrate that state-coupled volatility can be identified and estimated under partial observation when latent-state structure is explicitly modeled. The framework provides a practical methodological foundation for studying state-dependent variability and evaluating whether structured stochasticity contributes information about system dynamics beyond that contained in mean-state trajectories alone.

Periodic target updates in Q-learning and soft target updates in actor-critic methods are empirically well established stabilization mechanisms, but their precise theoretical explanation is still incomplete. This paper gives a rigorous and exact analysis of these mechanisms for Q-learning with linear function approximation (linear Q-learning) using the exact switched linear system (SLS) dynamics induced by the Bellman maximum and the joint spectral radius (JSR) of the resulting switching matrix families. Although linear Q-learning can fail to converge in general, we prove that, under explicit spectral and step-size conditions, periodic hard target updates and soft target updates can guarantee convergence to the exact projected Q-Bellman solution. The main analysis is carried out for deterministic linear Q-learning, where the target-update mechanism is most transparent. Once the corresponding JSR certificate is established for the mean recursion, the stochastic reinforcement-learning setting can be treated by replacing deterministic modes with sampled stochastic modes and adding the corresponding stochastic-noise analysis.

Modern Machine Learning (ML) and Artificial Intelligence (AI) models, especially large language models (LLMs), are increasingly used to generate scientific hypotheses and mechanistic explanations from observational data. This position paper argues that in the high-dimensional proxy regimes where modern ML excels, mechanistic learning is generically underdetermined: many incompatible mechanisms induce essentially the same observational relationships on the support of the data, so predictive success and coherent explanations are insufficient evidence of mechanism discovery. This underdetermination becomes uniquely hazardous with large language models (LLMs), which tend to collapse large equivalence classes of explanations into a single fluent narrative. This paper proposes concrete standards for ``mechanistic ML,'' and argues these norms are necessary if LLM-centered workflows are to support science rather than merely simulate it.

Urban nitrogen dioxide ($NO_2$) is a key indicator of combustion-related air pollution and exhibits strong spatial and temporal variability in cities. This study presents a satellite-based framework for tracking urban $NO_2$ pollution using tropospheric column observations from Sentinel-5P/TROPOMI over Guayas Province, Ecuador. Rather than estimating surface concentrations, the methodology emphasizes robust distributional metrics, including the median and upper-tail percentiles ($P_{90}$, $P_{95}$, and $P_{99}$), to characterize background conditions and localized pollution extremes at the canton scale. Multi-year satellite observations are aggregated annually and analyzed using unsupervised K-means clustering to identify characteristic pollution regimes without predefined thresholds. Results show that highly urbanized cantons consistently exhibit elevated extreme $NO_2$ values and greater variability, while less urbanized areas display lower and more homogeneous patterns. The proposed approach provides an interpretable and scalable tool for urban air-quality assessment in data-scarce regions using satellite observations alone. The implementation is publicly available on GitHub https://hvelesaca.github.io/sentinel-5P-clustering/.

Integrating over model uncertainty in factorial designs via Bayesian model averaging is hindered by the combinatorial explosion of interpretable interaction effects, often yielding a multimodal posterior, where standard Markov chain Monte Carlo algorithms encounter significant convergence issues. We propose a general computational framework that repurposes Rashomon sets, collections of high-performing models traditionally valued for prediction and interpretability, as a strategic "warm start" for estimating the full posterior. Our method, Rashomon-seeded annealing, initializes annealed importance sampling (AIS) by anchoring the starting density within these pre-identified, high-evidence regions while preserving global support over the entire model space. Rather than restricting inference to the Rashomon set and understating uncertainty, the AIS correction restores full posterior inference, turning the Rashomon certificate from an inferential truncation into a proposal mechanism. We demonstrate this approach using Rashomon Partition Sets (RPS) as a rigorous, certified seed constructor for factorial designs. The resulting algorithm yields consistent self-normalized posterior summaries, such as model-averaged cell means, credible intervals, and uncertainty summaries without exhaustive enumeration of the complete model space. This bridges the gap between high-evidence model discovery and rigorous Bayesian inference, and outlines a general strategy in which any high-posterior seed set can provide computational leverage for AIS-based model averaging.

Recently, the mapping of the replicator equation onto Bayes' theorem has been recognised, leading to an analogy between evolutionary dynamics and Bayesian learning. However, this analogy holds only for pure selection in infinite populations and breaks down when mutations -- a central mechanism of evolution -- are introduced. Here I propose that evolution by natural selection, at least for populations of haploid replicators in static environments, is best understood not as a learning process but as a process of causal inference. Each mutation event constitutes a natural experiment in which the parent serves as the control and the mutant offspring as the treated unit. Natural selection screens the causal effect of the mutation on fitness, retaining mutations with non-negative effects. I formalise this view within the Neyman-Rubin potential-outcomes framework. I first develop the general theory using a generic fitness outcome and show how the core identification assumptions in causal inference (Stable Unit Treatment Value Assumption, Consistency, Unconfoundedness, Positivity) map onto evolutionary biology. Using the unnormalised quasispecies equation, I prove that the intergenerational change in mean fitness decomposes exactly into a selection term -- recovering Fisher's Fundamental Theorem -- plus a mutation term that corresponds to a fitness-weighted average of the cumulated effect of all mutations over all parental genotypes. I show that this decomposition extends, under suitable assumptions, to the generalised replicator-mutator equation and that the frequencies of populations of matched parents-offspring update in proportion to the average causal effect of mutations on fitness.

This paper investigates non-stationary online learning using the metric of interval regret, which requires an online algorithm to perform well over every time interval. We propose the first online learning algorithm that achieves an interval regret bound scaling with gradient variation, a fundamental measure of the cumulative change in online function gradients, which relates to various problem-dependent quantities and is closely connected to stochastic optimization and other problems. Our method employs a simple and efficient two-layer online ensemble structure that achieves strong theoretical guarantees. Specifically, it enjoys a regret bound that simultaneously adapts to various problem-dependent quantities while also preserving the minimax-optimal rate in the worst case. Moreover, recognizing the challenge of hyperparameter tuning, we introduce a Lipschitz- and smoothness-agnostic variant that automatically adapts to these potentially unknown constants. This is primarily enabled by a novel Lipschitz-adaptive meta algorithm, which may be of independent interest. Beyond interval regret, our method also yields broader implications: it provides versatile bounds for interval dynamic regret, a stronger measure that competes with changing comparators over any interval, and yields the first piecewise characterization for stochastic extended adversarial optimization. Theoretical findings are validated by experiments.

Large Language Models remain plagued by hallucinations. Recent work has sought to tame their prevalence using statistical techniques based on conformal prediction, with both theoretical and empirical success. However, these methods operate in a post-hoc fashion, treating the sampling procedure itself as atomic and then surgically altering samples to remove hallucinated claims. This disconnect between filtering and generation can result in samples that are incoherent, inconsistent, or simply unlikely under the model itself. Moreover, post-hoc surgery is unable to shift probability mass towards more useful and helpful responses. To address these issues, we propose to instead sample from approximations to an LLM posterior, where the conditioning event corresponds to a calibrated, high-scoring region. We develop a calibration procedure tailored to the setting of conditional sequential generation that effectively identifies this region and achieves target risk control. Empirically, we apply our method to case studies focused on open-ended biography generation and mathematical problem solving; compared to prior work, we obtain the same statistical guarantees, with higher downstream utility.

Score-based diffusion models have emerged as prominent deep generative models; however, their application to tabular data remains challenging because their backbones assume fully specified inputs, whereas real-world tabular data often contain missing values. We propose AugMask, a plug-and-play training framework that adapts missing-unaware backbones to incomplete data by separating conditioning from supervision. AugMask 1) constructs numeric inputs via conditional stochastic augmentation using lightweight auxiliary models, and 2) applies denoising supervision only to observed coordinates. In effect, augmented missing entries serve as uncertain conditioning context rather than training targets. We connect this training rule to a Rao--Blackwellized objective and show that marginalizing missing entries yields a variance-weighted sensitivity penalty, discouraging over-reliance on uncertain completions. Across diverse datasets and missingness regimes, AugMask enables standard diffusion-based tabular generators to outperform specialized missing-aware baselines.

In nature, events that affect some individuals or groups but not others constitute an implicit intervention and are known as natural experiments. For example, the COVID-19 pandemic was an intervention by the coronavirus on the sub-population infected with COVID. We ask, do natural experiments occur in existing real-world datasets? If yes, how should we treat them? To detect natural experiments in data, we use causal discovery to recover the underlying causal graph and perform feature selection based on causal links. If downstream performance improves by treating the data as interventional rather than observational, we argue that this suggests the dataset contains natural experiments. We first validate this hypothesis by simulating datasets with and without natural experiments using synthetic graphs. We then perform a systematic empirical evaluation on a large suite of real-world datasets. Our results indicate that real-world datasets do contain natural experiments and we can take advantage of those natural experiments to improve model performance using causal inference. Our work represents the initial foray into this area, offering a preliminary exploration within a limited scope.

Understanding how structured internal structure emerges during neural network training is central to the study of deep learning. We investigate this phenomenon through the group composition task, where a two-layer neural network is trained to predict $g_1 \star g_2$ for elements of a finite group $G$. By lifting the projected gradient flow to the Fourier domain, we demonstrate that the training dynamics are governed by a Riemannian gradient ascent on a representation-theoretic energy functional. We prove that, under random initialization, this flow drives each neuron to converge almost surely toward a single irreducible representation, while the cross-layer Fourier coefficients achieve a rotational rank-one alignment. This framework provides a representation-theoretic account of feature learning and characterizes a novel low-rank compression phenomenon for matrix-valued group representations. Moreover, for Abelian groups, we provide a complete population-level description: random initialization promotes uniform diversification across nontrivial representations and induces Haar-uniform phases, jointly approximating the indicator via a majority-vote mechanism. We further prove that both phase alignment and representation competition emerge with exponential convergence rates.

We study the geometry of feasible value functions in infinite-horizon partially observable Markov decision processes (POMDPs) under memoryless stochastic policies. Our main contribution is a characterization of the feasible set of value functions as a semi-algebraic set, defined by explicit polynomial inequalities determined by the transition dynamics, observation kernel, and reward structure of the POMDP. This result extends prior work for fully observable Markov decision processes, where the feasible set is known to be a polytope, to the substantially more intricate partially observable setting. In contrast to the polyhedral structure arising in MDPs, partial observability induces fundamentally nonlinear constraints, leading to a richer and more complex geometric structure. Our geometric characterization provides new insight into the landscape of policy optimization in both MDPs and POMDPs, and reveals qualitative phenomena unique to partial observability, including the emergence of isolated local maximizers of the long-term reward and their dependence on the initial state distribution.

We develop a contraction-based framework for proving mixing-time bounds for Markov chain Monte Carlo algorithms. The framework is built around global and local contraction coefficients of Markov kernels under the $\mathsf E_γ$-divergence with $γ\ge1$. For projected Langevin Monte Carlo on a compact convex domain, we show that Gaussian smoothing yields an explicit global contraction coefficient for the $\mathsf E_γ$-divergence. This gives a direct proof of exponential convergence to the discretized stationary distribution for general smooth, possibly non-convex potentials. The rate is explicit, accommodates arbitrary random-batch sampling schemes, and yields convergence guarantees for several divergences, including KL, $χ^2$, and Rényi divergences. For independent Metropolis--Hastings with target $π$, proposal $q$, and unbounded importance weight $w=dπ/dq$, global contraction coefficients are typically trivial. We therefore introduce a local contraction coefficient on the core $C_R=\{w\le R\}$ and prove that it controls the rejection profile on the core. This yields warm-start convergence bounds governed by the local contraction coefficient and the tail profile $H_R=π(w>R)$, recovering sharp existing moment-based convergence rates when $\mathbb E_q[w^p]<\infty$ for some $p>1$, while remaining effective in heavy-tailed regimes where no finite moment of order $p>1$ exists.

Online debate platforms offer a unique window into the mechanisms driving opinion formation: they capture both explicit political preferences and the peer environment in which those preferences are expressed. In this work, I develop a Bayesian logistic regression model, inspired by ideal point models from political science, to disentangle two competing mechanisms of voting behaviour in online debates: conviction, driven by prior ideological beliefs, and conformity, driven by peer influence. I apply this framework to the Debate.org dataset, comprising approximately 341k votes across 78k debates on 48 socio-political topics. As the debate platform does not provide predefined topic labels for each debate, I infer the topic and stance from the debate text using large language models, and, with a Bayesian approach, I quantify the relative contribution of each mechanism. I find substantial heterogeneity across topics: conviction dominates on issues tied to personal freedoms and lifestyle choices, such as drug legalisation and legalised prostitution, while conformity dominates on several topics widely regarded as paradigmatic cases of moral conviction, including abortion, gun rights, and global warming. These results have implications for the stability of online political discourse and the design of deliberative platforms.

The discovery rate of fast radio bursts (FRBs) continues to increase with the advent of new radio facilities and yet extracting their astrophysical parameters such as scattering timescale ($τ$) remains a significant bottleneck. Current $τ$ measurement approaches like fitting analytic template models and scattering aware de-convolution are accurate but slow, sensitive to initialization, limited by low signal to noise and often require manual supervision. These limitations inspired us to explore fast, robust and scalable machine learning methods to estimate the astrophysical parameter value. We present a deep learning approach named Multimodal Transformer Based Generic Mixture Density Network (MT-GMDN) which ingests FRB dynamic spectrum and its corresponding timeseries profile through parallel transformer encoders, fuses their latent representations and predicts the distribution of $τ$ with probabilistic output derived from generic mixture-density formulation. This formulation not only estimates the value of $τ$ but also captures the (zero inflated) nature of FRB populations where a significant fraction of bursts exhibit unresolvable scattering. We trained MT-GMDN on $\sim3500$ FRBs from CHIME/FRB \cattwo while holding out some fraction of FRBs for validation during training and for testing after the training completes. The model achieves a coefficient of determination ($R^2$) value of $94\%$ on the expected value of $τ$ for the events with measurable scattering with an excellent recall value of $90\%$ on the test data set. The model was also able to incorporate heteroskedastic errors enabling us the construction of a confidence interval for the predictions.

We ask: when do Bayesian model averaging (BMA) weights over decision trees carry sufficient epistemic information to justify committed exploitation of the averaging distribution? We answer this question in closed form for Bayesian decision trees (BDTs) with Dirichlet-Multinomial leaf models and a Catalan-exponential tree-size prior (Schetinin&Jakaite, 2025), establishing a complete non-asymptotic theory of rational commitment thresholds.

Many interventions alter the structure of an outcome distribution rather than its mean: they can split a population into disconnected regimes, create loops or holes, generate branches, or reorganize an outcome cloud while leaving the average response nearly unchanged. In such settings, mean-based causal estimands such as the average treatment effect may miss important structural effects. We introduce topological-geometrical causal metrics based on summaries of interventional outcome laws, including density-superlevel Betti summaries, Euler signatures, and persistent-homology summaries. These metrics quantify structural differences between treated and untreated outcome laws beyond averages. We also study the assumptions needed for causal interpretation. We introduce topological ignorability, a topological analogue of conditional ignorability that requires invariance of the chosen structural feature rather than the full counterfactual distribution. When the chosen summary is injective, this condition coincides with weak ignorability; for noninjective summaries, it can identify the structural feature of interest without identifying the full interventional law. We define a covariate-standardized topological-geometrical causal effect and develop practical estimators. We validate the framework in two hidden-confounding benchmarks: a fully synthetic exact benchmark and a real-covariate semi-synthetic benchmark using Wisconsin breast-cancer covariates. In both, weak ignorability fails and balancing observed covariates nearly eliminates standardized mean differences, yet the coordinate-mean average treatment effect remains biased. By contrast, selected finite density-superlevel Betti and Euler contrasts remain stable across oracle, observational, and weighted analyses.

We propose CerT-MCMC, a framework that equips learned-transport Markov chain Monte Carlo with automatic, rigorous convergence certificates. A normalising flow maps a Gaussian reference to an approximation of the target posterior; the same flow then serves as both the independence Metropolis-Hastings proposal and the basis for a computable spectral-gap bound. We develop two complementary certificates. The covering certificate bounds the weight-ratio oscillation over the full proposal support via finite-sample covering arguments, yielding full-support spectral-gap bounds when a conservative gradient bound is available; its correction term scales as O(n^{-1/D}), making it rapidly weak and eventually vacuous as dimension increases. We prove a matching Omega(n^{-1/D}) lower bound, establishing that this barrier is intrinsic to pointwise Lipschitz certification. The quantile-core certificate restricts attention to a high-probability residual core on which the oscillation is controlled by one-dimensional empirical quantiles, with a finite-sample probability slack of O(n^{-1/2}), independent of the ambient dimension. On synthetic targets (D=2-20), structural-engineering posteriors (D=6,8), real-data logistic regression on the Heart Disease data set (D=13), and synthetic Bayesian logistic regression (D=20), the quantile-core certificate delivers non-vacuous spectral-gap bounds where the covering certificate is vacuous, and its spectral-gap proxy tracks empirical effective sample sizes within 7%. A negative control experiment confirms that the certificate discriminates flow quality by a factor exceeding 10x, whereas acceptance rates differ by only 1.15x. To our knowledge, the dual-certificate framework is the first to provide automatic, dimension-aware convergence certificates for learned-transport MCMC, distinguishing genuine transport failure from proof-technique limitations.

