Explicit Matrices with the Restricted Isometry Property: Breaking the Square-Root Bottleneck
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Matrices with the restricted isometry property (RIP) are of particular interest in compressed sensing. To date, the best known RIP matrices are constructed using random processes, while explicit constructions are notorious for performing at the “square-root bottleneck,” i.e., they only accept sparsity levels on the order of the square root of the number of measurements. The only known explicit matrix which surpasses this bottleneck was constructed by Bourgain, Dilworth, Ford, Konyagin, and Kutzarova in Bourgain et al. (Duke Math. J. 159:145–185, 2011). This chapter provides three contributions to advance the groundbreaking work of Bourgain et al.: (i) we develop an intuition for their matrix construction and underlying proof techniques; (ii) we prove a generalized version of their main result; and (iii) we apply this more general result to maximize the extent to which their matrix construction surpasses the square-root bottleneck.
The author thanks the anonymous referees for their helpful suggestions. This work was supported by NSF Grant No. DMS-1321779. The views expressed in this chapter are those of the author and do not reflect the official policy or position of the United States Air Force, Department of Defense, or the U.S. Government.
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