In this paper, we consider the problem of estimating multiple graphical models simultaneously using the fused lasso penalty, which encourages adjacent graphs to share similar structures. A motivating example is the analysis of brain networks of Alzheimer's disease using neuroimaging data. Specifically, we may wish to estimate a brain network for the normal controls (NC), a brain network for the patients with mild cognitive impairment (MCI), and a brain network for Alzheimer's patients (AD). We expect the two brain networks for NC and MCI to share common structures but not to be identical to each other; similarly for the two brain networks for MCI and AD. The proposed formulation can be solved using a second-order method. Our key technical contribution is to establish the necessary and sufficient condition for the graphs to be decomposable. Based on this key property, a simple screening rule is presented, which decomposes the large graphs into small subgraphs and allows an efficient estimation of multiple independent (small) subgraphs, dramatically reducing the computational cost. We perform experiments on both synthetic and real data; our results demonstrate the effectiveness and efficiency of the proposed approach.
The solution path of the 1D fused lasso for an $n$-dimensional input is piecewise linear with $\mathcal{O}(n)$ segments (Hoefling et al. 2010 and Tibshirani et al 2011). However, existing proofs of this bound do not hold for the weighted fused lasso. At the same time, results for the generalized lasso, of which the weighted fused lasso is a special case, allow $Ω(3^n)$ segments (Mairal et al. 2012). In this paper, we prove that the number of segments in the solution path of the weighted fused lasso is $\mathcal{O}(n^2)$, and that, for some instances, it is $Ω(n^2)$. We also give a new, very si…
Anomalies and outliers are common in real-world data, and they can arise from many sources, such as sensor faults. Accordingly, anomaly detection is important both for analyzing the anomalies themselves and for cleaning the data for further analysis of its ambient structure. Nonetheless, a precise definition of anomalies is important for automated detection and herein we approach such problems from the perspective of detecting sparse latent effects embedded in large collections of noisy data. Standard Graphical Lasso-based techniques can identify the conditional dependency structure of a colle…
The Lasso is a very well known penalized regression model, which adds an $L_{1}$ penalty with parameter $λ_{1}$ on the coefficients to the squared error loss function. The Fused Lasso extends this model by also putting an $L_{1}$ penalty with parameter $λ_{2}$ on the difference of neighboring coefficients, assuming there is a natural ordering. In this paper, we develop a fast path algorithm for solving the Fused Lasso Signal Approximator that computes the solutions for all values of $λ_1$ and $λ_2$. In the supplement, we also give an algorithm for the general Fused Lasso for the case with pred…
We consider the problem of Graphical lasso with an additional $\ell_{\infty}$ element-wise norm constraint on the precision matrix. This problem has applications in high-dimensional covariance decomposition such as in \citep{Janzamin-12}. We propose an ADMM algorithm to solve this problem. We also use a continuation strategy on the penalty parameter to have a fast implemenation of the algorithm.
We develop a new method called Discriminated Hub Graphical Lasso (DHGL) based on Hub Graphical Lasso (HGL) by providing prior information of hubs. We apply this new method in two situations: with known hubs and without known hubs. Then we compare DHGL with HGL using several measures of performance. When some hubs are known, we can always estimate the precision matrix better via DHGL than HGL. When no hubs are known, we use Graphical Lasso (GL) to provide information of hubs and find that the performance of DHGL will always be better than HGL if correct prior information is given and will seldo…
rdering of regression or classification coefficients occurs in many real-world applications. Fused Lasso exploits this ordering by explicitly regularizing the differences between neighboring coefficients through an $\ell_1$ norm regularizer. However, due to nonseparability and nonsmoothness of the regularization term, solving the fused Lasso problem is computationally demanding. Existing solvers can only deal with problems of small or medium size, or a special case of the fused Lasso problem in which the predictor matrix is identity matrix. In this paper, we propose an iterative algorithm base…
Central University of Finance and Economics（中央财经大学）；University of York（约克大学）；University of Illinois at Chicago（伊利诺伊大学芝加哥分校）
Feature selection has been proven a powerful preprocessing step for high-dimensional data analysis. However, most state-of-the-art methods tend to overlook the structural correlation information between pairwise samples, which may encapsulate useful information for refining the performance of feature selection. Moreover, they usually consider candidate feature relevancy equivalent to selected feature relevancy, and some less relevant features may be misinterpreted as salient features. To overcome these issues, we propose a new feature selection method using structural correlation between pairw…
We study the property of the Fused Lasso Signal Approximator (FLSA) for estimating a blocky signal sequence with additive noise. We transform the FLSA to an ordinary Lasso problem. By studying the property of the design matrix in the transformed Lasso problem, we find that the irrepresentable condition might not hold, in which case we show that the FLSA might not be able to recover the signal pattern. We then apply the newly developed preconditioning method -- Puffer Transformation [Jia and Rohe, 2012] on the transformed Lasso problem. We call the new method the preconditioned fused Lasso and …
