Learning the Structure for Structured Sparsity

Abstract—Structured sparsity has recently emerged in statistics, machine learning and signal processing as a promising paradigm for learning in high-dimensional settings. All existing methods for learning under the assumption of structured sparsity rely on prior knowledge on how to weight (or how to penalize) individual subsets of variables during the subset selection process, which is not avail-able in general. Inferring group weights from data is a key open research problem in structured sparsity. In this paper, we pro-pose a Bayesian approach to the problem of group weight learning. We model the group weights as hyperparameters of heavy-tailed priors on groups of variables and derive an approximate inference scheme to infer these hyperparameters. We empirically show that we are able to recover the model hyperparameters when the data are generated from the model, and we demonstrate the utility of learning weights in synthetic and real denoising problems. Index Terms — Structured sparsity, probabilistic modeling, Bayesian statistics, super-Gaussian prior, Gaussian scale mixture