The Empirical Bayes (EB) procedure of Hauer et al. (2002) is the workhorse of highway safety analysis: it combines a Safety Performance Function with observed crash counts to produce shrinkage estimates of segment-level crash rates. EB delivers practicality by holding several quantities fixed at calibration: SPF coefficients, per-type overdispersion, observed ADT, and a fixed exposure exponent. These assumptions strain when ADT is missing on a majority of segments. We present a fully Bayesian hierarchical model that moves beyond EB by relaxing each of these assumptions in a single joint inference. Fit on Ohio's road inventory (408,304 segments, 2.9 million crashes, 2013-2025), the model jointly imputes missing ADT and estimates per-segment crash rates with uncertainty. Posterior predictive checks of an initial fixed-exposure model expose a tail misfit; relaxing the exposure structure to a per-functional-class exposure exponent and an estimated length exponent, in place of a single scalar and a fixed offset, resolves it and improves out-of-sample predictive accuracy (PSIS-LOO $Δ\mathrm{elpd}$ = 9,394, SE 238). Crash count is sublinear in traffic in every class (exposure exponents 0.49-0.70, all $<1$, the safety-in-numbers effect) and sublinear in segment length ($β_{\mathrm{len}} = 0.69$). Partial pooling substantially improves out-of-sample predictive accuracy over complete pooling (PSIS-LOO $Δ\mathrm{elpd}$ = 4,780, SE 225). The Bayesian ADT submodel attains $R^2_{\log} = 0.756$ by encoding county and functional class as hierarchical priors, versus $0.653$ for a LightGBM restricted to the same continuous predictors. The output is a posterior crash rate distribution per segment, replacing the median-by-type point estimates used in our prior risk-aware routing framework.

We study the contraction in Wasserstein distance of the coordinate ascent variational inference algorithm. This is shown to hold under a transport-information inequality at the fixed points and a functional smoothness condition. The results are general and sharp, allow for local convergence guarantees, hold for general smooth manifolds, and also in some non-smooth spaces. We consider applications to Bayesian Gaussian Mixture Models, and high-dimensional Bayesian Probit Regression, and Logistic Regression with Pólya-Gamma random variables (i.e. Jaakkola-Jordan's algorithm).

It is increasingly common in machine learning to use learned models to label data and then employ such data to train more capable models. The phenomenon of weak-to-strong generalization exemplifies the advantage of this two-stage procedure: a strong student is trained on imperfect labels obtained from a weak teacher, and yet the strong student outperforms the weak teacher. In this paper, we show that the potential improvement is substantial, in the sense that it affects the scaling law followed by the test error. Specifically, we consider students and teachers trained via random feature ridge regression (RFRR). Our main technical contribution is to derive a deterministic equivalent for the excess test error of the student trained on labels obtained via the teacher. Via this deterministic equivalent, we then identify regimes in which the scaling law of the student improves upon that of the teacher, unveiling that the improvement can be achieved both in bias-dominated and variance-dominated settings. Strikingly, the student may attain the minimax optimal rate regardless of the scaling law of the teacher -- in fact, when the test error of the teacher does not even decay with the sample size.

Text-to-image models now generate graphic design at production scale, yet their supervision still comes primarily from photo-style preference datasets with a single overall verdict per comparison. Designers evaluate designs along several distinct axes (e.g., typography, layout, color harmony) that a single preference label collapses. We release \emph{TASTE} \textit{(Typography, Aesthetics, Spatial, Tone, Etc.)}, a multi-dimensional preference dataset in which two disjoint cohorts of five professional designers each ranked outputs from four current text-to-image models across nine criteria along with per-image hallucination flags. We pair the dataset with two contributions. First, a criterion-agnostic signal-validation framework based on Kendall's $τ$, majority-vote probability, and Condorcet cycles against exact iid-uniform nulls; the analysis reveals significant but moderate designer agreement, with every TASTE criterion rejecting the random-rater null. Second, we benchmark preference models on TASTE and find that off-the-shelf VLM judges and dedicated T2I scorers fail to reach majority agreement with the designer panel, while a small MLP head trained directly on TASTE substantially narrows the gap to the single-rater ceiling, setting a baseline for future TASTE-trained preference models.

Irregular multivariate time series impose a trade-off for long-horizon forecasting: discrete methods can distort temporal structure via re-gridding, while continuous-time models often require sequential solvers prone to drift. To bridge this gap, we present Latent Laplace Diffusion (LLapDiff), a generative framework that models the target as a low-dimensional latent trajectory, enabling horizon-wide generation without step-by-step integration over physical time. We guide the reverse process utilizing a stable modal parameterization motivated by stochastic port-Hamiltonian dynamics, and parameterize its mean evolution in the Laplace domain via learnable complex-conjugate poles, enabling direct evaluation over irregular timestamps. We also link continuous dynamics to irregular observations through renewal-averaging analysis, which maps sampling gaps to effective event-domain poles and motivates a gap-aware history summarizer. Extensive experiments show that LLapDiff improves over baselines in long-horizon forecasting, and its continuous-time generative nature supports missing-value imputation by querying the same model at historical timestamps. Code is available at https://github.com/pixelhero98/LLapDiffusion.

A striking geometric disparity has long persisted in the practice of deep learning. While modern neural network architectures naturally exhibit rich symmetry and equivariance properties, popular optimizers such as Adam and its variants operate inherently coordinate-wise, rendering them unable to respect the equivariance structures of the parameter space. We address this disparity by introducing a symmetry-compatible principle for optimizer design: the gradient update rule should be equivariant under the symmetry group acting on the corresponding weight block. Following this principle, we first provide a unified perspective on bi-orthogonally equivariant updates for general matrix layers, as employed by stochastic spectral descent, Muon, Scion, and polar gradient methods. More importantly, by moving from orthogonal groups to permutation and shared-shift symmetries, we derive symmetry-compatible optimizers for parameter blocks whose symmetries differ from those of general matrix layers: embedding and LM head matrices, SwiGLU MLP projections, and MoE router matrices. These constructions include one-sided spectral, row-norm, hybrid row-norm/spectral, row-aware, column-aware, centered row-norm, and left-spectral updates. They yield an end-to-end layerwise optimizer stack in which each major matrix-valued parameter class is assigned an update whose equivariance matches its symmetry group. We corroborate this principle through pre-training experiments on dense and sparse MoE language models, including Qwen3-0.6B-style, Gemma 3 1B-style, OLMoE-1B-7B-style, and downsized gpt-oss architectures. Across these experiments, symmetry-compatible update rules consistently improve final validation loss, reduce load imbalance in sparse MoE models, and in several cases improve training stability over the corresponding AdamW updates.

Probabilistic partial least squares (PPLS) is a central likelihood-based model for two-view learning when one needs both interpretable latent factors and calibrated uncertainty. Building on the identifiable parameterization of Bouhaddani et al.\ (2018), existing fitting pipelines still face two practical bottlenecks: noise--signal coupling under joint EM/ECM updates and nontrivial handling of orthogonality constraints. Following the fixed-noise scalar-likelihood protocol, we develop an end-to-end framework that combines noise pre-estimation, constrained likelihood optimization, and prediction calibration in one pipeline. We estimate the observation noise from the low-eigenvalue noise subspace and enforce orthogonality through exact Stiefel-manifold optimization. The noise-subspace estimator attains a signal-strength-independent leading finite-sample rate and matches a minimax lower bound, whereas a full-spectrum noise estimator carries a deterministic bias under the same model. We further extend the framework to sub-Gaussian settings via optional Gaussianization and provide closed-form standard errors through a block-structured Fisher analysis. Across synthetic high-noise settings and two multi-omics benchmarks (TCGA-BRCA and PBMC CITE-seq), the method achieves near-nominal coverage without post-hoc recalibration, reaches Ridge-level point accuracy on TCGA-BRCA at rank $r=3$, matches or exceeds PO2PLS on cross-view prediction while providing native calibrated uncertainty, and improves stability of parameter recovery.

Conformal prediction provides prediction sets with finite-sample marginal coverage, but many applications require coverage guarantees that adapt to individual test points, a subpopulation, or a structural component of the data. Existing methods targeting conditional coverage are largely analyzed case by case, leaving limited general theory for understanding where conditional miscoverage comes from, how different procedures should be compared, and how such guarantees can be extended beyond i.i.d.~data. We address these gaps through a unified framework and theory for conformal methods targeting conditional coverage. Our central contribution is a non-asymptotic decomposition of conditional miscoverage into three interpretable components: score-estimation error, finite-sample calibration error, and intrinsic conditional-mismatch error. This decomposition clarifies the mechanisms behind asymptotic conditional validity and places existing methods within a common analytical lens. Building on this framework, we derive principled guidance for conditional-coverage-oriented model selection, and develop localized methods with asymptotic conditional guarantees under covariate shift. Finally, we extend the framework to structured data, with concrete applications to graph-structured and hierarchical settings. Numerical experiments corroborate the theory and demonstrate the effectiveness of the proposed procedures.

We study the problem of learning generative models for discrete sequences in a continuous embedding space. Whereas prior approaches typically operate in Euclidean space or on the probability simplex, we instead work on the sphere $\mathbb S^{d-1}$. There the von Mises-Fisher (vMF) distribution induces a natural noise process and admits a closed-form conditional score. The conditional velocity is in general intractable. Exploiting the radial symmetry of the vMF density we reduce the continuity equation on $\mathbb S^{d-1}$ to a scalar ODE in the cosine similarity, whose unique bounded solution determines the velocity. The marginal velocity and marginal score on $(\mathbb S^{d-1})^L$ both decompose into posterior-weighted tangent sums that differ only by per-token scalar weights. This gives access to both ODE and predictor-corrector (PC) sampling. The posterior is the only learned object, trained by a cross-entropy loss. Experiments compare the vMF path against geodesic and Euclidean alternatives. The combination of vMF and PC sampling significantly improves results on Sudoku and language modeling.

In this paper we study continuum-marginal optimal transport. Given a time-continuous family of probability marginals, the problem is to recover the minimum-energy velocity field whose flow reproduces every marginal. This problem is the continuum limit of the classical two-marginal Benamou--Brenier formulation, and also the deterministic limit of the Nelson problem of stochastic optimal transport. We propose a practical mesh-free solver for this problem. The weak continuity equation is embedded in a reproducing kernel Hilbert space, yielding a sample-only objective that requires no spatial discretization. The velocity is parametrized by any linear-in-parameters dictionary or neural network, and is optimized by mini-batch stochastic methods. Synthetic experiments confirm that the method achieves accurate drift recovery and marginal consistency. The same computational framework also applies to the stochastic Nelson problem.

Evaluating generative AI models is increasingly resource-intensive due to slow inference, expensive raters, and a rapidly growing landscape of models and benchmarks. We propose ProEval, a proactive evaluation framework that leverages transfer learning to efficiently estimate performance and identify failure cases. ProEval employs pre-trained Gaussian Processes (GPs) as surrogates for the performance score function, mapping model inputs to metrics such as the severity of errors or safety violations. By framing performance estimation as Bayesian quadrature (BQ) and failure discovery as superlevel set sampling, we develop uncertainty-aware decision strategies that actively select or synthesize highly informative inputs for testing. Theoretically, we prove that our pre-trained GP-based BQ estimator is unbiased and bounded. Empirically, extensive experiments on reasoning, safety alignment, and classification benchmarks demonstrate that ProEval is significantly more efficient than competitive baselines. It requires 8-65x fewer samples to achieve estimates within 1% of the ground truth, while simultaneously revealing more diverse failure cases under a stricter evaluation budget.

We propose a novel conditional diffusion model for contextual portfolio optimization that learns the cross-sectional distribution of next-day stock returns conditioned on high-dimensional asset-specific factors. Our model leverages a Diffusion Transformer architecture with token-wise conditioning, which enables linking each asset's return to its own factor vector while capturing complex cross-asset dependencies. By drawing generative samples from the learned conditional return distribution, we perform daily mean-variance and mean-CVaR optimization, incorporating transaction costs and realistic constraints. Using data from the Chinese A-share market, we demonstrate that our approach consistently outperforms various standard benchmarks across multiple risk-adjusted performance metrics. Furthermore, we establish a 2-Wasserstein error bound for the conditional diffusion model and quantify how its distributional approximation errors propagate to the downstream portfolio optimization task. Our results demonstrate the potential of generative diffusion models for high-dimensional, risk-sensitive contextual stochastic optimization and financial decision making.

An individualized treatment rule (ITR) tailors treatments to a patient's specific characteristics. However, randomized controlled trials (RCTs) are often underpowered to detect the treatment effect heterogeneity needed for reliable ITR estimation. To address this limitation, there is growing interest in leveraging information from multiple studies to improve statistical power and support individualized decision-making. A key challenge in this context is that available RCTs may not evaluate the same set of treatments. In this paper, we propose an integrative learning framework that synthesizes evidence across multiple RCTs that share a common comparator but differ in their alternative treatment arms. Our method integrates information through a regularized weighted misclassification risk function and adaptively determines the contribution of each study to the ITRs of the others. We rigorously study the excess risk of the resulting estimator. Simulation studies demonstrate that the proposed approaches improve the estimation of both value and benefit functions. We illustrate the utility of our methodology using data from two landmark studies of major depressive disorder: the Establishing Moderators and Biosignatures of Antidepressant Response in Clinical Care study and the International Study to Predict Optimized Treatment in Depression study, both of which include a selective serotonin reuptake inhibitor as a common treatment arm. We find that the separate learning method outperforms one-size-fits-all methods, and our integrative methods further improve performance.

We develop horizon-aware anytime-valid tests and confidence sequences for bounded means under a strict deadline $N$. Using the betting/e-process framework, we cast horizon-aware betting as a finite-horizon optimal control problem with state space $(t, \log W_t)$, where $t$ is the time and $W_t$ is the test martingale value. We first show that in certain interior regions of the state space, policies that deviate significantly from Kelly betting are provably suboptimal, while Kelly betting reaches the threshold with high probability. We then identify sufficient conditions showing that outside this region, more aggressive betting than Kelly can be better if the bettor is behind schedule, and less aggressive can be better if the bettor is ahead. Taken together these results suggest a simple phase diagram in the $(t, \log W_t)$ plane, delineating regions where Kelly, fractional Kelly, and aggressive betting may be preferable. Guided by this phase diagram, we introduce a Deep Reinforcement Learning approach based on a universal Deep Q-Network (DQN) agent that learns a single policy from synthetic experience and maps simple statistics of past observations to bets across horizons and null values. In limited-horizon experiments, the learned DQN policy yields state-of-the-art results.

Robust Mixture Prior (RMP) is a popular Bayesian dynamic borrowing method, which combines an informative historical distribution with a less informative component (referred as robustification component) in a mixture prior to enhance the efficiency of hybrid-control randomized trials. Current practice typically focuses solely on the selection of the prior weight that governs the relative influence of these two components, often fixing the variance of the robustification component to that of a single observation. In this study we demonstrate that the performance of RMPs critically depends on the joint selection of both weight and variance of the robustification component. In particular, we show that a wide range of weight-variance pairs can yield practically identical posterior inferences (in particular regions of the parameter space) and that large variance robust components may be employed without incurring in the so called Lindley's paradox. We further show that the use of large variance robustification components leads to improved asymptotic Type I error control and enhanced robustness of the RMP to the specification of the location parameter of the robustification component. Finally, we leverage these theoretical results to propose a novel and practical hyper-parameter elicitation routine.