Smoothness of the subdiagonals of the Cholesky factor of large covariance matrices is closely related to the degrees of nonstationarity of autoregressive models for time series and longitudinal data. Heuristically, one expects for a nearly stationary covariance matrix the entries in each subdiagonal of the Cholesky factor of its inverse to be nearly the same in the sense that sum of absolute values of successive terms is small. Statistically such smoothness is achieved by regularizing each subdiagonal using fused-type lasso penalties. We rely on the standard Cholesky factor as the new paramete…
Toyota Technological Institute at Chicago（芝加哥丰田理工学院）；University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
This paper considers the problem of estimating multiple related Gaussian graphical models from a $p$-dimensional dataset consisting of different classes. Our work is based upon the formulation of this problem as group graphical lasso. This paper proposes a novel hybrid covariance thresholding algorithm that can effectively identify zero entries in the precision matrices and split a large joint graphical lasso problem into small subproblems. Our hybrid covariance thresholding method is superior to existing uniform thresholding methods in that our method can split the precision matrix of each in…
The graphical lasso \citep{FHT2007a} is an algorithm for learning the structure in an undirected Gaussian graphical model, using $\ell_1$ regularization to control the number of zeros in the precision matrix ${\BΘ}={\BΣ}^{-1}$ \citep{BGA2008,yuan_lin_07}. The {\texttt R} package \GL\ \citep{FHT2007a} is popular, fast, and allows one to efficiently build a path of models for different values of the tuning parameter. Convergence of \GL\ can be tricky; the converged precision matrix might not be the inverse of the estimated covariance, and occasionally it fails to converge with warm starts. In th…
We consider the task of estimating a Gaussian graphical model in the high-dimensional setting. The graphical lasso, which involves maximizing the Gaussian log likelihood subject to an l1 penalty, is a well-studied approach for this task. We begin by introducing a surprising connection between the graphical lasso and hierarchical clustering: the graphical lasso in effect performs a two-step procedure, in which (1) single linkage hierarchical clustering is performed on the variables in order to identify connected components, and then (2) an l1-penalized log likelihood is maximized on the subset …
We study regularized estimation in high-dimensional longitudinal classification problems, using the lasso and fused lasso regularizers. The constructed coefficient estimates are piecewise constant across the time dimension in the longitudinal problem, with adaptively selected change points (break points). We present an efficient algorithm for computing such estimates, based on proximal gradient descent. We apply our proposed technique to a longitudinal data set on Alzheimer's disease from the Cardiovascular Health Study Cognition Study, and use this data set to motivate and demonstrate several…
We propose a new algorithm for solving the graph-fused lasso (GFL), a method for parameter estimation that operates under the assumption that the signal tends to be locally constant over a predefined graph structure. Our key insight is to decompose the graph into a set of trails which can then each be solved efficiently using techniques for the ordinary (1D) fused lasso. We leverage these trails in a proximal algorithm that alternates between closed form primal updates and fast dual trail updates. The resulting techinque is both faster than previous GFL methods and more flexible in the choice …
This paper introduces the Sylvester graphical lasso (SyGlasso) that captures multiway dependencies present in tensor-valued data. The model is based on the Sylvester equation that defines a generative model. The proposed model complements the tensor graphical lasso (Greenewald et al., 2019) that imposes a Kronecker sum model for the inverse covariance matrix by providing an alternative Kronecker sum model that is generative and interpretable. A nodewise regression approach is adopted for estimating the conditional independence relationships among variables. The statistical convergence of the m…
We propose a convex formulation of the fused lasso signal approximation problem consisting of non-convex penalty functions. The fused lasso signal model aims to estimate a sparse piecewise constant signal from a noisy observation. Originally, the $\ell_1$ norm was used as a sparsity-inducing convex penalty function for the fused lasso signal approximation problem. However, the $\ell_1$ norm underestimates signal values. Non-convex sparsity-inducing penalty functions better estimate signal values. In this paper, we show how to ensure the convexity of the fused lasso signal approximation problem…
Gaussian Graphical Models (GGMs) are popular tools for studying network structures. However, many modern applications such as gene network discovery and social interactions analysis often involve high-dimensional noisy data with outliers or heavier tails than the Gaussian distribution. In this paper, we propose the Trimmed Graphical Lasso for robust estimation of sparse GGMs. Our method guards against outliers by an implicit trimming mechanism akin to the popular Least Trimmed Squares method used for linear regression. We provide a rigorous statistical analysis of our estimator in the high-dim…
In this paper we introduce and study fused lasso nearly-isotonic signal approximation, which is a combination of fused lasso and generalized nearly-isotonic regression. We show how these three estimators relate to each other, derive solution to the general problem, show that it is computationally feasible and provides a trade-off between piecewise monotonicity, sparsity and goodness-of-fit. Also, we derive an unbiased estimator of the degrees of freedom of the approximator.
