Explicit constructions of RIP matrices and related problems

TL;DR

This paper presents a new explicit method for constructing RIP matrices that surpasses previous limitations, along with improved constructions for sets with small moments, advancing compressed sensing and spherical code design.

Contribution

The paper introduces an explicit construction of RIP matrices that overcomes the natural coherence barrier and provides new bounds for sets with small moments, enhancing compressed sensing techniques.

Findings

Constructed RIP matrices for larger orders than previous explicit methods.

Developed new estimates for sumsets and exponential sums with structured sets.

Produced explicit matrices satisfying RIP with parameters matching existing constructions.

Abstract

We give a new explicit construction of $n\times N$ matrices satisfying the Restricted Isometry Property (RIP). Namely, for some c>0, large N and any n satisfying N^{1-c} < n < N, we construct RIP matrices of order k^{1/2+c}. This overcomes the natural barrier k=O(n^{1/2}) for proofs based on small coherence, which are used in all previous explicit constructions of RIP matrices. Key ingredients in our proof are new estimates for sumsets in product sets and for exponential sums with the products of sets possessing special additive structure. We also give a construction of sets of n complex numbers whose k-th moments are uniformly small for 1\le k\le N (Turan's power sum problem), which improves upon known explicit constructions when (\log N)^{1+o(1)} \le n\le (\log N)^{4+o(1)}. This latter construction produces elementary explicit examples of n by N matrices that satisfy RIP and whose columns constitute a new spherical code; for those problems the parameters closely match those of existing constructions in the range (\log N)^{1+o(1)} \le n\le (\log N)^{5/2+o(1)}.