Bounds of restricted isometry constants in extreme asymptotics: formulae for Gaussian matrices

TL;DR

This paper derives explicit formulas for the Restricted Isometry Constants of Gaussian matrices in various asymptotic regimes, providing insights into their behavior and implications for compressed sensing.

Contribution

It introduces new explicit formulae for RIC bounds of Gaussian matrices in three asymptotic settings, enhancing understanding of their properties in high-dimensional regimes.

Findings

RICs decay to zero when n/N fixed and k/n approaches zero

RICs become unbounded or approach bounds when k/n is fixed and n/N approaches zero

RICs approach a non-zero constant when n/N approaches zero with k/n decaying logarithmically

Abstract

Restricted Isometry Constants (RICs) provide a measure of how far from an isometry a matrix can be when acting on sparse vectors. This, and related quantities, provide a mechanism by which standard eigen-analysis can be applied to topics relying on sparsity. RIC bounds have been presented for a variety of random matrices and matrix dimension and sparsity ranges. We provide explicitly formulae for RIC bounds, of n by N Gaussian matrices with sparsity k, in three settings: a) n/N fixed and k/n approaching zero, b) k/n fixed and n/N approaching zero, and c) n/N approaching zero with k/n decaying inverse logrithmically in N/n; in these three settings the RICs a) decay to zero, b) become unbounded (or approach inherent bounds), and c) approach a non-zero constant. Implications of these results for RIC based analysis of compressed sensing algorithms are presented.