We consider the problem of predicting an outcome variable using $p$ covariates that are measured on $n$ independent observations, in the setting in which flexible and interpretable fits are desirable. We propose the fused lasso additive model (FLAM), in which each additive function is estimated to be piecewise constant with a small number of adaptively-chosen knots. FLAM is the solution to a convex optimization problem, for which a simple algorithm with guaranteed convergence to the global optimum is provided. FLAM is shown to be consistent in high dimensions, and an unbiased estimator of its …
Many Machine Learning algorithms are formulated as regularized optimization problems, but their performance hinges on a regularization parameter that needs to be calibrated to each application at hand. In this paper, we propose a general calibration scheme for regularized optimization problems and apply it to the graphical lasso, which is a method for Gaussian graphical modeling. The scheme is equipped with theoretical guarantees and motivates a thresholding pipeline that can improve graph recovery. Moreover, requiring at most one line search over the regularization path, the calibration schem…
The fused lasso, also known as total-variation denoising, is a locally-adaptive function estimator over a regular grid of design points. In this paper, we extend the fused lasso to settings in which the points do not occur on a regular grid, leading to an approach for non-parametric regression. This approach, which we call the $K$-nearest neighbors ($K$-NN) fused lasso, involves (i) computing the $K$-NN graph of the design points; and (ii) performing the fused lasso over this $K$-NN graph. We show that this procedure has a number of theoretical advantages over competing approaches: specificall…
Covariance graphical lasso applies a lasso penalty on the elements of the covariance matrix. This method is useful because it not only produces sparse estimation of covariance matrix but also discovers marginal independence structures by generating zeros in the covariance matrix. We propose and explore two new algorithms for solving the covariance graphical lasso problem. Our new algorithms are based on coordinate descent and ECM. We show that these two algorithms are more attractive than the only existing competing algorithm of Bien and Tibshirani (2011) in terms of simplicity, speed and stab…
We consider the sparse inverse covariance regularization problem or graphical lasso with regularization parameter $ρ$. Suppose the co- variance graph formed by thresholding the entries of the sample covariance matrix at $ρ$ is decomposed into connected components. We show that the vertex-partition induced by the thresholded covariance graph is exactly equal to that induced by the estimated concentration graph. This simple rule, when used as a wrapper around existing algorithms, leads to enormous performance gains. For large values of $ρ$, our proposal splits a large graphical lasso problem int…
Rice University（莱斯大学）；Baylor College of Medicine（贝勒医学院）
In neuroscience, researchers seek to uncover the connectivity of neurons from large-scale neural recordings or imaging; often people employ graphical model selection and estimation techniques for this purpose. But, existing technologies can only record from a small subset of neurons leading to a challenging problem of graph selection in the presence of extensive latent variables. Chandrasekaran et al. (2012) proposed a convex program to address this problem that poses challenges from both a computational and statistical perspective. To solve this problem, we propose an incredibly simple soluti…
We consider learning an undirected graphical model from sparse data. While several efficient algorithms have been proposed for graphical lasso (GL), the alternating direction method of multipliers (ADMM) is the main approach taken concerning for joint graphical lasso (JGL). We propose proximal gradient procedures with and without a backtracking option for the JGL. These procedures are first-order and relatively simple, and the subproblems are solved efficiently in closed form. We further show the boundedness for the solution of the JGL problem and the iterations in the algorithms. The numerica…
We show that the two-stage adaptive Lasso procedure (Zou, 2006) is consistent for high-dimensional model selection in linear and Gaussian graphical models. Our conditions for consistency cover more general situations than those accomplished in previous work: we prove that restricted eigenvalue conditions (Bickel et al., 2008) are also sufficient for sparse structure estimation.
