Compressed Sensing Using Binary Matrices of Nearly Optimal Dimensions

Mahsa Lotfi,Mathukumalli Vidyasagar

doi:10.1109/tsp.2020.2990154

Abstract

In this paper, we study the problem of compressed sensing using binary measurement matrices and $\ell _1$ -norm minimization (basis pursuit) as the recovery algorithm. We derive new upper and lower bounds on the number of measurements to achieve robust sparse recovery with binary matrices. We establish sufficient conditions for a column-regular binary matrix to satisfy the robust null space property (RNSP) and show that the associated sufficient conditions for robust sparse recovery obtained using the RNSP are better by a factor of $(3 \sqrt{3})/2 \approx 2.6$ compared to the sufficient conditions obtained using the restricted isometry property (RIP). Next we derive universal lower bounds on the number of measurements that any binary matrix needs to have in order to satisfy the weaker sufficient condition based on the RNSP and show that bipartite graphs of girth six are optimal. Then we display two classes of binary matrices, namely parity check matrices of array codes and Euler squares, which have girth six and are nearly optimal in the sense of almost satisfying the lower bound. In principle, randomly generated Gaussian measurement matrices are “order-optimal.” So we compare the phase transition behavior of the basis pursuit formulation using binary array codes and Gaussian matrices and show that (i) there is essentially no difference between the phase transition boundaries in the two cases and (ii) the CPU time of basis pursuit with binary matrices is hundreds of times faster than with Gaussian matrices and the storage requirements are less. Therefore it is suggested that binary matrices are a viable alternative to Gaussian matrices for compressed sensing using basis pursuit.

Highlights

C OMPRESSED sensing refers to the recovery of highdimensional but low-complexity entities from a limited number of measurements
We derive a sufficient condition for a binary matrix to satisfy the robust null space property (RNSP). In turn this leads to a new upper bound on the sparsity count k for which robust sparse recovery can be guaranteed using a column-regular binary matrix
The best known and the “best possible,” result relating restricted isometry property (RIP) and robust recovery is given below: Theorem 1: If A satisfies the RIP of order tk with constant δtk < (t − 1)/t for t ≥ 4/3, or δtk < t/(4 − t) for t ∈ (0, 4/3), (A, ΔBP) achieves robust sparse recovery of order k

Summary

INTRODUCTION

C OMPRESSED sensing refers to the recovery of highdimensional but low-complexity entities from a limited number of measurements. Popular compressed sensing approaches such as (1) can be applied effectively for far larger values of m and n and with greatly reduced CPU time, when A is a sparse binary matrix instead of a random Gaussian matrix. We derive a sufficient condition for a binary matrix to satisfy the robust null space property (RNSP) In turn this leads to a new upper bound on the sparsity count k for which robust sparse recovery can be guaranteed using a column-regular binary matrix..

BACKGROUND

Approaches to Compressed Sensing – I

Approaches to Compressed Sensing – II

Best Bounds on the Sparsity Count Using the RIP

Binary Matrices for Compressed Sensing: A Review

ROBUST NULL SPACE PROPERTY OF BINARY MATRICES

LOWER BOUNDS ON THE NUMBER OF MEASUREMENTS

CONSTRUCTION OF NEARLY OPTIMAL GRAPHS OF GIRTH SIX

NUMERICAL EXPERIMENTS

Guaranteed Recovery

Phase Transition Study

VIII. DISCUSSION