The Annals of Applied Probability

Improved bounds for sparse recovery from subsampled random convolutions

Shahar Mendelson, Holger Rauhut, and Rachel Ward

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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.

Article information

Ann. Appl. Probab., Volume 28, Number 6 (2018), 3491-3527.

Received: October 2016
Revised: March 2018
First available in Project Euclid: 8 October 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 94A20: Sampling theory
Secondary: 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52)

Circulant matrix compressive sensing generic chaining small ball estimates sparsity


Mendelson, Shahar; Rauhut, Holger; Ward, Rachel. Improved bounds for sparse recovery from subsampled random convolutions. Ann. Appl. Probab. 28 (2018), no. 6, 3491--3527. doi:10.1214/18-AAP1391.

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