In many real-world problems, complex dependencies are present both among samples and among features. The Kronecker sum or the Cartesian product of two graphs, each modeling dependencies across features and across samples, has been used as an inverse covariance matrix for a matrix-variate Gaussian distribution, as an alternative to a Kronecker-product inverse covariance matrix, due to its more intuitive sparse structure. However, the existing methods for sparse Kronecker-sum inverse covariance estimation are limited in that they do not scale to more than a few hundred features and samples and that the unidentifiable parameters pose challenges in estimation. In this paper, we introduce EiGLasso, a highly scalable method for sparse Kronecker-sum inverse covariance estimation, based on Newton's method combined with eigendecomposition of the two graphs for exploiting the structure of Kronecker sum. EiGLasso further reduces computation time by approximating the Hessian based on the eigendecomposition of the sample and feature graphs. EiGLasso achieves quadratic convergence with the exact Hessian and linear convergence with the approximate Hessian. We describe a simple new approach to estimating the unidentifiable parameters that generalizes the existing methods. On simulated and real-world data, we demonstrate that EiGLasso achieves two to three orders-of-magnitude speed-up compared to the existing methods.

This paper addresses the problem of scalable optimization for L1-regularized conditional Gaussian graphical models. Conditional Gaussian graphical models generalize the well-known Gaussian graphical models to conditional distributions to model the output network influenced by conditioning input variables. While highly scalable optimization methods exist for sparse Gaussian graphical model estimation, state-of-the-art methods for conditional Gaussian graphical models are not efficient enough and more importantly, fail due to memory constraints for very large problems. In this paper, we propose a new optimization procedure based on a Newton method that efficiently iterates over two sub-problems, leading to drastic improvement in computation time compared to the previous methods. We then extend our method to scale to large problems under memory constraints, using block coordinate descent to limit memory usage while achieving fast convergence. Using synthetic and genomic data, we show that our methods can solve one million dimensional problems to high accuracy in a little over a day on a single machine.

In this paper, we address the problem of learning the structure of Gaussian chain graph models in a high-dimensional space. Chain graph models are generalizations of undirected and directed graphical models that contain a mixed set of directed and undirected edges. While the problem of sparse structure learning has been studied extensively for Gaussian graphical models and more recently for conditional Gaussian graphical models (CGGMs), there has been little previous work on the structure recovery of Gaussian chain graph models. We consider linear regression models and a re-parameterization of the linear regression models using CGGMs as building blocks of chain graph models. We argue that when the goal is to recover model structures, there are many advantages of using CGGMs as chain component models over linear regression models, including convexity of the optimization problem, computational efficiency, recovery of structured sparsity, and ability to leverage the model structure for semi-supervised learning. We demonstrate our approach on simulated and genomic datasets.

We address the problem of learning a sparse Bayesian network structure for continuous variables in a high-dimensional space. The constraint that the estimated Bayesian network structure must be a directed acyclic graph (DAG) makes the problem challenging because of the huge search space of network structures. Most previous methods were based on a two-stage approach that prunes the search space in the first stage and then searches for a network structure that satisfies the DAG constraint in the second stage. Although this approach is effective in a low-dimensional setting, it is difficult to ensure that the correct network structure is not pruned in the first stage in a high-dimensional setting. In this paper, we propose a single-stage method, called A* lasso, that recovers the optimal sparse Bayesian network structure by solving a single optimization problem with A* search algorithm that uses lasso in its scoring system. Our approach substantially improves the computational efficiency of the well-known exact methods based on dynamic programming. We also present a heuristic scheme that further improves the efficiency of A* lasso without significantly compromising the quality of solutions and demonstrate this on benchmark Bayesian networks and real data.

