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Proximal thresholding algorithm for minimization over orthonormal bases (2007)
| Venue: | SIAM Journal on Optimization |
| Citations: | 62 - 14 self |
BibTeX
@ARTICLE{Combettes07proximalthresholding,
author = {Patrick L. Combettes and Jean-christophe Pesquet},
title = {Proximal thresholding algorithm for minimization over orthonormal bases},
journal = {SIAM Journal on Optimization},
year = {2007},
pages = {1351--1376}
}
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Abstract
The notion of soft thresholding plays a central role in problems from various areas of applied mathematics, in which the ideal solution is known to possess a sparse decomposition in some orthonormal basis. Using convex-analytical tools, we extend this notion to that of proximal thresholding and investigate its properties, providing in particular several characterizations of such thresholders. We then propose a versatile convex variational formulation for optimization over orthonormal bases that covers a wide range of problems, and establish the strong convergence of a proximal thresholding algorithm to solve it. Numerical applications to signal recovery are demonstrated. 1 Problem formulation Throughout this paper, H is a separable infinite-dimensional real Hilbert space with scalar product 〈 · | ·〉, norm ‖·‖, and distance d. Moreover, Γ0(H) denotes the class of proper lower semicontinuous convex functions from H to]−∞, +∞], and (ek)k∈N is an orthonormal basis of H. The standard denoising problem in signal theory consists of recovering the original form of a signal x ∈ H from an observation z = x + v, where v ∈ H is the realization of a noise process. In many instances, x is known to admit a sparse representation with respect to (ek)k∈N and an estimate x of x can be constructed by removing the coefficients of smallest magnitude in the 1 representation (〈z | ek〉)k∈N of z with respect to (ek)k∈N. A popular method consists of performing a so-called soft thresholding of each coefficient 〈z | ek 〉 at some predetermined level ωk ∈]0, +∞[, namely (see Fig. 1) (1.1) x = ∑ soft [−ωk,ωk] (〈z | ek〉)ek, where soft [−ωk,ωk] : ξ ↦ → sign(ξ) max{|ξ | − ωk, 0}. k∈N This approach has received considerable attention in various areas of applied mathematics ranging from nonlinear approximation theory to statistics, and from harmonic analysis to image processing;
Keyphrases
orthonormal base various area orthonormal basis applied mathematics signal theory problem formulation harmonic analysis strong convergence wide range ideal solution coefficient ek considerable attention image processing many instance proximal thresholding algorithm noise process nonlinear approximation theory popular method numerical application predetermined level convex-analytical tool particular several characterization semicontinuous convex function proximal thresholding versatile convex variational formulation separable infinite-dimensional real hilbert space sparse representation central role scalar product original form standard denoising problem so-called soft thresholding sparse decomposition
