A unified recovery bound estimation for noise-aware Lq optimization model in compressed sensing

TL;DR

This paper establishes a unified, noise-aware recovery bound for Lq optimization in compressed sensing, demonstrating improved bounds over L1 under certain sparsity conditions without relying on RIC.

Contribution

It provides a noise-aware, RIC-free recovery bound for Lq optimization, introducing a parameter {gamma} that links sparsity and error bounds, and compares Lq and L1 models.

Findings

Recovery error is bounded by a constant times noise level.

When {gamma} > 2, Lq model outperforms L1 in recovery bounds.

Lq model requires weaker sparsity conditions than L1.

Abstract

In this letter, we present a unified result for the stable recovery bound of Lq(0 < q < 1) optimization model in compressed sensing, which is a constrained Lq minimization problem aware of the noise in a linear system. Specifically, without using the restricted isometry constant (RIC), we show that the error between any global solution of the noise-aware Lq optimization model and the ideal sparse solution of the noiseless model is upper bounded by a constant times the noise level,given that the sparsity of the ideal solution is smaller than a certain number. An interesting parameter {gamma} is introduced, which indicates the sparsity level of the error vector and plays an important role in our analysis. In addition, we show that when {\gamma} > 2, the recovery bound of the Lq (0 < q < 1) model is smaller than that of the L1 model, and the sparsity requirement of the ideal solution in the Lq(0 < q < 1) model is weaker than that of the L1 model.