# Stability Analysis for a Class of Sparse Optimization Problems

## TL;DR

This paper establishes a stability result for $ ext{l}_1$-minimization in sparse optimization, generalizing previous results by introducing a new property of sensing matrices, which enhances understanding of signal recovery stability.

## Contribution

The paper introduces the restricted weak range space property (RSP) of sensing matrices, generalizing previous concepts, and establishes a stability result for $ ext{l}_1$-minimization in a broad class of $ ext{l}_0$-minimization problems.

## Key findings

- Introduces the restricted weak RSP of sensing matrices.
- Establishes a generalized stability theorem for $ ext{l}_1$-minimization.
- Includes several existing stability results as special cases.

## Abstract

The sparse optimization problems arise in many areas of science and engineering, such as compressed sensing, image processing, statistical and machine learning. The $\ell_{0}$-minimization problem is one of such optimization problems, which is typically used to deal with signal recovery. The $\ell_{1}$-minimization method is one of the plausible approaches for solving the $\ell_{0}$-minimization problems, and thus the stability of such a numerical method is vital for signal recovery. In this paper, we establish a stability result for the $\ell_{1}$-minimization problems associated with a general class of $\ell_{0}$-minimization problems. To this goal, we introduce the concept of restricted weak range space property (RSP) of a transposed sensing matrix, which is a generalized version of the weak RSP of the transposed sensing matrix introduced in [Zhao et al., Math. Oper. Res., 44(2019), 175-193]. The stability result established in this paper includes several existing ones as special cases.