We consider efficient implementations of the generalized lasso dual path algorithm of Tibshirani and Taylor (2011). We first describe a generic approach that covers any penalty matrix D and any (full column rank) matrix X of predictor variables. We then describe fast implementations for the special cases of trend filtering problems, fused lasso problems, and sparse fused lasso problems, both with X=I and a general matrix X. These specialized implementations offer a considerable improvement over the generic implementation, both in terms of numerical stability and efficiency of the solution path…
We propose a scalable, efficient and statistically motivated computational framework for Graphical Lasso (Friedman et al., 2007b) - a covariance regularization framework that has received significant attention in the statistics community over the past few years. Existing algorithms have trouble in scaling to dimensions larger than a thousand. Our proposal significantly enhances the state-of-the-art for such moderate sized problems and gracefully scales to larger problems where other algorithms become practically infeasible. This requires a few key new ideas. We operate on the primal problem an…
This thesis studies two problems in modern statistics. First, we study selective inference, or inference for hypothesis that are chosen after looking at the data. The motiving application is inference for regression coefficients selected by the lasso. We present the Condition-on-Selection method that allows for valid selective inference, and study its application to the lasso, and several other selection algorithms. In the second part, we consider the problem of learning the structure of a pairwise graphical model over continuous and discrete variables. We present a new pairwise model for grap…
We consider estimating an unknown signal, both blocky and sparse, which is corrupted by additive noise. We study three interrelated least squares procedures and their asymptotic properties. The first procedure is the fused lasso, put forward by Friedman et al. [Ann. Appl. Statist. 1 (2007) 302--332], which we modify into a different estimator, called the fused adaptive lasso, with better properties. The other two estimators we discuss solve least squares problems on sieves; one constrains the maximal $\ell_1$ norm and the maximal total variation seminorm, and the other restricts the number of …
We consider the problem of learning a structured multi-task regression, where the output consists of multiple responses that are related by a graph and the correlated response variables are dependent on the common inputs in a sparse but synergistic manner. Previous methods such as l1/l2-regularized multi-task regression assume that all of the output variables are equally related to the inputs, although in many real-world problems, outputs are related in a complex manner. In this paper, we propose graph-guided fused lasso (GFlasso) for structured multi-task regression that exploits the graph st…
A generalized gamification framework is introduced as a form of smart infrastructure with potential to improve sustainability and energy efficiency by leveraging humans-in-the-loop strategy. The proposed framework enables a Human-Centric Cyber-Physical System using an interface to allow building managers to interact with occupants. The interface is designed for occupant engagement-integration supporting learning of their preferences over resources in addition to understanding how preferences change as a function of external stimuli such as physical control, time or incentives. Towards intellig…
The graph fused lasso -- which includes as a special case the one-dimensional fused lasso -- is widely used to reconstruct signals that are piecewise constant on a graph, meaning that nodes connected by an edge tend to have identical values. We consider testing for a difference in the means of two connected components estimated using the graph fused lasso. A naive procedure such as a z-test for a difference in means will not control the selective Type I error, since the hypothesis that we are testing is itself a function of the data. In this work, we propose a new test for this task that contr…
L1 -penalized regression methods such as the Lasso (Tibshirani 1996) that achieve both variable selection and shrinkage have been very popular. An extension of this method is the Fused Lasso (Tibshirani and Wang 2007), which allows for the incorporation of external information into the model. In this article, we develop new and fast algorithms for solving the Fused Lasso which are based on coordinate-wise optimization. This class of algorithms has recently been applied very successfully to solve L1 -penalized problems very quickly (Friedman et al. 2007). As a straightforward coordinate-wise pr…
In this work, we consider to improve the model estimation efficiency by aggregating the neighbors' information as well as identify the subgroup membership for each node in the network. A tree-based $l_1$ penalty is proposed to save the computation and communication cost. We design a decentralized generalized alternating direction method of multiplier algorithm for solving the objective function in parallel. The theoretical properties are derived to guarantee both the model consistency and the algorithm convergence. Thorough numerical experiments are also conducted to back up our theory, which …
The fused lasso is an important method for signal processing when the hidden signals are sparse and blocky. It is often used in combination with the squared loss function. However, the squared loss is not suitable for heavy tail error distributions nor is robust against outliers which arise often in practice. The least absolute deviations (LAD) loss provides a robust alternative to the squared loss. In this paper, we study the asymptotic properties of the fused lasso estimator with the LAD loss for signal approximation. We refer to this estimator as the LAD fused lasso signal approximator, or …
