We study the multivariate square-root lasso, a method for fitting the multivariate response linear regression model with dependent errors. This estimator minimizes the nuclear norm of the residual matrix plus a convex penalty. Unlike existing methods that require explicit estimates of the error precision (inverse covariance) matrix, the multivariate square-root lasso implicitly accounts for error dependence and is the solution to a convex optimization problem. We establish error bounds which reveal that like the univariate square-root lasso, the multivariate square-root lasso is pivotal with respect to the unknown error covariance matrix. In addition, we propose a variation of the alternating direction method of multipliers algorithm to compute the estimator and discuss an accelerated first order algorithm that can be applied in certain cases. In both simulation studies and a genomic data application, we show that the multivariate square-root lasso can outperform more computationally intensive methods that require explicit estimation of the error precision matrix.
We propose the use of a robust covariance estimator based on multivariate Winsorization in the context of the Tarr-Muller-Weber framework for sparse estimation of the precision matrix of a Gaussian graphical model. Likewise Croux-Ollerer's precision matrix estimator, our proposed estimator attains the maximum finite sample breakdown point of 0.5 under cellwise contamination. We conduct an extensive Monte Carlo simulation study to assess the performance of ours and the currently existing proposals. We find that ours has a competitive behavior, regarding the the estimation of the precision matri…
We study a group lasso estimator for the multivariate linear regression model that accounts for correlated error terms. A block coordinate descent algorithm is used to compute this estimator. We perform a simulation study with categorical data and multivariate time series data, typical settings with a natural grouping among the predictor variables. Our simulation studies show the good performance of the proposed group lasso estimator compared to alternative estimators. We illustrate the method on a time series data set of gene expressions.
We propose a Bayesian procedure for simultaneous variable and covariance selection using continuous spike-and-slab priors in multivariate linear regression models where q possibly correlated responses are regressed onto p predictors. Rather than relying on a stochastic search through the high-dimensional model space, we develop an ECM algorithm similar to the EMVS procedure of Rockova & George (2014) targeting modal estimates of the matrix of regression coefficients and residual precision matrix. Varying the scale of the continuous spike densities facilitates dynamic posterior exploration and …
We describe an improved statistical downscaling method for Earth science applications using multivariate Basis Graphical Lasso (BGL). We demonstrate our method using a case study of sea surface temperature (SST) projections from CMIP6 Earth system models, which has direct applications for studies of multi-decadal projections of coral reef bleaching. We find that the BGL downscaling method is computationally tractable for large data sets, and that mean squared predictive error is roughly 8% lower than the current state-of-the-art interpolation-based statistical downscaling method. Finally, unli…
We consider a multivariate finite mixture of Gaussian regression models for high-dimensional data, where the number of covariates and the size of the response may be much larger than the sample size. We provide an $\ell_1$-oracle inequality satisfied by the Lasso estimator according to the Kullback-Leibler loss. This result is an extension of the $\ell_1$-oracle inequality established by Meynet in \cite{Meynet} in the multivariate case. We focus on the Lasso for its $\ell_1$-regularization properties rather than for the variable selection procedure, as it was done in Städler in \cite{Stadler}.
