Deriving RIP sensing matrices for sparsifying dictionaries

TL;DR

This paper introduces a novel method to derive sensing matrices for any sparsifying dictionary in compressive sensing, ensuring the restricted isometry property with high probability and achieving recovery performance comparable to traditional random matrices.

Contribution

It presents a new approach to generate sensing matrices for arbitrary dictionaries that likely satisfy the restricted isometry property, overcoming NP-hardness issues.

Findings

Recovery performance comparable to Gaussian and Bernoulli sensing matrices

Applicable to dictionaries like K-SVD, Parseval K-SVD, and wavelets

High probability of sensing matrix satisfying RIP for any dictionary

Abstract

Compressive sensing involves the inversion of a mapping $SD \in \mathbb{R}^{m \times n}$, where $m < n$, $S$ is a sensing matrix, and $D$ is a sparisfying dictionary. The restricted isometry property is a powerful sufficient condition for the inversion that guarantees the recovery of high-dimensional sparse vectors from their low-dimensional embedding into a Euclidean space via convex optimization. However, determining whether $SD$ has the restricted isometry property for a given sparisfying dictionary is an NP-hard problem, hampering the application of compressive sensing. This paper provides a novel approach to resolving this problem. We demonstrate that it is possible to derive a sensing matrix for any sparsifying dictionary with a high probability of retaining the restricted isometry property. In numerical experiments with sensing matrices for K-SVD, Parseval K-SVD, and wavelets, our recovery performance was comparable to that of benchmarks obtained using Gaussian and Bernoulli random sensing matrices for sparse vectors.