On the Sparsity Bound for the Existence of a Unique Solution in Compressive Sensing by the Gershgorin Theorem

TL;DR

This paper derives an improved sparsity bound for the existence of a unique solution in compressive sensing by applying the Gershgorin disk theorem, offering a more relaxed criterion than previous coherence-based methods.

Contribution

It introduces a novel sparsity bound for unique solution existence in compressive sensing using Gershgorin theorem, relaxing previous coherence index constraints.

Findings

Derived a new sparsity bound using Gershgorin disk theorem.

Improved the theoretical limit for unique solution existence.

Provides a computationally efficient criterion for compressive sensing.

Abstract

Since compressive sensing deals with a signal reconstruction using a reduced set of measurements, the existence of a unique solution is of crucial importance. The most important approach to this problem is based on the restricted isometry property which is computationally unfeasible. The coherence index-based uniqueness criteria are computationally efficient, however, they are pessimistic. An approach to alleviate this problem has been recently introduced by relaxing the coherence index condition for the unique signal reconstruction using the orthogonal matching pursuit approach. This approach can be further relaxed and the sparsity bound improved if we consider only the solution existence rather than its reconstruction. One such improved bound for the sparsity limit is derived in this paper using the Gershgorin disk theorem.