We consider the problem of estimating a sparse multi-response regression function, with an application to expression quantitative trait locus (eQTL) mapping, where the goal is to discover genetic variations that influence gene-expression levels. In particular, we investigate a shrinkage technique capable of capturing a given hierarchical structure over the responses, such as a hierarchical clustering tree with leaf nodes for responses and internal nodes for clusters of related responses at multiple granularity, and we seek to leverage this structure to recover covariates relevant to each hierarchically-defined cluster of responses. We propose a tree-guided group lasso, or tree lasso, for estimating such structured sparsity under multi-response regression by employing a novel penalty function constructed from the tree. We describe a systematic weighting scheme for the overlapping groups in the tree-penalty such that each regression coefficient is penalized in a balanced manner despite the inhomogeneous multiplicity of group memberships of the regression coefficients due to overlaps among groups. For efficient optimization, we employ a smoothing proximal gradient method that was originally developed for a general class of structured-sparsity-inducing penalties. Using simulated and yeast data sets, we demonstrate that our method shows a superior performance in terms of both prediction errors and recovery of true sparsity patterns, compared to other methods for learning a multivariate-response regression.

We study the problem of learning high dimensional regression models regularized by a structured-sparsity-inducing penalty that encodes prior structural information on either input or output sides. We consider two widely adopted types of such penalties as our motivating examples: 1) overlapping group lasso penalty, based on the l1/l2 mixed-norm penalty, and 2) graph-guided fusion penalty. For both types of penalties, due to their non-separability, developing an efficient optimization method has remained a challenging problem. In this paper, we propose a general optimization approach, called smoothing proximal gradient method, which can solve the structured sparse regression problems with a smooth convex loss and a wide spectrum of structured-sparsity-inducing penalties. Our approach is based on a general smoothing technique of Nesterov. It achieves a convergence rate faster than the standard first-order method, subgradient method, and is much more scalable than the most widely used interior-point method. Numerical results are reported to demonstrate the efficiency and scalability of the proposed method.

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 structure over the output variables. We introduce a novel penalty function based on fusion penalty to encourage highly correlated outputs to share a common set of relevant inputs. In addition, we propose a simple yet efficient proximal-gradient method for optimizing GFlasso that can also be applied to any optimization problems with a convex smooth loss and the general class of fusion penalty defined on arbitrary graph structures. By exploiting the structure of the non-smooth ''fusion penalty'', our method achieves a faster convergence rate than the standard first-order method, sub-gradient method, and is significantly more scalable than the widely adopted second-order cone-programming and quadratic-programming formulations. In addition, we provide an analysis of the consistency property of the GFlasso model. Experimental results not only demonstrate the superiority of GFlasso over the standard lasso but also show the efficiency and scalability of our proximal-gradient method.

Multitask learning addressed the problem of learning related tasks whose information can be shared each other. Traditional problem usually deal with homogeneous tasks such as regression, classification individually. In this paper we consider the problem learning multiple related tasks where tasks consist of both continuous and discrete outputs from a common set of input variables that lie in a high-dimensional space. All of the tasks are related in the sense that they share the same set of relevant input variables, but the amount of influence of each input on different outputs may vary. We formulate this problem as a combination of linear regression and logistic regression and model the joint sparsity as L1/Linf and L1/L2-norm of the model parameters. Among several possible applications, our approach addresses an important open problem in genetic association mapping, where we are interested in discovering genetic markers that influence multiple correlated traits jointly. In our experiments, we demonstrate our method in the scenario of association mapping, using simulated and asthma data, and show that the algorithm can effectively recover the relevant inputs with respect to all of the tasks.

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 model over the genotypes and traits, so that the genetic markers that jointly influence subgroups of highly correlated traits can be detected with high sensitivity and specificity. While most of the traditional methods examined each phenotype independently and combined the results afterwards, our approach analyzes all of the traits jointly in a single statistical method, and borrow information across correlated phenotypes to discover the genetic markers that perturbe a subset of correlated triats jointly rather than a single trait. Using simulated datasets based on the HapMap consortium data and an asthma dataset, we compare the performance of our method with the single-marker analysis, and other sparse regression methods such as the ridge regression and the lasso that do not use any structural information in the traits. Our results show that there is a significant advantage in detecting the true causal SNPs when we incorporate the correlation pattern in traits using our proposed methods.

In addition, each group is allowed to have its own component parameters coming from a prior described by a template mixture model.