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Explicit Construction of RIP Matrices Is Ramsey‐Hard
Name
1805.11238.pdf
Description
Submitted version
Size
107.4 KB
Format
Adobe PDF
Checksum (MD5)
418c469de58c6ac129b3f21b32c3ee06
Author(s)
Gamarnik, David
Date Issued
2020
Journal
Communications on Pure and Applied Mathematics
Publisher
Wiley
Version
Original manuscript
Abstract
© 2019 Wiley Periodicals, Inc. Matrices 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 (Formula presented.) 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.
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Creative Commons Attribution-Noncommercial-Share Alike
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DOI of Published Version
10.1002/CPA.21873