We present FlowSN, a statistical framework using simulation-based inference (SBI) with normalising flows to account for selection effects in observational astronomy. Failure to account for selection effects can lead to biased inference on global parameters. An example is Malmquist bias, where detection limits result in a sample skewed towards brighter objects. In Type Ia supernova (SN Ia) cosmology, these selection effects can systematically shift the inferred posterior distributions of cosmological parameters, necessitating the development of robust statistical frameworks to account for the biases. SBI enables us to implicitly learn probability distributions that are analytically intractable to calculate. In this work, we introduce a novel approach that employs a normalising flow to learn the non-analytic selected SN likelihood for a given survey from forward simulations, independent of the assumed cosmological model. The resulting likelihood approximation is incorporated into a hierarchical Bayesian framework and posterior sampling is performed using Hamiltonian Monte Carlo to obtain constraints on cosmological parameters conditioned on the observed data. The modular learnt likelihood approximation can be reused without retraining to evaluate different cosmological models, providing a key advantage over other SBI approaches. We demonstrate the performance of this methodology by training and testing the SBI technique using realistic LSST-like SNANA simulations for the first time. Our FlowSN approach yields accurate posterior estimates on cosmological parameters, including the dark energy equation of state $w_0$, that are an order of magnitude less biased than those obtained with conventional techniques and also exhibit improved frequentist calibration.

Market generators using deep generative models have shown promise for synthetic financial data generation, but existing approaches lack causal reasoning capabilities essential for counterfactual analysis and risk assessment. We propose a Time-series Neural Causal Model VAE (TNCM-VAE) that combines variational autoencoders with structural causal models to generate counterfactual financial time series while preserving both temporal dependencies and causal relationships. Our approach enforces causal constraints through directed acyclic graphs in the decoder architecture and employs the causal Wasserstein distance for training. We validate our method on synthetic autoregressive models inspired by the Ornstein-Uhlenbeck process, demonstrating superior performance in counterfactual probability estimation with L1 distances as low as 0.03-0.10 compared to ground truth. The model enables financial stress testing, scenario analysis, and enhanced backtesting by generating plausible counterfactual market trajectories that respect underlying causal mechanisms.

Small influential data subsets can dramatically impact model conclusions, with a few data points overturning key findings. While recent work identifies these most influential sets, there is no formal way to tell when maximum influence is excessive rather than expected under natural random sampling variation. We address this gap by developing a principled framework for most influential sets. Focusing on linear least-squares, we derive a convenient exact influence formula and identify the extreme value distributions of maximal influence - the heavy-tailed Fréchet for constant-size sets and heavy-tailed data, and the well-behaved Gumbel for growing sets or light tails. This allows us to conduct rigorous hypothesis tests for excessive influence. We demonstrate through applications across economics, biology, and machine learning benchmarks, resolving contested findings and replacing ad-hoc heuristics with rigorous inference.

This work introduces MAYA, a sequential imitation learning model based on multi-armed bandits, designed to reproduce and predict individual bees' decisions in contextualized foraging tasks. The model accounts for bees' limited memory through a temporal window $τ$, whose optimal value is around 7 trials, with a slight dependence on weather conditions. Experimental results on real, simulated, and complementary (mice) datasets show that MAYA (particularly with the Wasserstein distance) outperforms imitation baselines and classical statistical models, while providing interpretability of individual learning strategies and enabling the inference of realistic trajectories for prospective ecological applications.

Causal forests estimate how treatment effects vary across individuals, guiding personalized interventions in areas like marketing, operations, and public policy. A standard practice is honest estimation: dividing the data into two samples, one to define subgroups and another to estimate treatment effects within them. This is intended to reduce overfitting and is the default in many software packages. But is it the right choice? We show that honest estimation can reduce the accuracy of estimates of individual treatment effects, especially when effect heterogeneity is substantial and datasets are large enough to detect it. The reason is a bias-variance trade-off: honesty lowers the risk of overfitting but increases the risk of underfitting by limiting the data available to detect and model heterogeneity. Across more than 7,000 benchmark datasets, we find that the cost of using honesty by default can be as high as requiring 27% more data to match the performance of models trained without it. Honesty is best understood as a form of regularization. Whether to adopt it should depend on the goals of the application and its empirical performance, not on reflexive default use.

We study reinforcement learning with delayed state observation, where the agent observes the current state after some random number of time steps. We propose an algorithm that combines the augmentation method and the upper confidence bound approach. For tabular Markov decision processes (MDPs), we derive a regret bound of $\tilde{\mathcal{O}}(H \sqrt{D_{\max} SAK})$, where $S$ and $A$ are the cardinalities of the state and action spaces, $H$ is the time horizon, $K$ is the number of episodes, and $D_{\max}$ is the maximum length of the delay. We also provide a matching lower bound up to logarithmic factors, showing the optimality of our approach. Our analytical framework formulates this problem as a special case of a broader class of MDPs, where their transition dynamics decompose into a known component and an unknown but structured component. We establish general results for this abstract setting, which may be of independent interest.

Deep generative models based on neural differential equations have become state-of-the-art for many generation tasks. These models rely on ODE/SDE solvers that integrate from a prior distribution to the data distribution; in many applications it is also highly desirable to integrate in the inverse direction. Standard solvers, however, accumulate discretization errors that prohibit exact inversion, an inaccuracy that is unacceptable in precision-critical applications. Existing inversion methods suffer from poor stability and low order of convergence, and are strictly limited to the ODE setting. In this work, we propose Rex, a family of reversible exponential (stochastic) Runge-Kutta solvers obtained by applying Lawson methods to convert any explicit (stochastic) Runge-Kutta scheme into an algebraically reversible one for both diffusion ODEs and SDEs. Beyond a rigorous theoretical analysis -- establishing arbitrary-order convergence and a non-zero region of linear stability -- we empirically demonstrate that Rex achieves near-machine-precision reconstruction and improves Boltzmann sampling with flow models as well as image generation and editing with diffusion models.

This study introduces a computationally efficient algorithm, delayed acceptance Markov chain Monte Carlo (DA-MCMC), designed to improve posterior simulation in quasi-Bayesian inference. Quasi-Bayesian methods, which do not require fully specifying a probabilistic model, are often computationally expensive owing to the need to evaluate the inverse and determinant of large covariance matrices. DA-MCMC addresses this challenge by employing a two-stage process: In the first stage, proposals are screened using an approximate posterior, whereas a final acceptance or rejection decision is made in the second stage based on the exact target posterior. This reduces the need for costly matrix computations, thereby improving efficiency without sacrificing accuracy. We demonstrate the effectiveness of DA-MCMC through applications to both synthetic and real data. The results demonstrate that, although DA-MCMC slightly reduces the effective sample size per iteration compared with the standard MCMC, it achieves substantial improvement in terms of effective sample size per second, approximately doubling the efficiency. This makes DA-MCMC particularly useful for cases where posterior simulation is computationally intensive. Thus, the DA-MCMC algorithm offers a significant advancement in computational efficiency for quasi-Bayesian inference, making it a valuable tool for robust Bayesian analysis.

Effective initialization in deep networks requires an understanding of random neural networks. In this work, a rigorous probabilistic analysis of deep bias-free random Leaky ReLU networks is provided. We prove a Law of Large Numbers and a Central Limit Theorem for the logarithm of the norm of network activations, establishing that, as the number of layers increases, their growth is governed by a parameter called the Lyapunov exponent. This parameter characterizes a sharp phase transition between vanishing and exploding activations, and we calculate the Lyapunov exponent explicitly for Gaussian or orthogonal weight matrices. Our results reveal that standard methods, such as He initialization or orthogonal initialization, do not guarantee activation stability for deep networks of low width. Based on these theoretical insights, we propose a novel initialization method, referred to as Lyapunov initialization, which sets the Lyapunov exponent to zero and thereby ensures that the neural network is as stable as possible, leading empirically to improved learning.

The default Gaussian latent in flow-based generative models poses challenges when learning certain distributions such as heavy-tailed ones. We introduce a general framework for learning data-adaptive parametric prior distributions (latent noise) using one-dimensional quantile functions, optimized via the Wasserstein distance between noise and data. The quantile-based prior parameterization naturally adapts to both heavy-tailed and compactly supported distributions and shortens transport paths. Numerical results on heavy-tailed weather and image datasets confirm the method's flexibility and effectiveness achieved with negligible computational overhead.

We propose LoRA-MCL, a training scheme that extends next-token prediction in language models with a method designed to decode diverse, plausible sentence continuations at inference time. Traditional language modeling is an intrinsically ill-posed problem: given a context, multiple futures may be equally plausible. Our approach leverages Multiple Choice Learning (MCL) and the winner-takes-all loss to efficiently handle ambiguity through Low-Rank Adaptation. We provide a theoretical interpretation of applying MCL to language modeling, assuming the data is generated from a mixture of distributions. We illustrate the proposed approach using mixtures of Markov chains. We then demonstrate with experiments on audio and visual captioning, as well as machine translation, that our method achieves high diversity and relevance in generated outputs. We release the code for applying LoRA-MCL to a wide range of language models.

This work studies online episodic tabular Markov decision processes (MDPs) with known transitions and develops best-of-both-worlds algorithms that achieve refined data-dependent regret bounds in the adversarial regime and variance-dependent regret bounds in the stochastic regime. We quantify MDP complexity using a first-order quantity and several new data-dependent measures for the adversarial regime, including a second-order quantity and a path-length measure, as well as variance-based measures for the stochastic regime. To adapt to these measures, we develop algorithms based on global optimization and policy optimization, both built on optimistic follow-the-regularized-leader with log-barrier regularization. For global optimization, our algorithms achieve first-order, second-order, and path-length regret bounds in the adversarial regime, and in the stochastic regime, they achieve a variance-aware gap-independent bound and a variance-aware gap-dependent bound that is polylogarithmic in the number of episodes. For policy optimization, our algorithms achieve the same data- and variance-dependent adaptivity, up to a factor of the episode horizon, by exploiting a new optimistic $Q$-function estimator. Finally, we establish regret lower bounds in terms of data-dependent complexity measures for the adversarial regime and a variance measure for the stochastic regime, implying that the regret upper bounds achieved by the global-optimization approach are nearly optimal.

We propose causal preference elicitation, a Bayesian framework for expert-in-the-loop causal discovery that actively queries local edge relations to concentrate a posterior over directed acyclic graphs (DAGs). From any black-box observational posterior, we model noisy expert judgments with a three-way likelihood over edge existence and direction. Posterior inference uses a flexible particle approximation, and queries are selected by an efficient expected information gain criterion on the expert's categorical response. Experiments on synthetic graphs, protein signaling data, and a human gene perturbation benchmark show faster posterior concentration and improved recovery of directed effects under tight query budgets.

Can a diffusion model trained on bedrooms recover human faces? Diffusion models are widely used as priors for inverse problems, but standard approaches usually assume a high-fidelity model trained on data that closely match the unknown signal. In practice, one often must use a mismatched or low-fidelity diffusion prior. Surprisingly, these weak priors often perform nearly as well as full-strength, in-domain baselines. We study when and why inverse solvers are robust to weak diffusion priors. Through extensive experiments, we find that weak priors succeed when measurements are highly informative (e.g., many observed pixels), and we identify regimes where they fail. To explain this behavior, we combine Bayesian-consistency theory with local-correlation analysis: the theory gives conditions under which high-dimensional measurements make the posterior concentrate near the true signal, while the correlation analysis shows that weak and stronger natural-image priors can share similar local spatial structure. These results provide a principled justification on when weak diffusion priors can be used reliably. Code is available at https://github.com/jjia131/weak-diffusion-priors-inverse-problem.

Semi-supervised inference assumes access to a labeled dataset together with a large unlabeled dataset in which the outcome variable is missing, and it is widely used to improve statistical efficiency and support generalizability across populations. In many modern applications, however, individual-level unlabeled data may not be directly accessible due to privacy restrictions, data-sharing limits, or storage constraints, while summary statistics such as sample means and covariances from the unlabeled population are often available. In this work, we study this constrained semi-supervised setting where, in addition to labeled data with individualized observations, auxiliary information from the unlabeled population is available only through summary statistics. We propose new semi-supervised inference methods for mean estimation under both covariate-independent and covariate-dependent labeling and show that unlabeled summaries can still improve efficiency and help correct selection bias. The proposed methods apply in high dimensions and are robust to model misspecification. Valid inference is obtained under sparsity conditions comparable to those required by semi-supervised methods that assume access to individual-level unlabeled samples. Our approach relies on a specialized cross-fitting procedure, where sample splitting is applied only to the labeled data, which removes the need for individualized unlabeled covariates. We further extend this framework to average treatment effect estimation, enabling generalizability and transportability of causal conclusions in this constrained semi-supervised setting.

Conformal prediction quantifies the uncertainty of machine learning models by augmenting point predictions with valid prediction sets. For complex scenarios involving multiple trials, models, or data sources, conformal prediction sets can be aggregated to create a prediction set that captures the overall uncertainty, often improving precision. However, aggregating multiple prediction sets with individual $1-α$ coverage inevitably weakens the overall guarantee, typically resulting in $1-2α$ worst-case coverage. In this work, we propose a framework for the weighted aggregation of prediction sets, where weights are assigned to each prediction set based on their contribution. Our framework offers flexible control over how the sets are aggregated, achieving tighter coverage bounds that interpolate between the $1-2α$ guarantee of the combined models and the $1-α$ guarantee of an individual model depending on the distribution of weights. Importantly, our framework generalizes to data-dependent weights, as we derive a procedure for weighted aggregation that maintains finite-sample validity even when the weights depend on the data. This extension makes our framework broadly applicable to settings where weights are learned, such as mixture-of-experts (MoE), and we demonstrate through experiments in the MoE setting that our methods achieve adaptive coverage.

Random matrix theory has played an important role in various areas of pure mathematics, mathematical physics, and machine learning. From a practical perspective of data science, input data are usually normalized prior to processing. Thus, this study investigates the statistical properties of min-max normalized eigenvalues in random matrices. Previously, the effective distribution for such normalized eigenvalues has been proposed. In this study, we apply it to evaluate a scaling law of the cumulative distribution. Furthermore, we derive the residual error that arises during matrix factorization of random matrices. We conducted numerical experiments to verify these theoretical predictions.

In this study, a scalable online kernel learning framework is proposed for estimating bidirectional causal effects in systems characterized by mutual dependence and heteroskedasticity. Traditional causal inference often focuses on unidirectional effects, overlooking the common bidirectional relationships in real-world phenomena. Building on heteroskedasticity-based identification, the proposed method integrates a quasi-maximum likelihood estimator for simultaneous equation models with large scale online kernel learning. It employs random Fourier feature approximations to flexibly model nonlinear conditional means and variances, while an adaptive online gradient descent algorithm ensures computational efficiency for streaming and high-dimensional data. Results from extensive simulations demonstrate that the proposed method achieves superior accuracy and stability than single equation and polynomial approximation baselines, exhibiting lower bias and root mean squared error across various data-generating processes. These results confirm that the proposed approach effectively captures complex bidirectional causal effects with near-linear computational scaling. By combining econometric identification with modern machine learning techniques, the proposed framework offers a practical, scalable, and theoretically grounded solution for large scale causal inference in natural/social science, policy making, business, and industrial applications.

Computational modeling of single-cell gene expression is crucial for understanding cellular processes, but generating realistic expression profiles remains a major challenge. This difficulty arises from the count nature of gene expression data and complex latent dependencies among genes. Existing generative models often impose artificial gene orderings or rely on shallow neural network architectures. We introduce a scalable latent diffusion model for single-cell gene expression data, which we refer to as scLDM, that respects the fundamental exchangeability property of the data. Our VAE uses fixed-size latent variables leveraging a unified Multi-head Cross-Attention Block (MCAB) architecture, which serves dual roles: permutation-invariant pooling in the encoder and permutation-equivariant unpooling in the decoder. We enhance this framework by replacing the Gaussian prior with a latent diffusion model using Diffusion Transformers and linear interpolants, enabling high-quality generation with multi-conditional classifier-free guidance. We show its superior performance in a variety of experiments for both observational and perturbational single-cell data, as well as downstream tasks like cell-level classification.

Artificial intelligence (AI) is increasingly used to support renewable energy forecasting and grid operations. As renewable penetration grows, reliable probabilistic forecasting is becoming essential for managing uncertainty and supporting risk-aware operational decision-making. However, these forecasts often suffer from miscalibration due to temporal variability, changing weather conditions, and heterogeneous operating regimes. In many real-world settings, renewable energy forecasts are provided by external sources, vendors, or independently trained systems, making retraining infeasible because of limited model access or computational constraints. This creates a need for efficient and model-agnostic methods that can improve forecast reliability after they are produced. This paper presents Context-Aware Conformal Prediction (CACP), a framework for calibrating renewable energy forecasts. The proposed method relies on a weighting mechanism during the calibration procedure which assigns higher weights to historical observations that are more similar to the target forecasting condition. This enables adaptive prediction intervals that reflect local uncertainty regimes without requiring access to, or retraining of, the underlying forecasting model. Experiments are performed on a large-scale dataset from National Renewable Energy Laboratory (NREL) day-ahead solar forecasting, covering multiple systems including MISO, ERCTO, and SPP. The results show that CACP improves the reliability-efficiency tradeoff at both site and system levels compared to NREL's base forecasting model and the other conformal prediction baselines. These results suggest that CACP can serve as a practical reliability-enhancement layer for trustworthy AI-enabled renewable energy forecasting and operational decision support.

