# Sparse Optimization Problem with s-difference Regularization

## TL;DR

This paper introduces an s-difference regularization for sparse recovery, providing theoretical equivalence with L0 constraints, efficient solution methods, and demonstrating superior performance through numerical experiments.

## Contribution

It proposes a novel s-difference regularization, establishes its theoretical properties, and develops an efficient FBS algorithm with convergence guarantees.

## Key findings

- The s-difference regularization effectively promotes sparsity.
- The FBS method converges to a stationary point.
- Numerical results show improved recovery performance.

## Abstract

In this paper, a s-difference type regularization for sparse recovery problem is proposed, which is the difference of the normal penalty function R(x) and its corresponding struncated function R (xs). First, we show the equivalent conditions between the L0 constrained problem and the unconstrained s-difference penalty regularized problem. Next, we choose the forward-backward splitting (FBS) method to solve the nonconvex regularizes function and further derive some closed-form solutions for the proximal mapping of the s-difference regularization with some commonly used R(x), which makes the FBS easy and fast. We also show that any cluster point of the sequence generated by the proposed algorithm converges to a stationary point. Numerical experiments demonstrate the efficiency of the proposed s-difference regularization in comparison with some other existing penalty functions.