Sparsest Error Detection via Sparsity Invariant Transformation based ℓ1 Minimization

This paper presents a new method, referred to here as the sparsity invariant transformation based ‘1 minimization, to solve the ‘0 minimization problem for an over-determined linear system corrupted by additive sparse errors with arbitrary intensity. Many previous works have shown that ‘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 ‘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 ‘1 minimization method by using an SIT in conjunction with ‘1 relaxation on the ‘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 ‘1 minimization is a sparsest-errors solution. Besides, a randomized algorithm based on Monte Carlo simulation is proposed to search for a feasible SIT.