Randomized experiments (or A/B tests) are widely used to evaluate interventions in dynamic systems such as recommendation platforms, marketplaces, and digital health. In these settings, interventions affect both current and future system states, so estimating the global average treatment effect (GATE) requires accounting for temporal dynamics, which is especially challenging in the presence of nonstationarity; existing approaches suffer from high bias, high variance, or both. In this paper, we address this challenge via the novel Truncated Policy Gradient (TPG) estimator, which replaces instantaneous outcomes with short-horizon outcome trajectories. The estimator admits a policy gradient interpretation: it is a truncation of the first-order approximation to the GATE, yielding provable reductions in bias and variance in nonstationary Markovian settings. We further establish a central limit theorem for the TPG estimator and develop a consistent variance estimator that remains valid under nonstationarity with single-trajectory data. We validate our theory with two real-world case studies. The results show that relative to existing approaches, a well-calibrated TPG estimator can achieve a favorable balance between bias and variance in nonstationary settings, highlighting the value of the policy-gradient perspective for designing effective estimators under complex dynamics.

Higher-order spacing statistics in the $m$ superposed spectra of circular random matrices of the same class are studied numerically. We conjecture that for given $m$ (or order $k$) and $β$, the sequence of modified Dyson index $β'(k)$ (or $β'(m)$) obtained using the sum of absolute differences between the cumulative distribution functions method (denoted as $D(β')$) is unique. Also, for a given $k$, the distribution tends to the corresponding $k$-th order Poisson statistics in the limit $m\rightarrow \infty$. The quantum chaotic kicked top model for various Hilbert space dimensions is studied, and it is found to satisfy our conjecture. This involves the numerical verification of $m=2$ case of COE results. Our result can be used as a tool for the characterization of a system and to determine the symmetry structure of the system without desymmetrization of the spectra. Additionally, the comparative study of the higher-order spacing and ratio distributions in both $m=1$ and $m=2$ cases of COE as well as GOE is performed within and across these ensembles numerically using the $D(β')$ method. This study is carried out both by varying the dimension and keeping the number of realizations constant, and vice-versa. The same asymptotic higher-order statistics are observed across COE and GOE in terms of a given spectral fluctuation measure. But, within a given ensemble of COE or GOE, the results of higher-order spacing and ratio distributions agree with each other only up to some lower $k$, and beyond that, they start deviating from each other. Further, the spectral fluctuations of the intermediate map of various dimensions are studied. Various important observations and discussions from the analysis of our extensive numerical computations are presented.

Space-efficient streaming estimation of quantiles in massive datasets is a fundamental problem with numerous applications in data monitoring and analysis. While theoretical research led to optimal algorithms, such as the Greenwald-Khanna algorithm or the KLL sketch, practitioners often use other sketches that perform significantly better in practice but lack theoretical guarantees. Most notably, the widely used $t$-digest has unbounded worst-case error. In this paper, we seek to get the best of both worlds. We present a new quantile summary, SplineSketch, for numeric data, offering near-optimal theoretical guarantees, namely uniformly bounded rank error, and outperforming $t$-digest by a factor of 2-20 on a range of synthetic and real-world datasets. To achieve such performance, we develop a novel approach that maintains a dynamic subdivision of the input range into buckets while fitting the input distribution using monotone cubic spline interpolation. The core challenge is implementing this method in a space-efficient manner while ensuring strong worst-case guarantees.

Formula alpha mining, which generates predictive signals from financial data, is critical for quantitative investment. Although various algorithmic approaches-such as genetic programming, reinforcement learning, and large language models-have significantly expanded the capacity for alpha discovery, systematic evaluation remains a key challenge. Existing evaluation metrics predominantly include backtesting and correlation-based measures. Backtesting is computationally intensive, inherently sequential, and sensitive to specific strategy parameters. Correlation-based metrics, though efficient, assess only predictive ability and overlook other crucial properties such as temporal stability, robustness, diversity, and interpretability. Additionally, the closed-source nature of most existing alpha mining models hinders reproducibility and slows progress in this field. To address these issues, we propose AlphaEval, a unified, parallelizable, and backtest-free evaluation framework for automated alpha mining models. AlphaEval assesses the overall quality of generated alphas along five complementary dimensions: predictive power, stability, robustness to market perturbations, financial logic, and diversity. Extensive experiments across representative alpha mining algorithms demonstrate that AlphaEval achieves evaluation consistency comparable to comprehensive backtesting, while providing more comprehensive insights and higher efficiency. Furthermore, AlphaEval effectively identifies superior alphas compared to traditional single-metric screening approaches. All implementations and evaluation tools are open-sourced to promote reproducibility and community engagement.

In this work, we develop a scalable approach for a flexible latent factor model for high-dimensional dynamical systems. Each latent factor process has its own correlation and variance parameters, and the orthogonal factor loading matrix can be either fixed or estimated. We utilize an orthogonal factor loading matrix that avoids computing the inversion of the posterior covariance matrix at each time of the Kalman filter, and derive closed-form expressions in an expectation-maximization algorithm for parameter estimation, which substantially reduces the computational complexity without approximation. Our approach has several applications, including noise filtering for high-dimensional time series, estimating nonseparable covariance structure between different time series, and estimating latent physical processes from real-world measurements. Extensive simulated studies illustrate higher accuracy and scalability of our approach compared to alternatives. Furthermore, by applying our method to geodetic measurements to estimate slow slip events from geodetic data in the Cascadia region, our estimated slip better agrees with independently measured seismic data of tremor events. The substantial acceleration from our method enables the use of massive noisy data for geological hazard quantification and other applications.

Training-free guided generation is a widely used and powerful technique that allows the end user to exert further control over the generative process of flow/diffusion models. Generally speaking, two families of techniques have emerged for solving this problem for gradient-based guidance: namely, posterior guidance (i.e., guidance via projecting the current sample to the target distribution via the target prediction model) and end-to-end guidance (i.e., guidance by performing backpropagation throughout the entire ODE solve). In this work, we show that these two seemingly separate families can actually be unified by looking at posterior guidance as a greedy strategy of end-to-end guidance. We explore the theoretical connections between these two families and provide an in-depth theoretical of these two techniques relative to the continuous ideal gradients. Motivated by this analysis we then show a method for interpolating between these two families enabling a trade-off between compute and accuracy of the guidance gradients. We then validate this work on several inverse image problems and property-guided molecular generation.

In this paper, we investigate the data-driven identification of asymmetric interaction kernels in the Motsch-Tadmor model based on observed trajectory data. The model under consideration is governed by a class of semilinear evolution equations, where the interaction kernel defines a normalized, state-dependent Laplacian operator that governs collective dynamics. To address the resulting nonlinear inverse problem, we propose a variational framework that reformulates kernel identification using the implicit form of the governing equations, reducing it to a subspace identification problem. We establish an identifiability result that characterizes conditions under which the interaction kernel can be uniquely recovered up to scale. To solve the inverse problem robustly, we develop a sparse Bayesian learning algorithm that incorporates informative priors for regularization, quantifies uncertainty, and enables principled model selection. Extensive numerical experiments on representative interacting particle systems demonstrate the accuracy, robustness, and interpretability of the proposed framework across a range of noise levels and data regimes.

The G-Wishart distribution is a core component for the Bayesian analysis of Gaussian graphical models as the conjugate prior for the precision matrix. Evaluating the marginal likelihood of such models usually requires computing high-dimensional integrals to determine the G-Wishart normalising constant. Closed-form results are known for decomposable or chordal graphs, while an explicit representation as a formal series expansion has been derived recently for general graphs. The nested infinite sums, however, do not lend themselves to computation, remaining of limited practical value. Borrowing techniques from random matrix theory and Fourier analysis, we provide novel exact results well suited to the numerical evaluation of the normalising constant for classes of graphs beyond chordal graphs. We additionally develop a Monte Carlo scheme for general graphs, which can be orders of magnitude more efficient than current approaches.

We derive a novel canonical decomposition of factor models encompassing both the static factor model - where factors are loaded only contemporaneously - and the Generalised Dynamic Factor Model - where factors are loaded with lags. This decomposition features a new term: the weak common component, defined as the difference between the dynamic and static common components. It is driven by (possibly infinitely many) non-pervasive weak factors which belong to the dynamically common space. Through theoretical and empirical examples - both on U.S. macroeconomic indicators and global financial volatilities - we show that, in general, the weak common component shall not be neglected. Furthermore, we show that, by accounting for the presence of weak common components, we are likely to obtain Impulse Response Functions with more plausible shapes than those obtained from purely static approaches. In addition, we provide consistent estimators for all terms of the canonical decomposition and for the weak factors.

In sampling tasks, it is common for target distributions to be known up to a normalizing constant. However, in many situations, even evaluating the unnormalized distribution can be costly or infeasible. This issue arises in scenarios such as sampling from the Bayesian posterior for tall datasets and the `doubly-intractable' distributions. In this paper, we begin by observing that seemingly different Markov chain Monte Carlo (MCMC) algorithms, such as the exchange algorithm, PoissonMH, and TunaMH, can be unified under a simple common procedure. We then extend this procedure into a novel framework that allows the use of auxiliary variables in both the proposal and the acceptance--rejection step. Several new MCMC algorithms emerge from this framework that uses estimated gradients to guide the proposal moves. They have demonstrated significantly better performance than existing methods on both synthetic and real datasets. We also develop theory for the new framework and use it to simplify and extend results for existing algorithms. The code to reproduce the experimental results can be found at https://github.com/ywwes26/Auxiliary-MCMC.

We consider the community detection problem in sparse random hypergraphs under the non-uniform hypergraph stochastic block model (HSBM), a general model of random networks with community structure and higher-order interactions. When the random hypergraph has bounded expected degrees, we provide a spectral algorithm that outputs a partition with at least a $γ$ fraction of the vertices classified correctly, where $γ\in (0.5,1)$ depends on the signal-to-noise ratio (SNR) of the model. When the SNR grows slowly as the number of vertices goes to infinity, our algorithm achieves weak consistency, which improves the previous results in Ghoshdastidar and Dukkipati (2017) for non-uniform HSBMs. Our spectral algorithm consists of three major steps: (1) Hyperedge selection: select hyperedges of certain sizes to provide the maximal signal-to-noise ratio for the induced sub-hypergraph; (2) Spectral partition: construct a regularized adjacency matrix and obtain an approximate partition based on singular vectors; (3) Correction and merging: incorporate the hyperedge information from adjacency tensors to upgrade the error rate guarantee. The theoretical analysis of our algorithm relies on the concentration and regularization of the adjacency matrix for sparse non-uniform random hypergraphs, which can be of independent interest.

We derive universal approximation results for the class of (countably) $m$-rectifiable measures. Specifically, we prove that $m$-rectifiable measures can be approximated as push-forwards of the one-dimensional Lebesgue measure on $[0,1]$ using ReLU neural networks with arbitrarily small approximation error in terms of Wasserstein distance. What is more, the weights in the networks under consideration are quantized and bounded and the number of ReLU neural networks required to achieve an approximation error of $\varepsilon$ is no larger than $2^{b(\varepsilon)}$ with $b(\varepsilon)=\mathcal{O}(\varepsilon^{-m}\log^2(\varepsilon))$. This result improves Lemma IX.4 in Perekrestenko et al. as it shows that the rate at which $b(\varepsilon)$ tends to infinity as $\varepsilon$ tends to zero equals the rectifiability parameter $m$, which can be much smaller than the ambient dimension. We extend this result to countably $m$-rectifiable measures and show that this rate still equals the rectifiability parameter $m$ provided that, among other technical assumptions, the measure decays exponentially on the individual components of the countably $m$-rectifiable support set.

As the amount and complexity of available data increases, the need for robust statistical learning becomes more pressing. To enhance resilience against model misspecification, the generalized posterior inference method adjusts the likelihood term by exponentiating it with a learning rate, thereby fine-tuning the dispersion of the posterior distribution. This study proposes a computationally efficient strategy for selecting an appropriate learning rate. The proposed approach builds upon the generalized posterior calibration (GPC) algorithm, which is designed to select a learning rate that ensures nominal frequentist coverage. This algorithm, which evaluates the coverage probability using bootstrap samples, has high computational costs because of the repeated posterior simulations needed for bootstrap samples. To address this limitation, the study proposes an algorithm that combines elements of the GPC algorithm with the sequential Monte Carlo (SMC) sampler. By leveraging the similarity between the learning rate in generalized posterior inference and the inverse temperature in SMC sampling, the proposed algorithm efficiently calibrates the posterior distribution with a reduced computational cost. For demonstration, the proposed algorithm was applied to several statistical learning models and shown to be significantly faster than the original GPC.

Consider the community detection problem in random hypergraphs under the non-uniform hypergraph stochastic block model (HSBM), where each hyperedge appears independently with some given probability depending only on the labels of its vertices. We establish, for the first time in the literature, a sharp threshold for exact recovery under this non-uniform case, subject to minor constraints; in particular, we consider the model with multiple communities. One crucial point here is that by aggregating information from all the uniform layers, we may obtain exact recovery even in cases when this may appear impossible if each layer were considered alone. Besides that, we prove a wide-ranging, information-theoretic lower bound on the number of misclassified vertices \emph{for any algorithm}, depending on a \emph{generalized Chernoff-Hellinger} divergence involving model parameters. We provide two efficient algorithms which successfully achieve exact recovery when above the threshold, and attain the lowest possible mismatch ratio when the exact recovery is impossible, proved to be optimal. The theoretical analysis of our algorithms relies on the concentration and regularization of the adjacency matrix for non-uniform random hypergraphs, which could be of independent interest. We also address some open problems regarding parameter knowledge and estimation.

In this paper we develop a discrete Hierarchical Basis (HB) to efficiently solve the Radial Basis Function (RBF) interpolation problem with variable polynomial order. The HB forms an orthogonal set and is adapted to the kernel seed function and the placement of the interpolation nodes. Moreover, this basis is orthogonal to a set of polynomials up to a given order defined on the interpolating nodes. We are thus able to decouple the RBF interpolation problem for any order of the polynomial interpolation and solve it in two steps: (1) The polynomial orthogonal RBF interpolation problem is efficiently solved in the transformed HB basis with a GMRES iteration and a diagonal, or block SSOR preconditioner. (2) The residual is then projected onto an orthonormal polynomial basis. We apply our approach on several test cases to study its effectiveness, including an application to the Best Linear Unbiased Estimator regression problem.

Granger causal inference is a contentious but widespread method used in fields ranging from economics to neuroscience. The original definition addresses the notion of causality in time series by establishing functional dependence conditional on a specified model. Adaptation of Granger causality to nonlinear data remains challenging, and many methods apply in-sample tests that do not incorporate out-of-sample predictability, leading to concerns of model overfitting. To allow for out-of-sample comparison, a measure of functional connectivity is explicitly defined using permutations of the covariate set. Artificial neural networks serve as featurizers of the data to approximate any arbitrary, nonlinear relationship, and consistent estimation of the variance for each permutation is shown under certain conditions on the featurization process and the model residuals. Performance of the permutation method is compared to penalized variable selection, naive replacement, and omission techniques via simulation, and it is applied to neuronal responses of acoustic stimuli in the auditory cortex of anesthetized rats. Targeted use of the Granger causal framework, when prior knowledge of the causal mechanisms in a dataset are limited, can help to reveal potential predictive relationships between sets of variables that warrant further study.

The rapid rollout of COVID-19 vaccines in the United Kingdom in early 2021 differed markedly from that of many other European countries, providing a natural setting to assess the impact of vaccination speed on public health outcomes. We evaluate the impact of the accelerated UK vaccination rollout and associated policy transition on COVID-19 mortality and transmission dynamics by constructing a probabilistic reference trajectory for the UK under a slower vaccination and reopening trajectory. The proposed framework combines ideas from interrupted time series analysis and synthetic control methods with flexible probabilistic modelling based on multi-output Gaussian processes. These models capture non-linear and heterogeneous dependence structures across countries and over time, while providing uncertainty quantification through predictive distributions. A central feature of the methodology is a design-consistent validation strategy based on predictive performance in held-out pre-intervention periods, which is used both to guide model specification and to assess the plausibility of the reconstructed reference trajectory. The empirical results indicate a substantial reduction in COVID-19 mortality associated with the accelerated vaccination-policy transition, with little evidence of an effect on transmission rates. Generally, the framework illustrates how flexible probabilistic models and predictive validation can support causal and policy evaluation in complex time series settings.