The fused lasso, also known as (anisotropic) total variation denoising, is widely used for piecewise constant signal estimation with respect to a given undirected graph. The fused lasso estimate is highly nontrivial to compute when the underlying graph is large and has an arbitrary structure. But for a special graph structure, namely, the chain graph, the fused lasso---or simply, 1d fused lasso---can be computed in linear time. In this paper, we establish a surprising connection between the total variation of a generic signal defined over an arbitrary graph, and the total variation of this sig…
University of California, Santa Barbara（加州大学圣巴巴拉分校）
Building on a recent framework for distributionally robust optimization, we consider estimation of the inverse covariance matrix for multivariate data. We provide a novel notion of a Wasserstein ambiguity set specifically tailored to this estimation problem, leading to a tractable class of regularized estimators. Special cases include penalized likelihood estimators for Gaussian data, specifically the graphical lasso estimator. As a consequence of this formulation, the radius of the Wasserstein ambiguity set is directly related to the regularization parameter in the estimation problem. Using t…
Energy game-theoretic frameworks have emerged to be a successful strategy to encourage energy efficient behavior in large scale by leveraging human-in-the-loop strategy. A number of such frameworks have been introduced over the years which formulate the energy saving process as a competitive game with appropriate incentives for energy efficient players. However, prior works involve an incentive design mechanism which is dependent on knowledge of utility functions for all the players in the game, which is hard to compute especially when the number of players is high, common in energy game-theor…
In this paper, we propose a majorization-minimization (MM) algorithm for high-dimensional fused lasso regression (FLR) suitable for parallelization using graphics processing units (GPUs). The MM algorithm is stable and flexible as it can solve the FLR problems with various types of design matrices and penalty structures within a few tens of iterations. We also show that the convergence of the proposed algorithm is guaranteed. We conduct numerical studies to compare our algorithm with other existing algorithms, demonstrating that the proposed MM algorithm is competitive in many settings includi…
This work proposes a new algorithm for solving the graph-fused lasso (GFL), a method for parameter estimation that operates under the assumption that the signal tends to be locally constant over a predefined graph structure. The proposed method applies the alternating direction method of multipliers (ADMM) algorithm and is based on the decomposition of the objective function into two components. While ADMM has been widely used in this problem, existing works such as network lasso decompose the objective function into the loss function component and the total variation penalty component. In com…
A vital problem in solving classification or regression problem is to apply feature engineering and variable selection on data before fed into models.One of a most popular feature engineering method is to discretisize continous variable with some cutting points,which is refered to as bining processing.Good cutting points are important for improving model's ability, because wonderful bining may ignore some noisy variance in continous variable range and keep useful leveled information with good ordered encodings.However, to our best knowledge a majority of cutting point selection is done via res…
The main result of this article is that we obtain an elementwise error bound for the Fused Lasso estimator for any general convex loss function $ρ$. We then focus on the special cases when either $ρ$ is the square loss function (for mean regression) or is the quantile loss function (for quantile regression) for which we derive new pointwise error bounds. Even though error bounds for the usual Fused Lasso estimator and its quantile version have been studied before; our bound appears to be new. This is because all previous works bound a global loss function like the sum of squared error, or a su…
This article introduces the sparse group fused lasso (SGFL) as a statistical framework for segmenting sparse regression models with multivariate time series. To compute solutions of the SGFL, a nonsmooth and nonseparable convex program, we develop a hybrid optimization method that is fast, requires no tuning parameter selection, and is guaranteed to converge to a global minimizer. In numerical experiments, the hybrid method compares favorably to state-of-the-art techniques with respect to computation time and numerical accuracy; benefits are particularly substantial in high dimension. The meth…
Texas A&M University（得克萨斯农工大学）；University of Washington（华盛顿大学）
We present safe active incremental feature selection~(SAIF) to scale up the computation of LASSO solutions. SAIF does not require a solution from a heavier penalty parameter as in sequential screening or updating the full model for each iteration as in dynamic screening. Different from these existing screening methods, SAIF starts from a small number of features and incrementally recruits active features and updates the significantly reduced model. Hence, it is much more computationally efficient and scalable with the number of features. More critically, SAIF has the safe guarantee as it has t…
Lasso is a popular and efficient approach to simultaneous estimation and variable selection in high-dimensional regression models. In this paper, a robust LAD-lasso method for multiple outcomes is presented that addresses the challenges of non-normal outcome distributions and outlying observations. Measured covariate data from space or time, or spectral bands or genomic positions often have natural correlation structure arising from measuring distance between the covariates. The proposed multi-outcome approach includes handling of such covariate blocks by a group fusion penalty, which encourag…
