Worst Configurations (Instantons) for Compressed Sensing over Reals: a Channel Coding Approach

TL;DR

This paper introduces a novel algorithm to identify the worst-case error configurations, called instantons, that cause the failure of Basis Pursuit in compressed sensing, linking channel coding insights with sparse signal recovery.

Contribution

It develops a CS-Instanton Search Algorithm (ISA) to find minimal error patterns leading to Basis Pursuit failure, extending previous LDPC error-floor analysis to compressed sensing.

Findings

The CS-ISA converges quickly to instantons for dense random matrices.

The shortest instanton found has length 11 in a 120x512 matrix example.

The method effectively identifies worst-case error configurations for Basis Pursuit.

Abstract

We consider the Linear Programming (LP) solution of the Compressed Sensing (CS) problem over reals, also known as the Basis Pursuit (BasP) algorithm. The BasP allows interpretation as a channel-coding problem, and it guarantees error-free reconstruction with a properly chosen measurement matrix and sufficiently sparse error vectors. In this manuscript, we examine how the BasP performs on a given measurement matrix and develop an algorithm to discover the sparsest vectors for which the BasP fails. The resulting algorithm is a generalization of our previous results on finding the most probable error-patterns degrading performance of a finite size Low-Density Parity-Check (LDPC) code in the error-floor regime. The BasP fails when its output is different from the actual error-pattern. We design a CS-Instanton Search Algorithm (ISA) generating a sparse vector, called a CS-instanton, such that the BasP fails on the CS-instanton, while the BasP recovery is successful for any modification of the CS-instanton replacing a nonzero element by zero. We also prove that, given a sufficiently dense random input for the error-vector, the CS-ISA converges to an instanton in a small finite number of steps. The performance of the CS-ISA is illustrated on a randomly generated $120\times 512$ matrix. For this example, the CS-ISA outputs the shortest instanton (error vector) pattern of length 11.