In this paper we study a broad class of structured nonlinear programming (SNLP) problems. In particular, we first establish the first-order optimality conditions for them. Then we propose sequential convex programming (SCP) methods for solving them in which each iteration is obtained by solving a convex programming problem. Under some suitable assumptions, we establish that any accumulation point of the sequence generated by the methods is a KKT point of the SNLP problems. In addition, we propose a variant of the SCP method for SNLP in which nonmonotone scheme and ``local'' Lipschitz constants of the associated functions are used. A similar convergence result as mentioned above is established.

We consider the distribution of $\argζ(σ+it)$ on fixed lines $σ> \frac12$, and in particular the density \[d(σ) = \lim_{T \rightarrow +\infty} \frac{1}{2T} |\{t \in [-T,+T]: |\argζ(σ+it)| > π/2\}|\,,\] and the closely related density \[d_{-}(σ) = \lim_{T \rightarrow +\infty} \frac{1}{2T} |\{t \in [-T,+T]: \Reζ(σ+it) < 0\}|\,.\] Using classical results of Bohr and Jessen, we obtain an explicit expression for the characteristic function $ψ_σ(x)$ associated with $\argζ(σ+it)$. We give explicit expressions for $d(σ)$ and $d_{-}(σ)$ in terms of $ψ_σ(x)$. Finally, we give a practical algorithm for evaluating these expressions to obtain accurate numerical values of $d(σ)$ and $d_{-}(σ)$.

Nearest-site distances arise in many applications involving spherical or directional domains, including global geospatial analysis, wireless communications, spherical clustering, and cosine-similarity-based data analysis. In this paper, we study the distributional and computational properties of $L_2$, the minimal angular great-circle distance from a uniformly distributed random point on a sphere to a set of prespecified sites on the same sphere. We first derive the cumulative distribution function (CDF) and probability density function (PDF) of $L_0$, the angular great-circle distance from a fixed vertex of a spherical triangle to a random point uniformly distributed within that triangle. We then extend these triangle-level results to convex spherical polygons and use spherical Voronoi diagrams, triangulations of Voronoi cells, and numerical integration to obtain computable distributional and moment formulas for $L_2$. In addition, we derive explicit formulas for selected moments of $\cos(L_2)$, which are relevant to cosine similarity and spherical data analysis. Extensive Monte Carlo simulations validate the proposed CDF, PDF, and moment formulas and demonstrate computational efficiency of our method relative to generic numerical integration and simulation-based alternatives.

Removing noise from piecewise constant (PWC) signals, is a challenging signal processing problem arising in many practical contexts. For example, in exploration geosciences, noisy drill hole records need separating into stratigraphic zones, and in biophysics, jumps between molecular dwell states need extracting from noisy fluorescence microscopy signals. Many PWC denoising methods exist, including total variation regularization, mean shift clustering, stepwise jump placement, running medians, convex clustering shrinkage and bilateral filtering; conventional linear signal processing methods are fundamentally unsuited however. This paper shows that most of these methods are associated with a special case of a generalized functional, minimized to achieve PWC denoising. The minimizer can be obtained by diverse solver algorithms, including stepwise jump placement, convex programming, finite differences, iterated running medians, least angle regression, regularization path following, and coordinate descent. We introduce novel PWC denoising methods, which, for example, combine global mean shift clustering with local total variation smoothing. Head-to-head comparisons between these methods are performed on synthetic data, revealing that our new methods have a useful role to play. Finally, overlaps between the methods of this paper and others such as wavelet shrinkage, hidden Markov models, and piecewise smooth filtering are touched on.

Multi-target tracking is mainly challenged by the nonlinearity present in the measurement equation, and the difficulty in fast and accurate data association. To overcome these challenges, the present paper introduces a grid-based model in which the state captures target signal strengths on a known spatial grid (TSSG). This model leads to \emph{linear} state and measurement equations, which bypass data association and can afford state estimation via sparsity-aware Kalman filtering (KF). Leveraging the grid-induced sparsity of the novel model, two types of sparsity-cognizant TSSG-KF trackers are developed: one effects sparsity through $\ell_1$-norm regularization, and the other invokes sparsity as an extra measurement. Iterative extended KF and Gauss-Newton algorithms are developed for reduced-complexity tracking, along with accurate error covariance updates for assessing performance of the resultant sparsity-aware state estimators. Based on TSSG state estimates, more informative target position and track estimates can be obtained in a follow-up step, ensuring that track association and position estimation errors do not propagate back into TSSG state estimates. The novel TSSG trackers do not require knowing the number of targets or their signal strengths, and exhibit considerably lower complexity than the benchmark hidden Markov model filter, especially for a large number of targets. Numerical simulations demonstrate that sparsity-cognizant trackers enjoy improved root mean-square error performance at reduced complexity when compared to their sparsity-agnostic counterparts.

Coping with outliers contaminating dynamical processes is of major importance in various applications because mismatches from nominal models are not uncommon in practice. In this context, the present paper develops novel fixed-lag and fixed-interval smoothing algorithms that are robust to outliers simultaneously present in the measurements {\it and} in the state dynamics. Outliers are handled through auxiliary unknown variables that are jointly estimated along with the state based on the least-squares criterion that is regularized with the $\ell_1$-norm of the outliers in order to effect sparsity control. The resultant iterative estimators rely on coordinate descent and the alternating direction method of multipliers, are expressed in closed form per iteration, and are provably convergent. Additional attractive features of the novel doubly robust smoother include: i) ability to handle both types of outliers; ii) universality to unknown nominal noise and outlier distributions; iii) flexibility to encompass maximum a posteriori optimal estimators with reliable performance under nominal conditions; and iv) improved performance relative to competing alternatives at comparable complexity, as corroborated via simulated tests.

We propose an algorithm for evaluation of the cumulative bivariate normal distribution, building upon Marsaglia's ideas for evaluation of the cumulative univariate normal distribution. The algorithm is mathematically transparent, delivers competitive performance and can easily be extended to arbitrary precision.

Many structured data-fitting applications require the solution of an optimization problem involving a sum over a potentially large number of measurements. Incremental gradient algorithms offer inexpensive iterations by sampling a subset of the terms in the sum. These methods can make great progress initially, but often slow as they approach a solution. In contrast, full-gradient methods achieve steady convergence at the expense of evaluating the full objective and gradient on each iteration. We explore hybrid methods that exhibit the benefits of both approaches. Rate-of-convergence analysis shows that by controlling the sample size in an incremental gradient algorithm, it is possible to maintain the steady convergence rates of full-gradient methods. We detail a practical quasi-Newton implementation based on this approach. Numerical experiments illustrate its potential benefits.

This paper proposes a novel uncertainty quantification framework for computationally demanding systems characterized by a large vector of non-Gaussian uncertainties. It combines state-of-the-art techniques in advanced Monte Carlo sampling with Bayesian formulations. The key departure from existing works is the use of inexpensive, approximate computational models in a rigorous manner. Such models can readily be derived by coarsening the discretization size in the solution of the governing PDEs, increasing the time step when integration of ODEs is performed, using fewer iterations if a non-linear solver is employed or making use of lower order models. It is shown that even in cases where the inexact models provide very poor approximations of the exact response, statistics of the latter can be quantified accurately with significant reductions in the computational effort. Multiple approximate models can be used and rigorous confidence bounds of the estimates produced are provided at all stages.

We first introduce a class of divergence measures between power spectral density matrices. These are derived by comparing the suitability of different models in the context of optimal prediction. Distances between "infinitesimally close" power spectra are quadratic, and hence, they induce a differential-geometric structure. We study the corresponding Riemannian metrics and, for a particular case, provide explicit formulae for the corresponding geodesics and geodesic distances. The close connection between the geometry of power spectra and the geometry of the Fisher-Rao metric is noted.

This paper proposes a new framework for the optimization of excitation inputs for system identification. The optimization problem considered is to maximize a reduced Fisher information matrix in any of the classical D-, E-, or A-optimal senses. In contrast to the majority of published work on this topic, we consider the problem in the time domain and subject to constraints on the amplitude of the input signal. This optimization problem is nonconvex. The main result of the paper is a convex relaxation that gives an upper bound accurate to within $2/π$ of the true maximum. A randomized algorithm is presented for finding a feasible solution which, in a certain sense is expected to be at least $2/π$ as informative as the globally optimal input signal. In the case of a single constraint on input power, the proposed approach recovers the true global optimum exactly. Extensions to situations with both power and amplitude constraints on both inputs and outputs are given. A simple simulation example illustrates the technique.

Unions of subspaces provide a powerful generalization to linear subspace models for collections of high-dimensional data. To learn a union of subspaces from a collection of data, sets of signals in the collection that belong to the same subspace must be identified in order to obtain accurate estimates of the subspace structures present in the data. Recently, sparse recovery methods have been shown to provide a provable and robust strategy for exact feature selection (EFS)--recovering subsets of points from the ensemble that live in the same subspace. In parallel with recent studies of EFS with L1-minimization, in this paper, we develop sufficient conditions for EFS with a greedy method for sparse signal recovery known as orthogonal matching pursuit (OMP). Following our analysis, we provide an empirical study of feature selection strategies for signals living on unions of subspaces and characterize the gap between sparse recovery methods and nearest neighbor (NN)-based approaches. In particular, we demonstrate that sparse recovery methods provide significant advantages over NN methods and the gap between the two approaches is particularly pronounced when the sampling of subspaces in the dataset is sparse. Our results suggest that OMP may be employed to reliably recover exact feature sets in a number of regimes where NN approaches fail to reveal the subspace membership of points in the ensemble.

Rich and complex time-series data, such as those generated from engineering systems, financial markets, videos or neural recordings, are now a common feature of modern data analysis. Explaining the phenomena underlying these diverse data sets requires flexible and accurate models. In this paper, we promote Gaussian process dynamical systems (GPDS) as a rich model class that is appropriate for such analysis. In particular, we present a message passing algorithm for approximate inference in GPDSs based on expectation propagation. By posing inference as a general message passing problem, we iterate forward-backward smoothing. Thus, we obtain more accurate posterior distributions over latent structures, resulting in improved predictive performance compared to state-of-the-art GPDS smoothers, which are special cases of our general message passing algorithm. Hence, we provide a unifying approach within which to contextualize message passing in GPDSs.

Herding and kernel herding are deterministic methods of choosing samples which summarise a probability distribution. A related task is choosing samples for estimating integrals using Bayesian quadrature. We show that the criterion minimised when selecting samples in kernel herding is equivalent to the posterior variance in Bayesian quadrature. We then show that sequential Bayesian quadrature can be viewed as a weighted version of kernel herding which achieves performance superior to any other weighted herding method. We demonstrate empirically a rate of convergence faster than O(1/N). Our results also imply an upper bound on the empirical error of the Bayesian quadrature estimate.

Hodge theory is a beautiful synthesis of geometry, topology, and analysis, which has been developed in the setting of Riemannian manifolds. On the other hand, spaces of images, which are important in the mathematical foundations of vision and pattern recognition, do not fit this framework. This motivates us to develop a version of Hodge theory on metric spaces with a probability measure. We believe that this constitutes a step towards understanding the geometry of vision. The appendix by Anthony Baker provides a separable, compact metric space with infinite dimensional α-scale homology.

We consider the problem of providing optimal uncertainty quantification (UQ) --- and hence rigorous certification --- for partially-observed functions. We present a UQ framework within which the observations may be small or large in number, and need not carry information about the probability distribution of the system in operation. The UQ objectives are posed as optimization problems, the solutions of which are optimal bounds on the quantities of interest; we consider two typical settings, namely parameter sensitivities (McDiarmid diameters) and output deviation (or failure) probabilities. The solutions of these optimization problems depend non-trivially (even non-monotonically and discontinuously) upon the specified legacy data. Furthermore, the extreme values are often determined by only a few members of the data set; in our principal physically-motivated example, the bounds are determined by just 2 out of 32 data points, and the remainder carry no information and could be neglected without changing the final answer. We propose an analogue of the simplex algorithm from linear programming that uses these observations to offer efficient and rigorous UQ for high-dimensional systems with high-cardinality legacy data. These findings suggest natural methods for selecting optimal (maximally informative) next experiments.

We introduce a data-driven order reduction method for nonlinear control systems, drawing on recent progress in machine learning and statistical dimensionality reduction. The method rests on the assumption that the nonlinear system behaves linearly when lifted into a high (or infinite) dimensional feature space where balanced truncation may be carried out implicitly. This leads to a nonlinear reduction map which can be combined with a representation of the system belonging to a reproducing kernel Hilbert space to give a closed, reduced order dynamical system which captures the essential input-output characteristics of the original model. Empirical simulations illustrating the approach are also provided.

Markov chains can be used to generate samples whose distribution approximates a given target distribution. The quality of the samples of such Markov chains can be measured by the discrepancy between the empirical distribution of the samples and the target distribution. We prove upper bounds on this discrepancy under the assumption that the Markov chain is uniformly ergodic and the driver sequence is deterministic rather than independent $U(0,1)$ random variables. In particular, we show the existence of driver sequences for which the discrepancy of the Markov chain from the target distribution with respect to certain test sets converges with (almost) the usual Monte Carlo rate of $n^{-1/2}$.

Clustering is a fundamental problem in many scientific applications. Standard methods such as $k$-means, Gaussian mixture models, and hierarchical clustering, however, are beset by local minima, which are sometimes drastically suboptimal. Recently introduced convex relaxations of $k$-means and hierarchical clustering shrink cluster centroids toward one another and ensure a unique global minimizer. In this work we present two splitting methods for solving the convex clustering problem. The first is an instance of the alternating direction method of multipliers (ADMM); the second is an instance of the alternating minimization algorithm (AMA). In contrast to previously considered algorithms, our ADMM and AMA formulations provide simple and unified frameworks for solving the convex clustering problem under the previously studied norms and open the door to potentially novel norms. We demonstrate the performance of our algorithm on both simulated and real data examples. While the differences between the two algorithms appear to be minor on the surface, complexity analysis and numerical experiments show AMA to be significantly more efficient.

For the problems of low-rank matrix completion, the efficiency of the widely-used nuclear norm technique may be challenged under many circumstances, especially when certain basis coefficients are fixed, for example, the low-rank correlation matrix completion in various fields such as the financial market and the low-rank density matrix completion from the quantum state tomography. To seek a solution of high recovery quality beyond the reach of the nuclear norm, in this paper, we propose a rank-corrected procedure using a nuclear semi-norm to generate a new estimator. For this new estimator, we establish a non-asymptotic recovery error bound. More importantly, we quantify the reduction of the recovery error bound for this rank-corrected procedure. Compared with the one obtained for the nuclear norm penalized least squares estimator, this reduction can be substantial (around 50%). We also provide necessary and sufficient conditions for rank consistency in the sense of Bach (2008). Very interestingly, these conditions are highly related to the concept of constraint nondegeneracy in matrix optimization. As a byproduct, our results provide a theoretical foundation for the majorized penalty method of Gao and Sun (2010) and Gao (2010) for structured low-rank matrix optimization problems. Extensive numerical experiments demonstrate that our proposed rank-corrected procedure can simultaneously achieve a high recovery accuracy and capture the low-rank structure.

The classical shift retrieval problem considers two signals in vector form that are related by a shift. The problem is of great importance in many applications and is typically solved by maximizing the cross-correlation between the two signals. Inspired by compressive sensing, in this paper, we seek to estimate the shift directly from compressed signals. We show that under certain conditions, the shift can be recovered using fewer samples and less computation compared to the classical setup. Of particular interest is shift estimation from Fourier coefficients. We show that under rather mild conditions only one Fourier coefficient suffices to recover the true shift.

In this article, we analyze the SPICE method developed in [1], and establish its connections with other standard sparse estimation methods such as the Lasso and the LAD-Lasso. This result positions SPICE as a computationally efficient technique for the calculation of Lasso-type estimators. Conversely, this connection is very useful for establishing the asymptotic properties of SPICE under several problem scenarios and for suggesting suitable modifications in cases where the naive version of SPICE would not work.

This paper considers the nonparametric regression model with an additive error that is dependent on the explanatory variables. As is common in empirical studies in epidemiology and economics, it also supposes that valid instrumental variables are observed. A classical example in microeconomics considers the consumer demand function as a function of the price of goods and the income, both variables often considered as endogenous. In this framework, the economic theory also imposes shape restrictions on the demand function, like integrability conditions. Motivated by this illustration in microeconomics, we study an estimator of a nonparametric constrained regression function using instrumental variables by means of Tikhonov regularization. We derive rates of convergence for the regularized model both in a deterministic and stochastic setting under the assumption that the true regression function satisfies a projected source condition including, because of the non-convexity of the imposed constraints, an additional smallness condition.

