Proximal Thresholding Algorithm for Minimization over Orthonormal Bases
TLDR
This work proposes a versatile convex variational formulation for optimization over orthonormal bases that covers a wide range of problems, and establishes the strong convergence of a proximal thresholding algorithm to solve it.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 we establish the strong convergence of a proximal thresholding algorithm to solve it. Numerical applications to signal recovery are demonstrated.read more
Citations
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Book
Proximal Algorithms
Neal Parikh,Stephen Boyd +1 more
TL;DR: The many different interpretations of proximal operators and algorithms are discussed, their connections to many other topics in optimization and applied mathematics are described, some popular algorithms are surveyed, and a large number of examples of proxiesimal operators that commonly arise in practice are provided.
Posted Content
Proximal Splitting Methods in Signal Processing
Abstract: The proximity operator of a convex function is a natural extension of the notion of a projection operator onto a convex set. This tool, which plays a central role in the analysis and the numerical solution of convex optimization problems, has recently been introduced in the arena of signal processing, where it has become increasingly important. In this paper, we review the basic properties of proximity operators which are relevant to signal processing and present optimization methods based on these operators. These proximal splitting methods are shown to capture and extend several well-known algorithms in a unifying framework. Applications of proximal methods in signal recovery and synthesis are discussed.
Book ChapterDOI
Proximal Splitting Methods in Signal Processing
TL;DR: The basic properties of proximity operators which are relevant to signal processing and optimization methods based on these operators are reviewed and proximal splitting methods are shown to capture and extend several well-known algorithms in a unifying framework.
Journal ArticleDOI
Bregman Iterative Algorithms for $\ell_1$-Minimization with Applications to Compressed Sensing
TL;DR: In this paper, the authors proposed simple and extremely efficient methods for solving the basis pursuit problem, which is used in compressed sensing, using Bregman iterative regularization, and they gave a very accurate solution after solving only a very small number of instances of the unconstrained problem.
Journal ArticleDOI
Fixed-Point Continuation for $\ell_1$-Minimization: Methodology and Convergence
Elaine Hale,Wotao Yin,Yin Zhang +2 more
TL;DR: The structure of optimal solution sets is studied, finite convergence for important quantities is proved, and $q$-linear convergence rates for the fixed-point algorithm applied to problems with $f(x)$ convex, but not necessarily strictly convex are established.
References
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Book
A wavelet tour of signal processing
TL;DR: An introduction to a Transient World and an Approximation Tour of Wavelet Packet and Local Cosine Bases.
Book
Ten lectures on wavelets
TL;DR: This paper presents a meta-analyses of the wavelet transforms of Coxeter’s inequality and its applications to multiresolutional analysis and orthonormal bases.
Journal ArticleDOI
Atomic Decomposition by Basis Pursuit
TL;DR: Basis Pursuit (BP) is a principle for decomposing a signal into an "optimal" superposition of dictionary elements, where optimal means having the smallest l1 norm of coefficients among all such decompositions.
Journal ArticleDOI
Ideal spatial adaptation by wavelet shrinkage
TL;DR: In this article, the authors developed a spatially adaptive method, RiskShrink, which works by shrinkage of empirical wavelet coefficients, and achieved a performance within a factor log 2 n of the ideal performance of piecewise polynomial and variable-knot spline methods.
Journal ArticleDOI
Adapting to Unknown Smoothness via Wavelet Shrinkage
TL;DR: In this article, the authors proposed a smoothness adaptive thresholding procedure, called SureShrink, which is adaptive to the Stein unbiased estimate of risk (sure) for threshold estimates and is near minimax simultaneously over a whole interval of the Besov scale; the size of this interval depends on the choice of mother wavelet.