Spark Level Sparsity and the $\ell_1$ Tail Minimization

TL;DR

This paper reveals that measure-theoretic uniqueness allows for the recovery of nearly spark-level sparse signals in compressed sensing, challenging traditional sparsity limits, and demonstrates this through extensive empirical validation.

Contribution

It introduces a measure-theoretic perspective on uniqueness in compressed sensing, enabling recovery of signals with sparsity above half the number of measurements, supported by extensive experiments.

Findings

Measure-theoretic uniqueness for sparsity s > m/2

L1-tail minimization recovers signals beyond traditional limits

Standard L1-minimization fails for s > m/2

Abstract

Solving compressed sensing problems relies on the properties of sparse signals. It is commonly assumed that the sparsity s needs to be less than one half of the spark of the sensing matrix A, and then the unique sparsest solution exists, and recoverable by $\ell_1$-minimization or related procedures. We discover, however, a measure theoretical uniqueness exists for nearly spark-level sparsity from compressed measurements Ax = b. Specifically, suppose A is of full spark with m rows, and suppose $\frac{m}{2}$ < s < m. Then the solution to Ax = b is unique for x with $\|x\|_0 \leq s$ up to a set of measure 0 in every s-sparse plane. This phenomenon is observed and confirmed by an $\ell_1$-tail minimization procedure, which recovers sparse signals uniquely with s > $\frac{m}{2}$ in thousands and thousands of random tests. We further show instead that the mere $\ell_1$-minimization would actually fail if s > $\frac{m}{2}$ even from the same measure theoretical point of view.