Many inverse problems include nuisance parameters which, while not of direct interest, are required to recover primary parameters. Structure present in these problems allows efficient optimization strategies - a well known example is variable projection, where nonlinear least squares problems which are linear in some parameters can be very efficiently optimized. In this paper, we extend the idea of projecting out a subset over the variables to a broad class of maximum likelihood (ML) and maximum a posteriori likelihood (MAP) problems with nuisance parameters, such as variance or degrees of freedom. As a result, we are able to incorporate nuisance parameter estimation into large-scale constrained and unconstrained inverse problem formulations. We apply the approach to a variety of problems, including estimation of unknown variance parameters in the Gaussian model, degree of freedom (d.o.f.) parameter estimation in the context of robust inverse problems, automatic calibration, and optimal experimental design. Using numerical examples, we demonstrate improvement in recovery of primary parameters for several large- scale inverse problems. The proposed approach is compatible with a wide variety of algorithms and formulations, and its implementation requires only minor modifications to existing algorithms.

We show that the sensor self-localization problem can be cast as a static parameter estimation problem for Hidden Markov Models and we implement fully decentralized versions of the Recursive Maximum Likelihood and on-line Expectation-Maximization algorithms to localize the sensor network simultaneously with target tracking. For linear Gaussian models, our algorithms can be implemented exactly using a distributed version of the Kalman filter and a novel message passing algorithm. The latter allows each node to compute the local derivatives of the likelihood or the sufficient statistics needed for Expectation-Maximization. In the non-linear case, a solution based on local linearization in the spirit of the Extended Kalman Filter is proposed. In numerical examples we demonstrate that the developed algorithms are able to learn the localization parameters.

We show that Markov couplings can be used to improve the accuracy of Markov chain Monte Carlo calculations in some situations where the steady-state probability distribution is not explicitly known. The technique generalizes the notion of control variates from classical Monte Carlo integration. We illustrate it using two models of nonequilibrium transport.

The purpose of this paper is to study metrics suitable for assessing uncertainty of power spectra when these are based on finite second-order statistics. The family of power spectra which is consistent with a given range of values for the estimated statistics represents the uncertainty set about the "true" power spectrum. Our aim is to quantify the size of this uncertainty set using suitable notions of distance, and in particular, to compute the diameter of the set since this represents an upper bound on the distance between any choice of a nominal element in the set and the "true" power spectrum. Since the uncertainty set may contain power spectra with lines and discontinuities, it is natural to quantify distances in the weak topology---the topology defined by continuity of moments. We provide examples of such weakly-continuous metrics and focus on particular metrics for which we can explicitly quantify spectral uncertainty. We then consider certain high resolution techniques which utilize filter-banks for pre-processing, and compute worst-case a priori uncertainty bounds solely on the basis of the filter dynamics. This allows the a priori tuning of the filter-banks for improved resolution over selected frequency bands.

Motivated by the problems of computing sample covariance matrices, and of transforming a collection of vectors to a basis where they are sparse, we present a simple algorithm that computes an approximation of the product of two n-by-n real matrices A and B. Let ||AB||_F denote the Frobenius norm of AB, and b be a parameter determining the time/accuracy trade-off. Given 2-wise independent hash functions $_1,h_2: [n] -> [b], and s_1,s_2: [n] -> {-1,+1} the algorithm works by first "compressing" the matrix product into the polynomial p(x) = sum_{k=1}^n (sum_{i=1}^n A_{ik} s_1(i) x^{h_1(i)}) (sum_{j=1}^n B_{kj} s_2(j) x^{h_2(j)}) Using FFT for polynomial multiplication, we can compute c_0,...,c_{b-1} such that sum_i c_i x^i = (p(x) mod x^b) + (p(x) div x^b) in time Õ(n^2+ n b). An unbiased estimator of (AB)_{ij} with variance at most ||AB||_F^2 / b can then be computed as: C_{ij} = s_1(i) s_2(j) c_{(h_1(i)+h_2(j)) mod b. Our approach also leads to an algorithm for computing AB exactly, whp., in time Õ(N + nb) in the case where A and B have at most N nonzero entries, and AB has at most b nonzero entries. Also, we use error-correcting codes in a novel way to recover significant entries of AB in near-linear time.

Reliable spectrum sensing is a key functionality of a cognitive radio network. Cooperative spectrum sensing improves the detection reliability of a cognitive radio system but also increases the system energy consumption which is a critical factor particularly for low-power wireless technologies. A censored truncated sequential spectrum sensing technique is considered as an energy-saving approach. To design the underlying sensing parameters, the maximum energy consumption per sensor is minimized subject to a lower bounded global probability of detection and an upper bounded false alarm rate. This way both the interference to the primary user due to miss detection and the network throughput as a result of a low false alarm rate is controlled. We compare the performance of the proposed scheme with a fixed sample size censoring scheme under different scenarios. It is shown that as the sensing cost of the cognitive radios increases, the energy efficiency of the censored truncated sequential approach grows significantly.

Fast and accurate unveiling of power line outages is of paramount importance not only for preventing faults that may lead to blackouts, but also for routine monitoring and control tasks of the smart grid, including state estimation and optimal power flow. Existing approaches are either challenged by the \emph{combinatorial complexity} issues involved, and are thus limited to identifying single- and double-line outages; or, they invoke less pragmatic assumptions such as \emph{conditionally independent} phasor angle measurements available across the grid. Using only a subset of voltage phasor angle data, the present paper develops a near real-time algorithm for identifying multiple line outages at the affordable complexity of solving a quadratic program via block coordinate descent iterations. The novel approach relies on reformulating the DC linear power flow model as a \emph{sparse} overcomplete expansion, and leveraging contemporary advances in compressive sampling and variable selection using the least-absolute shrinkage and selection operator (Lasso). Analysis and simulated tests on the standard IEEE 118-bus system confirm the effectiveness of lassoing line changes in the smart power grid.

The $\ell$-1 norm based optimization is widely used in signal processing, especially in recent compressed sensing theory. This paper studies the solution path of the $\ell$-1 norm penalized least-square problem, whose constrained form is known as Least Absolute Shrinkage and Selection Operator (LASSO). A solution path is the set of all the optimizers with respect to the evolution of the hyperparameter (Lagrange multiplier). The study of the solution path is of great significance in viewing and understanding the profile of the tradeoff between the approximation and regularization terms. If the solution path of a given problem is known, it can help us to find the optimal hyperparameter under a given criterion such as the Akaike Information Criterion. In this paper we present a sufficient condition on $\ell$-1 norm penalized least-square problem. Under this sufficient condition, the number of nonzero entries in the optimizer or solution vector increases monotonically when the hyperparameter decreases. We also generalize the result to the often used total variation case, where the $\ell$-1 norm is taken over the first order derivative of the solution vector. We prove that the proposed condition has intrinsic connections with the condition given by Donoho, et al \cite{Donoho08} and the positive cone condition by Efron {\it el al} \cite{Efron04}. However, the proposed condition does not need to assume the sparsity level of the signal as required by Donoho et al's condition, and is easier to verify than Efron, et al's positive cone condition when being used for practical applications.

Much of uncertainty quantification to date has focused on determining the effect of variables modeled probabilistically, and with a known distribution, on some physical or engineering system. We develop methods to obtain information on the system when the distributions of some variables are known exactly, others are known only approximately, and perhaps others are not modeled as random variables at all. The main tool used is the duality between risk-sensitive integrals and relative entropy, and we obtain explicit bounds on standard performance measures (variances, exceedance probabilities) over families of distributions whose distance from a nominal distribution is measured by relative entropy. The evaluation of the risk-sensitive expectations is based on polynomial chaos expansions, which help keep the computational aspects tractable.

In this paper we are concerned with fully automatic and locally adaptive estimation of functions in a "signal + noise"-model where the regression function may additionally be blurred by a linear operator, e.g. by a convolution. To this end, we introduce a general class of statistical multiresolution estimators and develop an algorithmic framework for computing those. By this we mean estimators that are defined as solutions of convex optimization problems with supremum-type constraints. We employ a combination of the alternating direction method of multipliers with Dykstra's algorithm for computing orthogonal projections onto intersections of convex sets and prove numerical convergence. The capability of the proposed method is illustrated by various examples from imaging and signal detection.

Single fault sequential change point problems have become important in modeling for various phenomena in large distributed systems, such as sensor networks. But such systems in many situations present multiple interacting faults. For example, individual sensors in a network may fail and detection is performed by comparing measurements between sensors, resulting in statistical dependency among faults. We present a new formulation for multiple interacting faults in a distributed system. The formulation includes specifications of how individual subsystems composing the large system may fail, the information that can be shared among these subsystems and the interaction pattern between faults. We then specify a new sequential algorithm for detecting these faults. The main feature of the algorithm is that it uses composite stopping rules for a subsystem that depend on the decision of other subsystems. We provide asymptotic false alarm and detection delay analysis for this algorithm in the Bayesian setting and show that under certain conditions the algorithm is optimal. The analysis methodology relies on novel detailed comparison techniques between stopping times. We validate the approach with some simulations.

This paper formulates a generalized classification algorithm with an application to classifying (or `decoding') neural activity in the brain. Medical doctors and researchers have long been interested in how brain activity correlates to body movement. Experiments have been conducted on patients whom are unable to move, in order to gain insight as to how thinking about movements might generate discernable neural activity. Researchers are tasked with determining which neurons are responsible for different imagined movements and how the firing behavior changes, given neural firing data. For instance, imagined movements may include wrist flexion, elbow extension, or closing the hand. This is just one of many applications to data classification. Though this article deals with an application in neuroscience, the generalized algorithm proposed in this article has applications in scientific areas ranging from neuroscience to acoustic and medical imaging.

In this paper, we propose three approaches for the estimation of the Tucker decomposition of multi-way arrays (tensors) from partial observations. All approaches are formulated as convex minimization problems. Therefore, the minimum is guaranteed to be unique. The proposed approaches can automatically estimate the number of factors (rank) through the optimization. Thus, there is no need to specify the rank beforehand. The key technique we employ is the trace norm regularization, which is a popular approach for the estimation of low-rank matrices. In addition, we propose a simple heuristic to improve the interpretability of the obtained factorization. The advantages and disadvantages of three proposed approaches are demonstrated through numerical experiments on both synthetic and real world datasets. We show that the proposed convex optimization based approaches are more accurate in predictive performance, faster, and more reliable in recovering a known multilinear structure than conventional approaches.

The goal of decentralized optimization over a network is to optimize a global objective formed by a sum of local (possibly nonsmooth) convex functions using only local computation and communication. It arises in various application domains, including distributed tracking and localization, multi-agent co-ordination, estimation in sensor networks, and large-scale optimization in machine learning. We develop and analyze distributed algorithms based on dual averaging of subgradients, and we provide sharp bounds on their convergence rates as a function of the network size and topology. Our method of analysis allows for a clear separation between the convergence of the optimization algorithm itself and the effects of communication constraints arising from the network structure. In particular, we show that the number of iterations required by our algorithm scales inversely in the spectral gap of the network. The sharpness of this prediction is confirmed both by theoretical lower bounds and simulations for various networks. Our approach includes both the cases of deterministic optimization and communication, as well as problems with stochastic optimization and/or communication.

Denoising by frame thresholding is one of the most basic and efficient methods for recovering a discrete signal or image from data that are corrupted by additive Gaussian white noise. The basic idea is to select a frame of analyzing elements that separates the data in few large coefficients due to the signal and many small coefficients mainly due to the noise ε_n. Removing all data coefficients being in magnitude below a certain threshold yields a reconstruction of the original signal. In order to properly balance the amount of noise to be removed and the relevant signal features to be kept, a precise understanding of the statistical properties of thresholding is important. For that purpose we derive the asymptotic distribution of max_{ω\in Ω_n} |<ϕ_ω^n,ε_n>| for a wide class of redundant frames (ϕ_ω^n: ω\in Ω_n}. Based on our theoretical results we give a rationale for universal extreme value thresholding techniques yielding asymptotically sharp confidence regions and smoothness estimates corresponding to prescribed significance levels. The results cover many frames used in imaging and signal recovery applications, such as redundant wavelet systems, curvelet frames, or unions of bases. We show that `generically' a standard Gumbel law results as it is known from the case of orthonormal wavelet bases. However, for specific highly redundant frames other limiting laws may occur. We indeed verify that the translation invariant wavelet transform shows a different asymptotic behaviour.

Identifying and understanding modular organizations is centrally important in the study of complex systems. Several approaches to this problem have been advanced, many framed in information-theoretic terms. Our treatment starts from the complementary point of view of statistical modeling and prediction of dynamical systems. It is known that for finite amounts of training data, simpler models can have greater predictive power than more complex ones. We use the trade-off between model simplicity and predictive accuracy to generate optimal multiscale decompositions of dynamical networks into weakly-coupled, simple modules. State-dependent and causal versions of our method are also proposed.

In this paper we solve support vector machines in reproducing kernel Banach spaces with reproducing kernels defined on nonsymmetric domains instead of the traditional methods in reproducing kernel Hilbert spaces. Using the orthogonality of semi-inner-products, we can obtain the explicit representations of the dual (normalized-duality-mapping) elements of support vector machine solutions. In addition, we can introduce the reproduction property in a generalized native space by Fourier transform techniques such that it becomes a reproducing kernel Banach space, which can be even embedded into Sobolev spaces, and its reproducing kernel is set up by the related positive definite function. The representations of the optimal solutions of support vector machines (regularized empirical risks) in these reproducing kernel Banach spaces are formulated explicitly in terms of positive definite functions, and their finite numbers of coefficients can be computed by fixed point iteration. We also give some typical examples of reproducing kernel Banach spaces induced by Matérn functions (Sobolev splines) so that their support vector machine solutions are well computable as the classical algorithms. Moreover, each of their reproducing bases includes information from multiple training data points. The concept of reproducing kernel Banach spaces offers us a new numerical tool for solving support vector machines.

Monte Carlo methods are used to approximate the means, $μ$, of random variables $Y$, whose distributions are not known explicitly. The key idea is that the average of a random sample, $Y_1, ..., Y_n$, tends to $μ$ as $n$ tends to infinity. This article explores how one can reliably construct a confidence interval for $μ$ with a prescribed half-width (or error tolerance) $\varepsilon$. Our proposed two-stage algorithm assumes that the kurtosis of $Y$ does not exceed some user-specified bound. An initial independent and identically distributed (IID) sample is used to confidently estimate the variance of $Y$. A Berry-Esseen inequality then makes it possible to determine the size of the IID sample required to construct the desired confidence interval for $μ$. We discuss the important case where $Y=f(\vX)$ and $\vX$ is a random $d$-vector with probability density function $ρ$. In this case $μ$ can be interpreted as the integral $\int_{\reals^d} f(\vx) ρ(\vx) \dif \vx$, and the Monte Carlo method becomes a method for multidimensional cubature.

We perform a non-asymptotic analysis on the singular vector distribution under Gaussian noise. In particular, we provide sufficient conditions on a matrix for its first few singular vectors to have near normal distribution. Our result can be used to facilitate the error analysis in linear dimension reduction.

Sparsity has become a key concept for solving of high-dimensional inverse problems using variational regularization techniques. Recently, using similar sparsity-constraints in the Bayesian framework for inverse problems by encoding them in the prior distribution has attracted attention. Important questions about the relation between regularization theory and Bayesian inference still need to be addressed when using sparsity promoting inversion. A practical obstacle for these examinations is the lack of fast posterior sampling algorithms for sparse, high-dimensional Bayesian inversion: Accessing the full range of Bayesian inference methods requires being able to draw samples from the posterior probability distribution in a fast and efficient way. This is usually done using Markov chain Monte Carlo (MCMC) sampling algorithms. In this article, we develop and examine a new implementation of a single component Gibbs MCMC sampler for sparse priors relying on L1-norms. We demonstrate that the efficiency of our Gibbs sampler increases when the level of sparsity or the dimension of the unknowns is increased. This property is contrary to the properties of the most commonly applied Metropolis-Hastings (MH) sampling schemes: We demonstrate that the efficiency of MH schemes for L1-type priors dramatically decreases when the level of sparsity or the dimension of the unknowns is increased. Practically, Bayesian inversion for L1-type priors using MH samplers is not feasible at all. As this is commonly believed to be an intrinsic feature of MCMC sampling, the performance of our Gibbs sampler also challenges common beliefs about the applicability of sample based Bayesian inference.

This work considers a computationally and statistically efficient parameter estimation method for a wide class of latent variable models---including Gaussian mixture models, hidden Markov models, and latent Dirichlet allocation---which exploits a certain tensor structure in their low-order observable moments (typically, of second- and third-order). Specifically, parameter estimation is reduced to the problem of extracting a certain (orthogonal) decomposition of a symmetric tensor derived from the moments; this decomposition can be viewed as a natural generalization of the singular value decomposition for matrices. Although tensor decompositions are generally intractable to compute, the decomposition of these specially structured tensors can be efficiently obtained by a variety of approaches, including power iterations and maximization approaches (similar to the case of matrices). A detailed analysis of a robust tensor power method is provided, establishing an analogue of Wedin's perturbation theorem for the singular vectors of matrices. This implies a robust and computationally tractable estimation approach for several popular latent variable models.