Radiogenomics is an emerging field in cancer research that combines medical imaging data with genomic data to predict patients clinical outcomes. In this paper, we propose a multivariate sparse group lasso joint model to integrate imaging and genomic data for building prediction models. Specifically, we jointly consider two models, one regresses imaging features on genomic features, and the other regresses patients clinical outcomes on genomic features. The regularization penalties through sparse group lasso allow incorporation of intrinsic group information, e.g. biological pathway and imagin…
There has been much recent work on inference after model selection when the noise level is known, however, $σ$ is rarely known in practice and its estimation is difficult in high-dimensional settings. In this work we propose using the square-root LASSO (also known as the scaled LASSO) to perform selective inference for the coefficients and the noise level simultaneously. The square-root LASSO has the property that choosing a reasonable tuning parameter is scale-free, namely it does not depend on the noise level in the data. We provide valid p-values and confidence intervals for the coefficient…
In this paper, we investigate a multivariate multi-response (MVMR) linear regression problem, which contains multiple linear regression models with differently distributed design matrices, and different regression and output vectors. The goal is to recover the support union of all regression vectors using $l_1/l_2$-regularized Lasso. We characterize sufficient and necessary conditions on sample complexity \emph{as a sharp threshold} to guarantee successful recovery of the support union. Namely, if the sample size is above the threshold, then $l_1/l_2$-regularized Lasso correctly recovers the s…
When it comes to datasets with a tremendous amount of predictors, variable reduction techniques such as PCA or FA are often used. In this paper, the elastic net, which lies in between the LASSO method and ridge regression, is used as a variable reduction technique followed by further analysis with multivariate regression. Specifically, a messy only dataset is used to show how it can be 'tidied' up and broken down into sensible subsets using the aforementioned method.
We propose a pivotal method for estimating high-dimensional sparse linear regression models, where the overall number of regressors $p$ is large, possibly much larger than $n$, but only $s$ regressors are significant. The method is a modification of the lasso, called the square-root lasso. The method is pivotal in that it neither relies on the knowledge of the standard deviation $σ$ or nor does it need to pre-estimate $σ$. Moreover, the method does not rely on normality or sub-Gaussianity of noise. It achieves near-oracle performance, attaining the convergence rate $σ\{(s/n)\log p\}^{1/2}$ in …
Zero-inflated explanatory variables are common in fields such as ecology and finance. In this paper we address the problem of having excess of zero values in some explanatory variables which are subject to multioutcome lasso-regularized variable selection. Briefly, the problem results from the failure of the lasso-type of shrinkage methods to recognize any difference between zero value occurring either in the regression coefficient or in the corresponding value of the explanatory variable. This kind of confounding will obviously increase number of false positives - all non-zero regression coef…
Due to its low computational cost, Lasso is an attractive regularization method for high-dimensional statistical settings. In this paper, we consider multivariate counting processes depending on an unknown function parameter to be estimated by linear combinations of a fixed dictionary. To select coefficients, we propose an adaptive $\ell_1$-penalization methodology, where data-driven weights of the penalty are derived from new Bernstein type inequalities for martingales. Oracle inequalities are established under assumptions on the Gram matrix of the dictionary. Nonasymptotic probabilistic resu…
Université Paris Saclay（巴黎萨克雷大学）；Inria（法国国家信息与自动化研究所）；CEA（法国原子能和替代能源委员会）；Univ. Montpellier（蒙彼利埃大学）；CNRS（法国国家科学研究中心）
Sparsity promoting norms are frequently used in high dimensional regression. A limitation of such Lasso-type estimators is that the optimal regularization parameter depends on the unknown noise level. Estimators such as the concomitant Lasso address this dependence by jointly estimating the noise level and the regression coefficients. Additionally, in many applications, the data is obtained by averaging multiple measurements: this reduces the noise variance, but it dramatically reduces sample sizes and prevents refined noise modeling. In this work, we propose a concomitant estimator that can c…
Extending the results of Bellec, Lecué and Tsybakov to the setting of sparse high-dimensional linear regression with unknown variance, we show that two estimators, the Square-Root Lasso and the Square-Root Slope can achieve the optimal minimax prediction rate, which is $(s/n) \log (p/s)$, up to some constant, under some mild conditions on the design matrix. Here, $n$ is the sample size, $p$ is the dimension and $s$ is the sparsity parameter. We also prove optimality for the estimation error in the $l_q$-norm, with $q \in [1,2]$ for the Square-Root Lasso, and in the $l_2$ and sorted $l_1$ norms…
We propose a self-tuning $\sqrt{\mathrm {Lasso}}$ method that simultaneously resolves three important practical problems in high-dimensional regression analysis, namely it handles the unknown scale, heteroscedasticity and (drastic) non-Gaussianity of the noise. In addition, our analysis allows for badly behaved designs, for example, perfectly collinear regressors, and generates sharp bounds even in extreme cases, such as the infinite variance case and the noiseless case, in contrast to Lasso. We establish various nonasymptotic bounds for $\sqrt{\mathrm {Lasso}}$ including prediction norm rate …
