Sparsest Error Detection via Sparsity Invariant Transformation based $\ell_1$ Minimization

TL;DR

This paper introduces a novel sparsity invariant transformation approach combined with $\, ext{l}_1$ minimization to reliably detect sparse errors in over-determined linear systems, overcoming limitations of previous methods.

Contribution

It proposes the concept of sparsity invariant transformations (SIT) and demonstrates their effectiveness in ensuring $\, ext{l}_1$ minimization finds the sparsest error solution in any over-determined system.

Findings

Existence of specific SITs guaranteeing sparsest solutions

Development of a randomized Monte Carlo algorithm to find feasible SITs

Theoretical proof of the method's effectiveness for all over-determined systems

Abstract

This paper presents a new method, referred to here as the sparsity invariant transformation based $\ell_1$ minimization, to solve the $\ell_0$ minimization problem for an over-determined linear system corrupted by additive sparse errors with arbitrary intensity. Many previous works have shown that $\ell_1$ minimization can be applied to realize sparse error detection in many over-determined linear systems. However, performance of this approach is strongly dependent on the structure of the measurement matrix, which limits application possibility in practical problems. Here, we present a new approach based on transforming the $\ell_0$ minimization problem by a linear transformation that keeps sparsest solutions invariant. We call such a property a sparsity invariant property (SIP), and a linear transformation with SIP is referred to as a sparsity invariant transformation (SIT). We propose the SIT-based $\ell_1$ minimization method by using an SIT in conjunction with $\ell_1$ relaxation on the $\ell_0$ minimization problem. We prove that for any over-determined linear system, there always exists a specific class of SIT's that guarantees a solution to the SIT-based $\ell_1$ minimization is a sparsest-errors solution. Besides, a randomized algorithm based on Monte Carlo simulation is proposed to search for a feasible SIT.