Model Selection with Piecewise Regular Gauges

Abstract

Regularization plays a pivotal role when facing the challenge of solving ill-posed inverse problems, where the number of observations is smaller than the ambient dimension of the object to be estimated. A line of recent work has studied regularization models with various types of low-dimensional structures. In such settings, the general approach is to solve a regularized optimization problem, which combines a data fidelity term and some regularization penalty that promotes the assumed low-dimensional/simple structure. This paper provides a general framework to capture this low-dimensional structure through what we coin piecewise regular gauges. These are convex, non-negative, closed, bounded and positively homogenous functions that will promote objects living on low-dimensional subspaces. This class of regularizers encompasses many popular examples such as the L^1 norm, L^1-L^2 norm (group sparsity), nuclear norm, as well as several others including the L^inf norm. We will show that the set of piecewise regular gauges is closed under addition and pre-composition by a linear operator, which allows to cover mixed regularization (e.g. sparse+low-rank), and the so-called analysis-type priors (e.g. total variation, fused Lasso, trace Lasso, bounded polyhedral gauges). Our main result presents a unified sharp analysis of exact and robust recovery of the low-dimensional subspace model associated to the object to recover from partial measurements. This analysis is illustrated on a number of special and previously studied cases.no