Improved bounds for sparse recovery from subsampled random convolutions
(1610.04983)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 $m \geq s \log2(s) \log(\log(s)) \log(n)$ 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.
We're not able to analyze this paper right now due to high demand.
Please check back later (sorry!).
Generate a summary of this paper on our Pro plan:
We ran into a problem analyzing this paper.