SVD method for sparse recovery

TL;DR

This paper introduces efficient SVD-based algorithms for sparse recovery that outperform traditional iterative methods in speed and success rate, especially for diagonal and general linear operators.

Contribution

The paper develops two novel inversion schemes for $oldsymbol{ ext{ell}_p}$ regularization with specific operators, extending SVD methods beyond classical quadratic regularization.

Findings

Algorithms operate faster than traditional methods

Higher success rate in sparse recovery tasks

Effective for both diagonal and general linear operators

Abstract

Sparsity regularization has garnered significant interest across multiple disciplines, including statistics, imaging, and signal processing. Standard techniques for addressing sparsity regularization include iterative soft thresholding algorithms and their accelerated variants. However, these algorithms rely on Landweber iteration, which can be computationally intensive. Therefore, there is a pressing need to develop a more efficient algorithm for sparsity regularization. The Singular Value Decomposition (SVD) method serves as a regularization strategy that does not require Landweber iterations; however, it is confined to classical quadratic regularization. This paper introduces two inversion schemes tailored for situations where the operator $K$ is diagonal within a specific orthogonal basis, focusing on $\ell_{p}$ regularization when $p=1$ and $p=1/2$. Furthermore, we demonstrate that for a general linear compact operator $K$, the SVD method serves as an effective regularization strategy. To assess the efficacy of the proposed methodologies, We conduct several numerical experiments to evaluate the proposed method's effectiveness. The results indicate that our algorithms not only operate faster but also achieve a higher success rate than traditional iterative methods.