We introduce and study the Group Square-Root Lasso (GSRL) method for estimation in high dimensional sparse regression models with group structure. The new estimator minimizes the square root of the residual sum of squares plus a penalty term proportional to the sum of the Euclidean norms of groups of the regression parameter vector. The net advantage of the method over the existing Group Lasso (GL)-type procedures consists in the form of the proportionality factor used in the penalty term, which for GSRL is independent of the variance of the error terms. This is of crucial importance in models…
In multivariate regression, a $K$-dimensional response vector is regressed upon a common set of $p$ covariates, with a matrix $B^*\in\mathbb{R}^{p\times K}$ of regression coefficients. We study the behavior of the multivariate group Lasso, in which block regularization based on the $\ell_1/\ell_2$ norm is used for support union recovery, or recovery of the set of $s$ rows for which $B^*$ is nonzero. Under high-dimensional scaling, we show that the multivariate group Lasso exhibits a threshold for the recovery of the exact row pattern with high probability over the random design and noise that …
This paper introduces a multi-way tensor generalization of the Bigraphical Lasso (BiGLasso), which uses a two-way sparse Kronecker-sum multivariate-normal model for the precision matrix to parsimoniously model conditional dependence relationships of matrix-variate data based on the Cartesian product of graphs. We call this generalization the {\bf Te}nsor g{\bf ra}phical Lasso (TeraLasso). We demonstrate using theory and examples that the TeraLasso model can be accurately and scalably estimated from very limited data samples of high dimensional variables with multiway coordinates such as space,…
Southwest Jiaotong University（西南交通大学）；Zhejiang University City College（浙江大学城市学院）
Square-root Lasso problems are proven robust regression problems. Furthermore, square-root regression problems with structured sparsity also plays an important role in statistics and machine learning. In this paper, we focus on the numerical computation of large-scale linearly constrained sparse group square-root Lasso problems. In order to overcome the difficulty that there are two nonsmooth terms in the objective function, we propose a dual semismooth Newton (SSN) based augmented Lagrangian method (ALM) for it. That is, we apply the ALM to the dual problem with the subproblem solved by the S…
We propose a method for detecting significant interactions in very large multivariate spatial point patterns. This methodology develops high dimensional data understanding in the point process setting. The method is based on modelling the patterns using a flexible Gibbs point process model to directly characterise point-to-point interactions at different spatial scales. By using the Gibbs framework significant interactions can also be captured at small scales. Subsequently, the Gibbs point process is fitted using a pseudo-likelihood approximation, and we select significant interactions automat…
In many applied fields, such as genomics, different types of data are collected on the same system, and it is not uncommon that some of these datasets are subject to censoring as a result of the measurement technologies used, such as data generated by polymerase chain reactions and flow cytometer. When the overall objective is that of network inference, at possibly different levels of a system, information coming from different sources and/or different steps of the analysis can be integrated into one model with the use of conditional graphical models. In this paper, we develop a doubly penaliz…
There have been many attempts to identify high-dimensional network features via multivariate approaches. Specifically, when the number of voxels or nodes, denoted as p, are substantially larger than the number of images, denoted as n, it produces an under-determined model with infinitely many possible solutions. The small-n large-p problem is often remedied by regularizing the under-determined system with additional sparse penalties. Popular sparse network models include sparse correlations, LASSO, sparse canonical correlations and graphical-LASSO. These popular sparse models require optimizin…
Lasso is a seminal contribution to high-dimensional statistics, but it hinges on a tuning parameter that is difficult to calibrate in practice. A partial remedy for this problem is Square-Root Lasso, because it inherently calibrates to the noise variance. However, Square-Root Lasso still requires the calibration of a tuning parameter to all other aspects of the model. In this study, we introduce TREX, an alternative to Lasso with an inherent calibration to all aspects of the model. This adaptation to the entire model renders TREX an estimator that does not require any calibration of tuning par…
The precision matrix that encodes conditional linear dependency relations among a set of variables forms an important object of interest in multivariate analysis. Sparse estimation procedures for precision matrices such as the graphical lasso (Glasso) gained popularity as they facilitate interpretability, thereby separating pairs of variables that are conditionally dependent from those that are independent (given all other variables). Glasso lacks, however, robustness to outliers. To overcome this problem, one typically applies a robust plug-in procedure where the Glasso is computed from a rob…
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…
Two extensions of generalized linear models are considered. In the first one, response variables depend on multiple linear combinations of covariates. In the second one, only response variables are observed while the linear covariates are missing. We derive stochastic Lipschitz continuity results for the loss functions involved in the regression problems and apply them to get bounds on estimation error for Lasso. Multivariate comparison results on Rademacher complexity are obtained as tools to establish the stochastic Lipschitz continuity results.
