Explicit Construction of RIP Matrices Is Ramsey‐Hard

Abstract

AbstractMatrices Φ ∈ ℝn × 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 = logO(1)p, the explicit deteministic construction of such matrices defied the repeated efforts, and most of the known approaches hit the so‐called sparsity bottleneck. The notable exception is the work by Bourgain et al. constructing an n × p RIP matrix with sparsity s = Θ(n1/2 + ϵ), but in the regime n = Ω(p1 − δ).In this short note we resolve this open question by showing that an explicit construction of a matrix satisfying the RIP in the regime n = O(log2p) and s = Θ(n1/2) implies an explicit construction of a three‐colored Ramsey graph on p nodes with clique sizes bounded by O(log2p) — a question in the field of extremal combinatorics that has been open for decades. © 2019 Wiley Periodicals, Inc.