Nowadays, the numerical models of real-world structures are more precise, more complex and, of course, more time-consuming. Despite the growth of a computational effort, the exploration of model behaviour remains a complex task. The sensitivity analysis is a basic tool for investigating the sensitivity of the model to its inputs. One widely used strategy to assess the sensitivity is based on a finite set of simulations for a given sets of input parameters, i.e. points in the design space. An estimate of the sensitivity can be then obtained by computing correlations between the input parameters and the chosen response of the model. The accuracy of the sensitivity prediction depends on the choice of design points called the design of experiments. The aim of the presented paper is to review and compare available criteria determining the quality of the design of experiments suitable for sampling-based sensitivity analysis.

We introduce a novel strategy to address the issue of demand estimation in single-item single-period stochastic inventory optimisation problems. Our strategy analytically combines confidence interval analysis and inventory optimisation. We assume that the decision maker is given a set of past demand samples and we employ confidence interval analysis in order to identify a range of candidate order quantities that, with prescribed confidence probability, includes the real optimal order quantity for the underlying stochastic demand process with unknown stationary parameter(s). In addition, for each candidate order quantity that is identified, our approach can produce an upper and a lower bound for the associated cost. We apply our novel approach to three demand distribution in the exponential family: binomial, Poisson, and exponential. For two of these distributions we also discuss the extension to the case of unobserved lost sales. Numerical examples are presented in which we show how our approach complements existing frequentist - e.g. based on maximum likelihood estimators - or Bayesian strategies.

We present a novel algebraic combinatorial view on low-rank matrix completion based on studying relations between a few entries with tools from algebraic geometry and matroid theory. The intrinsic locality of the approach allows for the treatment of single entries in a closed theoretical and practical framework. More specifically, apart from introducing an algebraic combinatorial theory of low-rank matrix completion, we present probability-one algorithms to decide whether a particular entry of the matrix can be completed. We also describe methods to complete that entry from a few others, and to estimate the error which is incurred by any method completing that entry. Furthermore, we show how known results on matrix completion and their sampling assumptions can be related to our new perspective and interpreted in terms of a completability phase transition.

We are introducing two methods for revealing the true inflection point of data that contains or not error. The starting point is a set of geometrical properties that follow the existence of an inflection point p for a smooth function. These properties connect the concept of convexity/concavity before and after p respectively with three chords defined properly. Finally a set of experiments is presented for the class of sigmoid curves and for the third order polynomials.

The reduced basis method is a powerful model reduction technique designed to speed up the computation of multiple numerical solutions of parametrized partial differential equations. We consider a quantity of interest, which is a linear functional of the PDE solution. A new probabilistic error bound for the reduced model is proposed. It is efficiently and explicitly computable, and we show on different examples that this error bound is sharper than existing ones. We include application of our work to sensitivity analysis studies.

We describe several algorithms for matrix completion and matrix approximation when only some of its entries are known. The approximation constraint can be any whose approximated solution is known for the full matrix. For low rank approximations, similar algorithms appears recently in the literature under different names. In this work, we introduce new theorems for matrix approximation and show that these algorithms can be extended to handle different constraints such as nuclear norm, spectral norm, orthogonality constraints and more that are different than low rank approximations. As the algorithms can be viewed from an optimization point of view, we discuss their convergence to global solution for the convex case. We also discuss the optimal step size and show that it is fixed in each iteration. In addition, the derived matrix completion flow is robust and does not require any parameters. This matrix completion flow is applicable to different spectral minimizations and can be applied to physics, mathematics and electrical engineering problems such as data reconstruction of images and data coming from PDEs such as Helmholtz equation used for electromagnetic waves.

We discuss two new methods of recovery of sparse signals from noisy observation based on $\ell_1$- minimization. They are closely related to the well-known techniques such as Lasso and Dantzig Selector. However, these estimators come with efficiently verifiable guaranties of performance. By optimizing these bounds with respect to the method parameters we are able to construct the estimators which possess better statistical properties than the commonly used ones. We also show how these techniques allow to provide efficiently computable accuracy bounds for Lasso and Dantzig Selector. We link our performance estimations to the well known results of Compressive Sensing and justify our proposed approach with an oracle inequality which links the properties of the recovery algorithms and the best estimation performance when the signal support is known. We demonstrate how the estimates can be computed using the Non-Euclidean Basis Pursuit algorithm.

We generalize Newton-type methods for minimizing smooth functions to handle a sum of two convex functions: a smooth function and a nonsmooth function with a simple proximal mapping. We show that the resulting proximal Newton-type methods inherit the desirable convergence behavior of Newton-type methods for minimizing smooth functions, even when search directions are computed inexactly. Many popular methods tailored to problems arising in bioinformatics, signal processing, and statistical learning are special cases of proximal Newton-type methods, and our analysis yields new convergence results for some of these methods.

In this report a derivation of the MAP state estimator objective function for general (possibly non-square) discrete time causal/non-causal descriptor systems is presented. The derivation made use of the Kronecker Canonical Transformation to extract the prior distribution on the descriptor state vector so that Maximum a Posteriori (MAP) point estimation can be used. The analysis indicates that the MAP estimate for index 1 causal descriptor systems does not require any model transformations and can be found recursively. Furthermore, if the descriptor system is of index 2 or higher and the noise free system is causal, then the MAP estimate can also be found recursively without model transformations provided that model causality is accounted for in designing the stochastic model.

This work is concerned with the ill-posed inverse problem of estimating turbulent flows from the observation of an image sequence. From a Bayesian perspective, a divergence-free isotropic fractional Brownian motion (fBm) is chosen as a prior model for instantaneous turbulent velocity fields. This self-similar prior characterizes accurately second-order statistics of velocity fields in incompressible isotropic turbulence. Nevertheless, the associated maximum a posteriori involves a fractional Laplacian operator which is delicate to implement in practice. To deal with this issue, we propose to decompose the divergent-free fBm on well-chosen wavelet bases. As a first alternative, we propose to design wavelets as whitening filters. We show that these filters are fractional Laplacian wavelets composed with the Leray projector. As a second alternative, we use a divergence-free wavelet basis, which takes implicitly into account the incompressibility constraint arising from physics. Although the latter decomposition involves correlated wavelet coefficients, we are able to handle this dependence in practice. Based on these two wavelet decompositions, we finally provide effective and efficient algorithms to approach the maximum a posteriori. An intensive numerical evaluation proves the relevance of the proposed wavelet-based self-similar priors.

This paper discusses a methodology for determining a functional representation of a random process from a collection of scattered pointwise samples. The present work specifically focuses onto random quantities lying in a high dimensional stochastic space in the context of limited amount of information. The proposed approach involves a procedure for the selection of an approximation basis and the evaluation of the associated coefficients. The selection of the approximation basis relies on the a priori choice of the High-Dimensional Model Representation format combined with a modified Least Angle Regression technique. The resulting basis then provides the structure for the actual approximation basis, possibly using different functions, more parsimonious and nonlinear in its coefficients. To evaluate the coefficients, both an alternate least squares and an alternate weighted total least squares methods are employed. Examples are provided for the approximation of a random variable in a high-dimensional space as well as the estimation of a random field. Stochastic dimensions up to 100 are considered, with an amount of information as low as about 3 samples per dimension, and robustness of the approximation is demonstrated w.r.t. noise in the dataset. The computational cost of the solution method is shown to scale only linearly with the cardinality of the a priori basis and exhibits a (N_q)^s, 2 <= s <= 3, dependence with the number N_q of samples in the dataset. The provided numerical experiments illustrate the ability of the present approach to derive an accurate approximation from scarce scattered data even in the presence of noise.

Constructing an efficient parameterization of a large, noisy data set of points lying close to a smooth manifold in high dimension remains a fundamental problem. One approach consists in recovering a local parameterization using the local tangent plane. Principal component analysis (PCA) is often the tool of choice, as it returns an optimal basis in the case of noise-free samples from a linear subspace. To process noisy data samples from a nonlinear manifold, PCA must be applied locally, at a scale small enough such that the manifold is approximately linear, but at a scale large enough such that structure may be discerned from noise. Using eigenspace perturbation theory and non-asymptotic random matrix theory, we study the stability of the subspace estimated by PCA as a function of scale, and bound (with high probability) the angle it forms with the true tangent space. By adaptively selecting the scale that minimizes this bound, our analysis reveals an appropriate scale for local tangent plane recovery. We also introduce a geometric uncertainty principle quantifying the limits of noise-curvature perturbation for stable recovery. With the purpose of providing perturbation bounds that can be used in practice, we propose plug-in estimates that make it possible to directly apply the theoretical results to real data sets.

We first review existing sequential methods for estimating a binomial proportion. Afterward, we propose a new family of group sequential sampling schemes for estimating a binomial proportion with prescribed margin of error and confidence level. In particular, we establish the uniform controllability of coverage probability and the asymptotic optimality for such a family of sampling schemes. Our theoretical results establish the possibility that the parameters of this family of sampling schemes can be determined so that the prescribed level of confidence is guaranteed with little waste of samples. Analytic bounds for the cumulative distribution functions and expectations of sample numbers are derived. Moreover, we discuss the inherent connection of various sampling schemes. Numerical issues are addressed for improving the accuracy and efficiency of computation. Computational experiments are conducted for comparing sampling schemes. Illustrative examples are given for applications in clinical trials.

In this paper, we develop a computational approach for estimating the mean value of a quantity in the presence of uncertainty. We demonstrate that, under some mild assumptions, the upper and lower bounds of the mean value are efficiently computable via a sample reuse technique, of which the computational complexity is shown to posses a Poisson distribution.

This paper offers a critical view of the "worst-case" approach that is the cornerstone of robust control design. It is our contention that a blind acceptance of worst-case scenarios may lead to designs that are actually more dangerous than designs based on probabilistic techniques with a built-in risk factor. The real issue is one of modeling. If one accepts that no mathematical model of uncertainties is perfect then a probabilistic approach can lead to more reliable control even if it cannot guarantee stability for all possible cases. Our presentation is based on case analysis. We first establish that worst-case is not necessarily "all-encompassing." In fact, we show that for some uncertain control problems to have a conventional robust control solution it is necessary to make assumptions that leave out some feasible cases. Once we establish that point, we argue that it is not uncommon for the risk of unaccounted cases in worst-case design to be greater than that of the accepted risk in a probabilistic approach. With an example, we quantify the risks and show that worst-case can be significantly more risky. Finally, we join our analysis with existing results on computational complexity and probabilistic robustness to argue that the deterministic worst-case analysis is not necessarily the better tool.

This paper derives new algorithms for signomial programming, a generalization of geometric programming. The algorithms are based on a generic principle for optimization called the MM algorithm. In this setting, one can apply the geometric-arithmetic mean inequality and a supporting hyperplane inequality to create a surrogate function with parameters separated. Thus, unconstrained signomial programming reduces to a sequence of one-dimensional minimization problems. Simple examples demonstrate that the MM algorithm derived can converge to a boundary point or to one point of a continuum of minimum points. Conditions under which the minimum point is unique or occurs in the interior of parameter space are proved for geometric programming. Convergence to an interior point occurs at a linear rate. Finally, the MM framework easily accommodates equality and inequality constraints of signomial type. For the most important special case, constrained quadratic programming, the MM algorithm involves very simple updates.

The non-negative solution to an underdetermined linear system can be uniquely recovered sometimes, even without imposing any additional sparsity constraints. In this paper, we derive conditions under which a unique non-negative solution for such a system can exist, based on the theory of polytopes. Furthermore, we develop the paradigm of combined sparse representations, where only a part of the coefficient vector is constrained to be non-negative, and the rest is unconstrained (general). We analyze the recovery of the unique, sparsest solution, for combined representations, under three different cases of coefficient support knowledge: (a) the non-zero supports of non-negative and general coefficients are known, (b) the non-zero support of general coefficients alone is known, and (c) both the non-zero supports are unknown. For case (c), we propose the combined orthogonal matching pursuit algorithm for coefficient recovery and derive the deterministic sparsity threshold under which recovery of the unique, sparsest coefficient vector is possible. We quantify the order complexity of the algorithms, and examine their performance in exact and approximate recovery of coefficients under various conditions of noise. Furthermore, we also obtain their empirical phase transition characteristics. We show that the basis pursuit algorithm, with partial non-negative constraints, and the proposed greedy algorithm perform better in recovering the unique sparse representation when compared to their unconstrained counterparts. Finally, we demonstrate the utility of the proposed methods in recovering images corrupted by saturation noise.

We introduce GP-FNARX: a new model for nonlinear system identification based on a nonlinear autoregressive exogenous model (NARX) with filtered regressors (F) where the nonlinear regression problem is tackled using sparse Gaussian processes (GP). We integrate data pre-processing with system identification into a fully automated procedure that goes from raw data to an identified model. Both pre-processing parameters and GP hyper-parameters are tuned by maximizing the marginal likelihood of the probabilistic model. We obtain a Bayesian model of the system's dynamics which is able to report its uncertainty in regions where the data is scarce. The automated approach, the modeling of uncertainty and its relatively low computational cost make of GP-FNARX a good candidate for applications in robotics and adaptive control.

This paper presents a finite-time analysis of the KL-UCB algorithm, an online, horizon-free index policy for stochastic bandit problems. We prove two distinct results: first, for arbitrary bounded rewards, the KL-UCB algorithm satisfies a uniformly better regret bound than UCB or UCB2; second, in the special case of Bernoulli rewards, it reaches the lower bound of Lai and Robbins. Furthermore, we show that simple adaptations of the KL-UCB algorithm are also optimal for specific classes of (possibly unbounded) rewards, including those generated from exponential families of distributions. A large-scale numerical study comparing KL-UCB with its main competitors (UCB, UCB2, UCB-Tuned, UCB-V, DMED) shows that KL-UCB is remarkably efficient and stable, including for short time horizons. KL-UCB is also the only method that always performs better than the basic UCB policy. Our regret bounds rely on deviations results of independent interest which are stated and proved in the Appendix. As a by-product, we also obtain an improved regret bound for the standard UCB algorithm.

Although nonnegative matrix factorization (NMF) is NP-hard in general, it has been shown very recently that it is tractable under the assumption that the input nonnegative data matrix is close to being separable (separability requires that all columns of the input matrix belongs to the cone spanned by a small subset of these columns). Since then, several algorithms have been designed to handle this subclass of NMF problems. In particular, Bittorf, Recht, Ré and Tropp (`Factoring nonnegative matrices with linear programs', NIPS 2012) proposed a linear programming model, referred to as Hottopixx. In this paper, we provide a new and more general robustness analysis of their method. In particular, we design a provably more robust variant using a post-processing strategy which allows us to deal with duplicates and near duplicates in the dataset.

Sparsity-constrained optimization has wide applicability in machine learning, statistics, and signal processing problems such as feature selection and compressive Sensing. A vast body of work has studied the sparsity-constrained optimization from theoretical, algorithmic, and application aspects in the context of sparse estimation in linear models where the fidelity of the estimate is measured by the squared error. In contrast, relatively less effort has been made in the study of sparsity-constrained optimization in cases where nonlinear models are involved or the cost function is not quadratic. In this paper we propose a greedy algorithm, Gradient Support Pursuit (GraSP), to approximate sparse minima of cost functions of arbitrary form. Should a cost function have a Stable Restricted Hessian (SRH) or a Stable Restricted Linearization (SRL), both of which are introduced in this paper, our algorithm is guaranteed to produce a sparse vector within a bounded distance from the true sparse optimum. Our approach generalizes known results for quadratic cost functions that arise in sparse linear regression and Compressive Sensing. We also evaluate the performance of GraSP through numerical simulations on synthetic data, where the algorithm is employed for sparse logistic regression with and without $\ell_2$-regularization.

