Dictionary-sparse recovery from heavy-tailed measurements

Abstract

Abstract The recovery of signals that are sparse not in a given basis, but rather sparse with respect to an over-complete dictionary is one of the most flexible settings in the field of compressed sensing with numerous applications. As in the standard compressed sensing setting, it is possible that the signal can be reconstructed efficiently from a few linear measurements, for example, by the so-called $\ell _1$-synthesis method. However, it has been less well understood which measurement matrices provably work for this setting. Whereas in the standard setting, it has been shown that even certain heavy-tailed measurement matrices can be used in the same sample complexity regime as Gaussian matrices, comparable results are only available for the restrictive class of sub-Gaussian measurement matrices as far as the recovery of dictionary-sparse signals via $\ell _1$-synthesis is concerned. In this work, we fill this gap and establish optimal guarantees for the recovery of vectors that are (approximately) sparse with respect to a dictionary via the $\ell _1$-synthesis method from linear, potentially noisy measurements for a large class of random measurement matrices. In particular, we show that random measurements that fulfill only a small-ball assumption and a weak moment assumption, such as random vectors with i.i.d. Student-$t$ entries with a logarithmic number of degrees of freedom, lead to comparable guarantees as (sub-)Gaussian measurements. Our results apply for a large class of both random and deterministic dictionaries. As a technical tool, we show a bound on the expectation of the sum of squared order statistics under very general assumptions, which might be of independent interest. As a corollary of our results, we also obtain a slight improvement on the weakest assumption on a measurement matrix with i.i.d. rows sufficient for uniform recovery in standard compressed sensing, improving on results by Lecué and Mendelson and by Dirksen, Lecué and Rauhut.