A fast approach for overcomplete sparse decomposition based on smoothed L0 norm

TL;DR

This paper introduces a fast algorithm called SL0 for overcomplete sparse decomposition that directly minimizes the L0 norm, significantly outperforming traditional LP-based methods in speed while maintaining accuracy.

Contribution

The paper presents a novel SL0 algorithm that directly minimizes the L0 norm for sparse solutions, offering a much faster alternative to LP-based methods.

Findings

SL0 is 100 to 1000 times faster than LP solvers.

SL0 achieves comparable or better accuracy in sparse decomposition.

The method is applicable to various problems like compressed sensing and atomic decomposition.

Abstract

In this paper, a fast algorithm for overcomplete sparse decomposition, called SL0, is proposed. The algorithm is essentially a method for obtaining sparse solutions of underdetermined systems of linear equations, and its applications include underdetermined Sparse Component Analysis (SCA), atomic decomposition on overcomplete dictionaries, compressed sensing, and decoding real field codes. Contrary to previous methods, which usually solve this problem by minimizing the L1 norm using Linear Programming (LP) techniques, our algorithm tries to directly minimize the L0 norm. It is experimentally shown that the proposed algorithm is about two to three orders of magnitude faster than the state-of-the-art interior-point LP solvers, while providing the same (or better) accuracy.