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G.: Local linear convergence of Forward–Backward under partial smoothness. NIPS
, 1970
"... In this paper, we consider the Forward–Backward proximal splitting algorithm to minimize the sum of two proper convex functions, one of which having a Lipschitz continuous gradient and the other being partly smooth relative to an active manifold M. We propose a generic framework under which we sho ..."
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Cited by 4 (2 self)
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In this paper, we consider the Forward–Backward proximal splitting algorithm to minimize the sum of two proper convex functions, one of which having a Lipschitz continuous gradient and the other being partly smooth relative to an active manifold M. We propose a generic framework under which we show that the Forward–Backward (i) correctly identifies the active manifold M in a finite number of iterations, and then (ii) enters a local linear convergence regime that we characterize precisely. This gives a grounded and unified explanation to the typical behaviour that has been observed numerically for many problems encompassed in our framework, including the Lasso, the group Lasso, the fused Lasso and the nuclear norm regularization to name a few. These results may have numerous applications including in signal/image processing processing, sparse recovery and machine learning. 1
Journal of Machine Learning Research (2014) Submitted; Published Model Consistency of Partly Smooth Regularizers
, 2014
"... HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte p ..."
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HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et a ̀ la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
Sparse Spikes Deconvolution on Thin Grids
, 2015
"... HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte p ..."
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HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et a ̀ la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
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"... Regularization plays a pivotal role when facing the challenge of solving illposed 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 lowdimensional str ..."
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Regularization plays a pivotal role when facing the challenge of solving illposed 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 lowdimensional 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 lowdimensional/simple structure. This paper provides a general framework to capture this lowdimensional structure through what we coin partly smooth functions relative to a linear manifold. These are convex, nonnegative, closed and finitevalued functions that will promote objects living on lowdimensional subspaces. This class of regularizers encompasses many popular examples such as the `1 norm, `1 − `2 norm (group sparsity), as well as several others including the ` ∞ norm. We also show that the set of partly smooth functions relative to a linear manifold is closed under addition and precomposition by a linear operator, which allows to cover mixed regularization, and the socalled analysistype priors (e.g. total variation, fused Lasso, finitevalued polyhedral gauges). Our main result presents a unified sharp analysis of exact and robust recovery of the lowdimensional 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 ` ∞ regularization in a compressed sensing scenario.