### Structured sparsity through convex optimization

Sparse estimation methods are aimed at using or obtaining parsimonious representations of data or models. While naturally cast as a combinatorial optimization problem, variable or feature selection admits a convex relaxation through the regularization by the $\ell_1$-norm. In this paper, we consider situations where we are not only interested in sparsity, but where some structural prior knowledge is available as well. We show that the $\ell_1$-norm can then be extended to structured norms built on either disjoint or overlapping groups of variables, leading to a flexible framework that can deal with various structures. We present applications to unsupervised learning, for structured sparse principal component analysis and hierarchical dictionary learning, and to supervised learning in the context of non-linear variable selection.

### Group and Graph Joint Sparsity for Linked Data Classification

Various sparse regularizers have been applied to machine learning problems, among which structured sparsity has been proposed for a better adaption to structured data. In this paper, motivated by effectively classifying linked data (e.g. Web pages, tweets, articles with references, and biological network data) where a group structure exists over the whole dataset and links exist between specific samples, we propose a joint sparse representation model that combines group sparsity and graph sparsity, to select a small number of connected components from the graph of linked samples, meanwhile promoting the sparsity of edges that link samples from different groups in each connected component. Consequently, linked samples are selected from a few sparsely-connected groups. Both theoretical analysis and experimental results on four benchmark datasets show that the joint sparsity model outperforms traditional group sparsity model and graph sparsity model, as well as the latest group-graph sparsity model.

### Supervised Feature Selection in Graphs with Path Coding Penalties and Network Flows

We consider supervised learning problems where the features are embedded in a graph, such as gene expressions in a gene network. In this context, it is of much interest to automatically select a subgraph with few connected components; by exploiting prior knowledge, one can indeed improve the prediction performance or obtain results that are easier to interpret. Regularization or penalty functions for selecting features in graphs have recently been proposed, but they raise new algorithmic challenges. For example, they typically require solving a combinatorially hard selection problem among all connected subgraphs. In this paper, we propose computationally feasible strategies to select a sparse and well-connected subset of features sitting on a directed acyclic graph (DAG). We introduce structured sparsity penalties over paths on a DAG called "path coding" penalties. Unlike existing regularization functions that model long-range interactions between features in a graph, path coding penalties are tractable. The penalties and their proximal operators involve path selection problems, which we efficiently solve by leveraging network flow optimization. We experimentally show on synthetic, image, and genomic data that our approach is scalable and leads to more connected subgraphs than other regularization functions for graphs.

### Convex and Network Flow Optimization for Structured Sparsity

We consider a class of learning problems regularized by a structured sparsity-inducing norm defined as the sum of l_2- or l_infinity-norms over groups of variables. Whereas much effort has been put in developing fast optimization techniques when the groups are disjoint or embedded in a hierarchy, we address here the case of general overlapping groups. To this end, we present two different strategies: On the one hand, we show that the proximal operator associated with a sum of l_infinity-norms can be computed exactly in polynomial time by solving a quadratic min-cost flow problem, allowing the use of accelerated proximal gradient methods. On the other hand, we use proximal splitting techniques, and address an equivalent formulation with non-overlapping groups, but in higher dimension and with additional constraints. We propose efficient and scalable algorithms exploiting these two strategies, which are significantly faster than alternative approaches. We illustrate these methods with several problems such as CUR matrix factorization, multi-task learning of tree-structured dictionaries, background subtraction in video sequences, image denoising with wavelets, and topographic dictionary learning of natural image patches.

### A Communication-Efficient Parallel Method for Group-Lasso

Group-Lasso (gLasso) identifies important explanatory factors in predicting the response variable by considering the grouping structure over input variables. However, most existing algorithms for gLasso are not scalable to deal with large-scale datasets, which are becoming a norm in many applications. In this paper, we present a divide-and-conquer based parallel algorithm (DC-gLasso) to scale up gLasso in the tasks of regression with grouping structures. DC-gLasso only needs two iterations to collect and aggregate the local estimates on subsets of the data, and is provably correct to recover the true model under certain conditions. We further extend it to deal with overlappings between groups. Empirical results on a wide range of synthetic and real-world datasets show that DC-gLasso can significantly improve the time efficiency without sacrificing regression accuracy.