AUTHOR=Jung Peter , Kueng Richard , Mixon Dustin G. 

TITLE=Derandomizing Compressed Sensing With Combinatorial Design

JOURNAL=Frontiers in Applied Mathematics and Statistics

VOLUME=Volume 5 - 2019

YEAR=2019

URL=https://www.frontiersin.org/journals/applied-mathematics-and-statistics/articles/10.3389/fams.2019.00026

DOI=10.3389/fams.2019.00026

ISSN=2297-4687

ABSTRACT=<p>Compressed sensing is the art of effectively reconstructing structured <italic>n</italic>-dimensional vectors from substantially fewer measurements than naively anticipated. A plethora of analytical 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 a considerable reduction in the randomness required for such proof strategies. More precisely, we establish uniform <italic>s</italic>-sparse reconstruction guarantees for <italic>Cs</italic> log(<italic>n</italic>) measurements that are chosen independently from strength-4 orthogonal arrays and maximal sets of mutually unbiased bases, respectively. These are highly structured families of <inline-formula><mml:math id="M1" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mover accent="true"><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mo>~</mml:mo></mml:mover><mml:msup><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula> vectors that imitate signed Bernoulli and standard Gaussian vectors in a (partially) derandomized fashion.</p>