We study Bayesian inversion for a model elliptic PDE with unknown diffusion coefficient. We provide complexity analyses of several Markov Chain-Monte Carlo (MCMC) methods for the efficient numerical evaluation of expectations under the Bayesian posterior distribution, given data $δ$. Particular attention is given to bounds on the overall work required to achieve a prescribed error level $\varepsilon$. Specifically, we first bound the computational complexity of "plain" MCMC, based on combining MCMC sampling with linear complexity multilevel solvers for elliptic PDE. Our (new) work versus accuracy bounds show that the complexity of this approach can be quite prohibitive. Two strategies for reducing the computational complexity are then proposed and analyzed: first, a sparse, parametric and deterministic generalized polynomial chaos (gpc) "surrogate" representation of the forward response map of the PDE over the entire parameter space, and, second, a novel Multi-Level Markov Chain Monte Carlo (MLMCMC) strategy which utilizes sampling from a multilevel discretization of the posterior and of the forward PDE. For both of these strategies we derive asymptotic bounds on work versus accuracy, and hence asymptotic bounds on the computational complexity of the algorithms. In particular we provide sufficient conditions on the regularity of the unknown coefficients of the PDE, and on the approximation methods used, in order for the accelerations of MCMC resulting from these strategies to lead to complexity reductions over "plain" MCMC algorithms for Bayesian inversion of PDEs.}

We discuss structured Schatten norms for tensor decomposition that includes two recently proposed norms ("overlapped" and "latent") for convex-optimization-based tensor decomposition, and connect tensor decomposition with wider literature on structured sparsity. Based on the properties of the structured Schatten norms, we mathematically analyze the performance of "latent" approach for tensor decomposition, which was empirically found to perform better than the "overlapped" approach in some settings. We show theoretically that this is indeed the case. In particular, when the unknown true tensor is low-rank in a specific mode, this approach performs as good as knowing the mode with the smallest rank. Along the way, we show a novel duality result for structures Schatten norms, establish the consistency, and discuss the identifiability of this approach. We confirm through numerical simulations that our theoretical prediction can precisely predict the scaling behavior of the mean squared error.

We present a Kalman smoothing framework based on modeling errors using the heavy tailed Student's t distribution, along with algorithms, convergence theory, open-source general implementation, and several important applications. The computational effort per iteration grows linearly with the length of the time series, and all smoothers allow nonlinear process and measurement models. Robust smoothers form an important subclass of smoothers within this framework. These smoothers work in situations where measurements are highly contaminated by noise or include data unexplained by the forward model. Highly robust smoothers are developed by modeling measurement errors using the Student's t distribution, and outperform the recently proposed L1-Laplace smoother in extreme situations with data containing 20% or more outliers. A second special application we consider in detail allows tracking sudden changes in the state. It is developed by modeling process noise using the Student's t distribution, and the resulting smoother can track sudden changes in the state. These features can be used separately or in tandem, and we present a general smoother algorithm and open source implementation, together with convergence analysis that covers a wide range of smoothers. A key ingredient of our approach is a technique to deal with the non-convexity of the Student's t loss function. Numerical results for linear and nonlinear models illustrate the performance of the new smoothers for robust and tracking applications, as well as for mixed problems that have both types of features.

Non-convex sparsity-inducing penalties have recently received considerable attentions in sparse learning. Recent theoretical investigations have demonstrated their superiority over the convex counterparts in several sparse learning settings. However, solving the non-convex optimization problems associated with non-convex penalties remains a big challenge. A commonly used approach is the Multi-Stage (MS) convex relaxation (or DC programming), which relaxes the original non-convex problem to a sequence of convex problems. This approach is usually not very practical for large-scale problems because its computational cost is a multiple of solving a single convex problem. In this paper, we propose a General Iterative Shrinkage and Thresholding (GIST) algorithm to solve the nonconvex optimization problem for a large class of non-convex penalties. The GIST algorithm iteratively solves a proximal operator problem, which in turn has a closed-form solution for many commonly used penalties. At each outer iteration of the algorithm, we use a line search initialized by the Barzilai-Borwein (BB) rule that allows finding an appropriate step size quickly. The paper also presents a detailed convergence analysis of the GIST algorithm. The efficiency of the proposed algorithm is demonstrated by extensive experiments on large-scale data sets.

Seasonality (or periodicity) and trend are features describing an observed sequence, and extracting these features is an important issue in many scientific fields. However, it is not an easy task for existing methods to analyze simultaneously the trend and {\it dynamics} of the seasonality such as time-varying frequency and amplitude, and the {\it adaptivity} of the analysis to such dynamics and robustness to heteroscedastic, dependent errors is not guaranteed. These tasks become even more challenging when there exist multiple seasonal components. We propose a nonparametric model to describe the dynamics of multi-component seasonality, and investigate the recently developed Synchrosqueezing transform (SST) in extracting these features in the presence of a trend and heteroscedastic, dependent errors. The identifiability problem of the nonparametric seasonality model is studied, and the adaptivity and robustness properties of the SST are theoretically justified in both discrete- and continuous-time settings. Consequently we have a new technique for de-coupling the trend, seasonality and heteroscedastic, dependent error process in a general nonparametric setup. Results of a series of simulations are provided, and the incidence time series of varicella and herpes zoster in Taiwan and respiratory signals observed from a sleep study are analyzed.

In the Multi-Armed Bandit (MAB) problem, there is a given set of arms with unknown reward models. At each time, a player selects one arm to play, aiming to maximize the total expected reward over a horizon of length T. An approach based on a Deterministic Sequencing of Exploration and Exploitation (DSEE) is developed for constructing sequential arm selection policies. It is shown that for all light-tailed reward distributions, DSEE achieves the optimal logarithmic order of the regret, where regret is defined as the total expected reward loss against the ideal case with known reward models. For heavy-tailed reward distributions, DSEE achieves O(T^1/p) regret when the moments of the reward distributions exist up to the pth order for 1<p<=2 and O(T^1/(1+p/2)) for p>2. With the knowledge of an upperbound on a finite moment of the heavy-tailed reward distributions, DSEE offers the optimal logarithmic regret order. The proposed DSEE approach complements existing work on MAB by providing corresponding results for general reward distributions. Furthermore, with a clearly defined tunable parameter-the cardinality of the exploration sequence, the DSEE approach is easily extendable to variations of MAB, including MAB with various objectives, decentralized MAB with multiple players and incomplete reward observations under collisions, MAB with unknown Markov dynamics, and combinatorial MAB with dependent arms that often arise in network optimization problems such as the shortest path, the minimum spanning, and the dominating set problems under unknown random weights.

In this paper we study the performance of the Projected Gradient Descent(PGD) algorithm for $\ell_{p}$-constrained least squares problems that arise in the framework of Compressed Sensing. Relying on the Restricted Isometry Property, we provide convergence guarantees for this algorithm for the entire range of $0\leq p\leq1$, that include and generalize the existing results for the Iterative Hard Thresholding algorithm and provide a new accuracy guarantee for the Iterative Soft Thresholding algorithm as special cases. Our results suggest that in this group of algorithms, as $p$ increases from zero to one, conditions required to guarantee accuracy become stricter and robustness to noise deteriorates.

In nonparametric statistical problems, we wish to find an estimator of an unknown function f. We can split its error into bias and variance terms; Smirnov, Bickel and Rosenblatt have shown that, for a histogram or kernel estimate, the supremum norm of the variance term is asymptotically distributed as a Gumbel random variable. In the following, we prove a version of this result for estimators using compactly-supported wavelets, a popular tool in nonparametric statistics. Our result relies on an assumption on the nature of the wavelet, which must be verified by provably-good numerical approximations. We verify our assumption for Daubechies wavelets and symlets, with N = 6, ..., 20 vanishing moments; larger values of N, and other wavelet bases, are easily checked, and we conjecture that our assumption holds also in those cases.

We consider the problem of estimating the division rate of a size-structured population in a nonparametric setting. The size of the system evolves according to a transport-fragmentation equation: each individual grows with a given transport rate, and splits into two offsprings of the same size, following a binary fragmentation process with unknown division rate that depends on its size. In contrast to a deterministic inverse problem approach, as in (Perthame, Zubelli, 2007) and (Doumic, Perthame, Zubelli, 2009), we take in this paper the perspective of statistical inference: our data consists in a large sample of the size of individuals, when the evolution of the system is close to its time-asymptotic behavior, so that it can be related to the eigenproblem of the considered transport-fragmentation equation (see \cite{PR} for instance). By estimating statistically each term of the eigenvalue problem and by suitably inverting a certain linear operator (see previously quoted articles), we are able to construct a more realistic estimator of the division rate that achieves the same optimal error bound as in related deterministic inverse problems. Our procedure relies on kernel methods with automatic bandwidth selection. It is inspired by model selection and recent results of Goldenschluger and Lepski.

We discuss construction of coverings of the unit ball of a finite dimensional Banach space. The well known technique of comparing volumes gives upper and lower bounds on covering numbers. This technique does not provide a construction of good coverings. Here we apply incoherent dictionaries for construction of good coverings. We use the following strategy. First, we build a good covering by balls with a radius close to one. Second, we iterate this construction to obtain a good covering for any radius. We mostly concentrate on the first step of this strategy.

Given a set of mixtures, blind source separation attempts to retrieve the source signals without or with very little information of the the mixing process. We present a geometric approach for blind separation of nonnegative linear mixtures termed {\em facet component analysis} (FCA). The approach is based on facet identification of the underlying cone structure of the data. Earlier works focus on recovering the cone by locating its vertices (vertex component analysis or VCA) based on a mutual sparsity condition which requires each source signal to possess a stand-alone peak in its spectrum. We formulate alternative conditions so that enough data points fall on the facets of a cone instead of accumulating around the vertices. To find a regime of unique solvability, we make use of both geometric and density properties of the data points, and develop an efficient facet identification method by combining data classification and linear regression. For noisy data, we show that denoising methods may be employed, such as the total variation technique in imaging processing, and principle component analysis. We show computational results on nuclear magnetic resonance spectroscopic data to substantiate our method.

We discuss a general technique that can be used to form a differentiable bound on the optima of non-differentiable or discrete objective functions. We form a unified description of these methods and consider under which circumstances the bound is concave. In particular we consider two concrete applications of the method, namely sparse learning and support vector classification.

This paper describes a general approach to the compartmental modeling of nuclear data based on spectral analysis and statistical optimization. We utilize the renal physiology as test case and validate the method against both synthetic data and real measurements acquired during two micro-PET experiments with murine models.

For a power system operating in the vicinity of the power transfer limit of its transmission system, effect of stochastic fluctuations of power loads can become critical as a sufficiently strong such fluctuation may activate voltage instability and lead to a large scale collapse of the system. Considering the effect of these stochastic fluctuations near a codimension 1 saddle-node bifurcation, we explicitly calculate the autocorrelation function of the state vector and show how its behavior explains the phenomenon of critical slowing-down often observed for power systems on the threshold of blackout. We also estimate the collapse probability/mean clearing time for the power system and construct a new indicator function signaling the proximity to a large scale collapse. The new indicator function is easy to estimate in real time using PMU data feeds as well as SCADA information about fluctuations of power load on the nodes of the power grid. We discuss control strategies leading to the minimization of the collapse probability.

Nonnegative matrix factorization (NMF) has become a very popular technique in machine learning because it automatically extracts meaningful features through a sparse and part-based representation. However, NMF has the drawback of being highly ill-posed, that is, there typically exist many different but equivalent factorizations. In this paper, we introduce a completely new way to obtaining more well-posed NMF problems whose solutions are sparser. Our technique is based on the preprocessing of the nonnegative input data matrix, and relies on the theory of M-matrices and the geometric interpretation of NMF. This approach provably leads to optimal and sparse solutions under the separability assumption of Donoho and Stodden (NIPS, 2003), and, for rank-three matrices, makes the number of exact factorizations finite. We illustrate the effectiveness of our technique on several image datasets.

Many problems in sequential decision making and stochastic control often have natural multiscale structure: sub-tasks are assembled together to accomplish complex goals. Systematically inferring and leveraging hierarchical structure, particularly beyond a single level of abstraction, has remained a longstanding challenge. We describe a fast multiscale procedure for repeatedly compressing, or homogenizing, Markov decision processes (MDPs), wherein a hierarchy of sub-problems at different scales is automatically determined. Coarsened MDPs are themselves independent, deterministic MDPs, and may be solved using existing algorithms. The multiscale representation delivered by this procedure decouples sub-tasks from each other and can lead to substantial improvements in convergence rates both locally within sub-problems and globally across sub-problems, yielding significant computational savings. A second fundamental aspect of this work is that these multiscale decompositions yield new transfer opportunities across different problems, where solutions of sub-tasks at different levels of the hierarchy may be amenable to transfer to new problems. Localized transfer of policies and potential operators at arbitrary scales is emphasized. Finally, we demonstrate compression and transfer in a collection of illustrative domains, including examples involving discrete and continuous statespaces.

We present an algorithm, AROFAC2, which detects the (CP-)rank of a degree 3 tensor and calculates its factorization into rank-one components. We provide generative conditions for the algorithm to work and demonstrate on both synthetic and real world data that AROFAC2 is a potentially outperforming alternative to the gold standard PARAFAC over which it has the advantages that it can intrinsically detect the true rank, avoids spurious components, and is stable with respect to outliers and non-Gaussian noise.

This note gives a simple analysis of a randomized approximation scheme for matrix multiplication proposed by Sarlos (2006) based on a random rotation followed by uniform column sampling. The result follows from a matrix version of Bernstein's inequality and a tail inequality for quadratic forms in subgaussian random vectors.

A novel Mathematical Random Number Generator (MRNG) is presented here. In this case, "mathematical" refers to the fact that to construct that generator it is not necessary to resort to a physical phenomenon, such as the thermal noise of an electronic device, but rather to a mathematical procedure. The MRNG generates binary strings - in principle, as long as desired - which may be considered genuinely random in the sense that they pass the statistical tests currently accepted to evaluate the randomness of those strings. From those strings, the MRNG also generates random numbers expressed in base 10. An MRNG has been installed as a facility on the following web page: http://www.appliedmathgroup.org. This generator may be used for applications in tasks in: a) computational simulation of probabilistic-type systems, and b) the random selection of samples of different populations. Users interested in applications in cryptography can build another MRNG, but they would have to withhold information - specified in section 5 - from people who are not authorized to decode messages encrypted using that resource.

We consider the problem of positioning a cloud of points in the Euclidean space $\mathbb{R}^d$, using noisy measurements of a subset of pairwise distances. This task has applications in various areas, such as sensor network localization and reconstruction of protein conformations from NMR measurements. Also, it is closely related to dimensionality reduction problems and manifold learning, where the goal is to learn the underlying global geometry of a data set using local (or partial) metric information. Here we propose a reconstruction algorithm based on semidefinite programming. For a random geometric graph model and uniformly bounded noise, we provide a precise characterization of the algorithm's performance: In the noiseless case, we find a radius $r_0$ beyond which the algorithm reconstructs the exact positions (up to rigid transformations). In the presence of noise, we obtain upper and lower bounds on the reconstruction error that match up to a factor that depends only on the dimension $d$, and the average degree of the nodes in the graph.

We study a random sampling technique to approximate integrals $\int_{[0,1]^s}f(\mathbf{x})\,\mathrm{d}\mathbf{x}$ by averaging the function at some sampling points. We focus on cases where the integrand is smooth, which is a problem which occurs in statistics. The convergence rate of the approximation error depends on the smoothness of the function $f$ and the sampling technique. For instance, Monte Carlo (MC) sampling yields a convergence of the root mean square error (RMSE) of order $N^{-1/2}$ (where $N$ is the number of samples) for functions $f$ with finite variance. Randomized QMC (RQMC), a combination of MC and quasi-Monte Carlo (QMC), achieves a RMSE of order $N^{-3/2+\varepsilon}$ under the stronger assumption that the integrand has bounded variation. A combination of RQMC with local antithetic sampling achieves a convergence of the RMSE of order $N^{-3/2-1/s+\varepsilon}$ (where $s\ge1$ is the dimension) for functions with mixed partial derivatives up to order two. Additional smoothness of the integrand does not improve the rate of convergence of these algorithms in general. On the other hand, it is known that without additional smoothness of the integrand it is not possible to improve the convergence rate. This paper introduces a new RQMC algorithm, for which we prove that it achieves a convergence of the root mean square error (RMSE) of order $N^{-α-1/2+\varepsilon}$ provided the integrand satisfies the strong assumption that it has square integrable partial mixed derivatives up to order $α>1$ in each variable. Known lower bounds on the RMSE show that this rate of convergence cannot be improved in general for integrands with this smoothness. We provide numerical examples for which the RMSE converges approximately with order $N^{-5/2}$ and $N^{-7/2}$, in accordance with the theoretical upper bound.

We describe regularized methods for image reconstruction and focus on the question of hyperparameter and instrument parameter estimation, i.e. unsupervised and myopic problems. We developed a Bayesian framework that is based on the \post density for all unknown quantities, given the observations. This density is explored by a Markov Chain Monte-Carlo sampling technique based on a Gibbs loop and including a Metropolis-Hastings step. The numerical evaluation relies on the SPIRE instrument of the Herschel observatory. Using simulated and real observations, we show that the hyperparameters and instrument parameters are correctly estimated, which opens up many perspectives for imaging in astrophysics.