Sparse Regularization via Convex Analysis

Sparse approximate solutions to linear equations are classically obtained via L1 norm regularized least squares, but this method often underestimates the true solution. As an alternative to the L1 norm, this paper proposes a class of nonconvex penalty functions that maintain the convexity of the lea...

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Bibliographic Details
Published in:IEEE transactions on signal processing Vol. 65; no. 17; pp. 4481 - 4494
Main Author: Selesnick, Ivan
Format: Journal Article
Language:English
Published: IEEE 01.09.2017
Subjects:
convex function
Convex functions
Convolution
Cost function
denoising
Noise reduction
optimization
Signal processing algorithms
sparse approximation
Sparse regularization
ISSN:1053-587X, 1941-0476
Online Access:Get full text
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Summary:Sparse approximate solutions to linear equations are classically obtained via L1 norm regularized least squares, but this method often underestimates the true solution. As an alternative to the L1 norm, this paper proposes a class of nonconvex penalty functions that maintain the convexity of the least squares cost function to be minimized, and avoids the systematic underestimation characteristic of L1 norm regularization. The proposed penalty function is a multivariate generalization of the minimax-concave penalty. It is defined in terms of a new multivariate generalization of the Huber function, which in turn is defined via infimal convolution. The proposed sparse-regularized least squares cost function can be minimized by proximal algorithms comprising simple computations.
ISSN:1053-587X
1941-0476
DOI:10.1109/TSP.2017.2711501