Improved bounds for sparse recovery from subsampled random convolutions

TL;DR

This paper improves theoretical bounds for recovering sparse signals from subsampled random convolutions using -minimization, matching Gaussian measurement conditions for small sparsity and providing better bounds for larger sparsity.

Contribution

The paper introduces improved measurement bounds for sparse recovery from subsampled random convolutions, extending known results to larger sparsity levels with novel proof techniques.

Findings

For small sparsity, -minimization requires m  s    s    n measurements, matching Gaussian bounds.

For larger sparsity, -minimization requires m  s    s        n measurements, improving previous estimates.

The proof employs a novel combination of small ball probability estimates and chaining techniques.

Abstract

We study the recovery of sparse vectors from subsampled random convolutions via $\ell_1$-minimization. We consider the setup in which both the subsampling locations as well as the generating vector are chosen at random. For a subgaussian generator with independent entries, we improve previously known estimates: if the sparsity $s$ is small enough, i.e., $s \lesssim \sqrt{n/\log(n)}$, we show that $m \gtrsim s \log(en/s)$ measurements are sufficient to recover $s$-sparse vectors in dimension $n$ with high probability, matching the well-known condition for recovery from standard Gaussian measurements. If $s$ is larger, then essentially $m \geq s \log^2(s) \log(\log(s)) \log(n)$ measurements are sufficient, again improving over previous estimates. Our results are shown via the so-called robust null space property which is weaker than the standard restricted isometry property. Our method of proof involves a novel combination of small ball estimates with chaining techniques {which should be of independent interest.