Model Selection with Low Complexity Priors

Samuel Vaiter 1 Mohammad Golbabaee 1 Jalal M. Fadili 2 Gabriel Peyré 1
2 Equipe Image - Laboratoire GREYC - UMR6072
GREYC - Groupe de Recherche en Informatique, Image, Automatique et Instrumentation de Caen
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 partly smooth functions relative to a linear manifold. These are convex, non-negative, closed and finite-valued functions that will promote objects living on low-dimensional subspaces. This class of regularizers encompasses many popular examples such as the L1 norm, L1-L2 norm (group sparsity), as well as several others including the Linfty norm. We also show that the set of partly smooth functions relative to a linear manifold is closed under addition and pre-composition by a linear operator, which allows to cover mixed regularization, and the so-called analysis-type priors (e.g. total variation, fused Lasso, finite-valued 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, and on an analysis of the performance of Linfty regularization in a compressed sensing scenario.
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  • HAL Id : hal-00842603, version 3
  • ARXIV : 1307.2342

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Samuel Vaiter, Mohammad Golbabaee, Jalal M. Fadili, Gabriel Peyré. Model Selection with Low Complexity Priors. Information and Inference, Oxford University Press (OUP), 2015, 52 p. ⟨hal-00842603v3⟩

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