Volatility forecasts are key inputs in financial analysis. While lasso based forecasts have shown to perform well in many applications, their use to obtain volatility forecasts has not yet received much attention in the literature. Lasso estimators produce parsimonious forecast models. Our forecast combination approach hedges against the risk of selecting a wrong degree of model parsimony. Apart from the standard lasso, we consider several lasso extensions that account for the dynamic nature of the forecast model. We apply forecast combined lasso estimators in a comprehensive forecasting exerc…
We study a high-dimensional regression model. Aim is to construct a confidence set for a given group of regression coefficients, treating all other regression coefficients as nuisance parameters. We apply a one-step procedure with the square-root Lasso as initial estimator and a multivariate square-root Lasso for constructing a surrogate Fisher information matrix. The multivariate square-root Lasso is based on nuclear norm loss with $\ell_1$-penalty. We show that this procedure leads to an asymptotically $χ^2$-distributed pivot, with a remainder term depending only on the $\ell_1$-error of the…
Large-scale {\it in vitro} drug sensitivity screens are an important tool in personalized oncology to predict the effectiveness of potential cancer drugs. The prediction of the sensitivity of cancer cell lines to a panel of drugs is a multivariate regression problem with high-dimensional heterogeneous multi-omics data as input data and with potentially strong correlations between the outcome variables which represent the sensitivity to the different drugs. We propose a joint penalized regression approach with structured penalty terms which allow us to utilize the correlation structure between …
In this paper we study the properties of the Lasso estimator of the drift component in the diffusion setting. More specifically, we consider a multivariate parametric diffusion model $X$ observed continuously over the interval $[0,T]$ and investigate drift estimation under sparsity constraints. We allow the dimensions of the model and the parameter space to be large. We obtain an oracle inequality for the Lasso estimator and derive an error bound for the $L^2$-distance using concentration inequalities for linear functionals of diffusion processes. The probabilistic part is based upon elements …
Shrinkage algorithms are of great importance in almost every area of statistics due to the increasing impact of big data. Especially time series analysis benefits from efficient and rapid estimation techniques such as the lasso. However, currently lasso type estimators for autoregressive time series models still focus on models with homoscedastic residuals. Therefore, an iteratively reweighted adaptive lasso algorithm for the estimation of time series models under conditional heteroscedasticity is presented in a high-dimensional setting. The asymptotic behaviour of the resulting estimator is a…
In this paper, we consider the nonparametric regression problem with multivariate predictors. We provide a characterization of the degrees of freedom and divergence for estimators of the unknown regression function, which are obtained as outputs of linearly constrained quadratic optimization procedures, namely, minimizers of the least squares criterion with linear constraints and/or quadratic penalties. As special cases of our results, we derive explicit expressions for the degrees of freedom in many nonparametric regression problems, e.g., bounded isotonic regression, multivariate (penalized)…
In high dimension, it is customary to consider Lasso-type estimators to enforce sparsity. For standard Lasso theory to hold, the regularization parameter should be proportional to the noise level, yet the latter is generally unknown in practice. A possible remedy is to consider estimators, such as the Concomitant/Scaled Lasso, which jointly optimize over the regression coefficients as well as over the noise level, making the choice of the regularization independent of the noise level. However, when data from different sources are pooled to increase sample size, or when dealing with multimodal …
