Explicit construction of RIP matrices is Ramsey-hard

TL;DR

This paper links the explicit construction of RIP matrices with small dimensions to a long-standing open problem in extremal combinatorics, showing that such constructions are as hard as constructing certain Ramsey graphs.

Contribution

It establishes a complexity equivalence between constructing RIP matrices in small regimes and creating specific Ramsey graphs, resolving an open question in the field.

Findings

Explicit RIP matrices imply Ramsey graphs with bounded clique sizes

Constructing small RIP matrices is as hard as a long-standing combinatorial problem

Connects compressive sensing matrix design to extremal graph theory

Abstract

Matrices $\Phi\in\R^{n\times p}$ satisfying the Restricted Isometry Property (RIP) are an important ingredient of the compressive sensing methods. While it is known that random matrices satisfy the RIP with high probability even for $n=\log^{O(1)}p$, the explicit construction of such matrices defied the repeated efforts, and the most known approaches hit the so-called $\sqrt{n}$ sparsity bottleneck. The notable exception is the work by Bourgain et al \cite{bourgain2011explicit} constructing an $n\times p$ RIP matrix with sparsity $s=\Theta(n^{{1\over 2}+\epsilon})$, but in the regime $n=\Omega(p^{1-\delta})$. In this short note we resolve this open question in a sense by showing that an explicit construction of a matrix satisfying the RIP in the regime $n=O(\log^2 p)$ and $s=\Theta(n^{1\over 2})$ implies an explicit construction of a three-colored Ramsey graph on $p$ nodes with clique sizes bounded by $O(\log^2 p)$ -- a question in the extremal combinatorics which has been open for decades.