Improved bounds for sparse recovery from subsampled random convolutions

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 sub-Gaussian generator with independent entries, we improve previously known estimates: if the sparsity $s$ is small enough, that is, $s\lesssim\sqrt{n/\log(n)}$, 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.