Sparse Recovery

Dictionary-Sparse Recovery From Heavy-Tailed Measurements

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 …

Iteratively Reweighted Least Squares for $\ell_1$-minimization with Global Linear Convergence Rate

Iteratively Reweighted Least Squares (IRLS), whose history goes back more than 80 years, represents an important family of algorithms for non-smooth optimization as it is able to optimize these problems by solving a sequence of linear systems. In …

Understanding and Enhancing Data Recovery Algorithms - From Noise-Blind Sparse Recovery to Reweighted Methods for Low-Rank Matrix Optimization

We prove new results about the robustness of noise-blind decoders for the problem of re- constructing a sparse vector from underdetermined linear measurements. Our results imply provable robustness of equality-constrained l1-minimization for random …

On the geometry of polytopes generated by heavy-tailed random vectors

We study the geometry of centrally-symmetric random polytopes, generated by $N$ independent copies of a random vector $X$ taking values in ${\mathbb{R}^n}$. We show that under minimal assumptions on $X$, for $N \gtrsim n$ and with high probability, …

A Quotient Property for Matrices with Heavy-Tailed Entries and its Application to Noise-Blind Compressed Sensing

For a large class of random matrices $A$ with i.i.d. entries we show that the $\ell_1$-quotient property holds with probability exponentially close to $1$. In contrast to previous results, our analysis does not require concentration of the entrywise …