Sparse Reconstruction with Multiple Walsh matrices

TL;DR

This paper introduces a new class of random matrices based on Walsh matrices that satisfy the restricted isometry property, enabling effective sparse signal reconstruction in compressed sensing and dimensionality reduction.

Contribution

The paper presents a novel class of Walsh-based random matrices that satisfy the restricted isometry property with high probability for sparse signal reconstruction.

Findings

Matrices satisfy the restricted isometry property with high probability

Applicable to compressed sensing and dimensionality reduction

Enables accurate sparse vector reconstruction

Abstract

The problem of how to find a sparse representation of a signal is an important one in applied and computational harmonic analysis. It is closely related to the problem of how to reconstruct a sparse vector from its projection in a much lower-dimensional vector space. This is the setting of compressed sensing, where the projection is given by a matrix with many more columns than rows. We introduce a class of random matrices that can be used to reconstruct sparse vectors in this paradigm. These matrices satisfy the restricted isometry property with overwhelming probability. We also discuss an application in dimensionality reduction where we initially discovered this class of matrices.