Abstract
Numerical algorithms for the l0-norm regularized non-smooth non-convex minimization problems have recently became a topic of great interest within signal processing, compressive sensing, statistics, and machine learning. Nevertheless, the l0-norm makes the problem combinatorial and generally computationally intractable. In this paper, we construct a new surrogate function to approximate l0-norm regularization, and subsequently make the discrete optimization problem continuous and smooth. Then we use the well-known spectral gradient algorithm to solve the resulting smooth optimization problem. Experiments are provided which illustrate this method is very promising.
Highlights
IntroductionThe focus of this paper is on the following structured minimization: min x∈Rn f (x). denotes the cardinality or the number of nonzero components in x
The focus of this paper is on the following structured minimization: min x∈Rn f (x) := 1 ∥Ax − b∥22 + μ∥x∥0, (1.1) where ∥x∥0 =
We show that the proposed smooth function trends to the l0-norm as β → ∞
Summary
The focus of this paper is on the following structured minimization: min x∈Rn f (x). denotes the cardinality or the number of nonzero components in x. A closely related method is the fixed-point continuation and active set [16, 17], which solves a smooth subproblem to determine the magnitudes of the nonzero components of x based on an active set. Another closely related method is the sparse reconstruction algorithm [14], which involves minimizing a non-smooth convex problem with separable structures. We do numerical experiments to recover a large sparse signal from its limited measurement, which illustrate that the constructed function works well, and indicate that the proposed algorithm is encouraging.
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