Derandomizing Compressed Sensing With Combinatorial Design - Jung, Peter; Kueng, Richard; Mixon, Dustin G. (Frontiers in Applied Mathematics and Statistics, 2019)
Compressed sensing is the art of efficiently reconstructing structured n-dimensional vectors from substantially fewer measurements than naively anticipated. A plethora of analytic reconstruction guarantees support this credo. The strongest among them are based on deep results from large-dimensional probability theory and require a considerable amount of randomness in the measurement design. Here, we demonstrate that derandomization techniques allow for considerably reducing the amount of randomness that is required for such proof strategies. More precisely, we establish uniform s-sparse reconstruction guarantees for Cs log(n) measurements that are chosen independently from strength-4 orthogonal arrays and maximal sets of mutually unbiased bases, respectively. These are highly structured families of Cn˜ 2 vectors that imitate signed Bernoulli and standard Gaussian vectors in a (partially) derandomized fashion.