The graphical lasso (glasso) is a widely-used fast algorithm for estimating sparse inverse covariance matrices. The glasso solves an L1 penalized maximum likelihood problem and is available as an R library on CRAN. The output from the glasso, a regularized covariance matrix estimate a sparse inverse covariance matrix estimate, not only identify a graphical model but can also serve as intermediate inputs into multivariate procedures such as PCA, LDA, MANOVA, and others. The glasso indeed produces a covariance matrix estimate which solves the L1 penalized optimization problem in a dual sense; ho…
This work aims to give non-asymptotic results for estimating the first principal component of a multivariate random process. We first define the covariance function and the covariance operator in the multivariate case. We then define a projection operator. This operator can be seen as a reconstruction step from the raw data in the functional data analysis context. Next, we show that the eigenelements can be expressed as the solution to an optimization problem, and we introduce the LASSO variant of this optimization problem and the associated plugin estimator. Finally, we assess the estimator's…
We consider the least-squares regression problem with unknown noise variance, where the observed data points are allowed to be corrupted by outliers. Building on the median-of-means (MOM) method introduced by Lecue and Lerasle Ann.Statist.48(2):906-931(April 2020) in the case of known noise variance, we propose a general MOM approach for simultaneous inference of both the regression function and the noise variance, requiring only an upper bound on the noise level. Interestingly, this generalization requires care due to regularity issues that are intrinsic to the underlying convex-concave optim…
The time-evolving precision matrix of a piecewise-constant Gaussian graphical model encodes the dynamic conditional dependency structure of a multivariate time-series. Traditionally, graphical models are estimated under the assumption that data is drawn identically from a generating distribution. Introducing sparsity and sparse-difference inducing priors we relax these assumptions and propose a novel regularized M-estimator to jointly estimate both the graph and changepoint structure. The resulting estimator possesses the ability to therefore favor sparse dependency structures and/or smoothly …
Many complex disease syndromes such as asthma consist of a large number of highly related, rather than independent, clinical phenotypes, raising a new technical challenge in identifying genetic variations associated simultaneously with correlated traits. In this study, we propose a new statistical framework called graph-guided fused lasso (GFlasso) to address this issue in a principled way. Our approach explicitly represents the dependency structure among the quantitative traits as a network, and leverages this trait network to encode structured regularizations in a multivariate regression mod…
In this article we study variable selection problem using LASSO with new improvisations. LASSO uses $\ell_{1}$ penalty, it shrinks most of the coefficients to zero when number of explanatory variables $(p)$ are much larger the number of observations $(N)$. Novelty of the approach developed in this article blends basic ideas behind resampling and LASSO together which provides a significant variable reduction and improved prediction accuracy in terms of mean squared error in the test sample. Different weighting schemes have been explored using Bootstrapped LASSO, the basic methodology developed …
In this paper, we propose a novel variable selection approach in the framework of multivariate linear models taking into account the dependence that may exist between the responses. It consists in estimating beforehand the covariance matrix of the responses and to plug this estimator in a Lasso criterion, in order to obtain a sparse estimator of the coefficient matrix. The properties of our approach are investigated both from a theoretical and a numerical point of view. More precisely, we give general conditions that the estimators of the covariance matrix and its inverse have to satisfy in or…
Exploring the genetic basis of heritable traits remains one of the central challenges in biomedical research. In simple cases, single polymorphic loci explain a significant fraction of the phenotype variability. However, many traits of interest appear to be subject to multifactorial control by groups of genetic loci instead. Accurate detection of such multivariate associations is nontrivial and often hindered by limited power. At the same time, confounding influences such as population structure cause spurious association signals that result in false positive findings if they are not accounted…
