An Improved Estimate in the Restricted Isometry Problem

Abstract

It is shown that for the n × n-Hadamard matrix (or, more generally, a bounded orthogonal matrix) the RIP-property for r-space vectors holds, with row restriction to a set S of size

$$\displaystyle{\vert S\vert < C(\varepsilon )(\log n)^{2}(\log r)r.}$$

This bound represents a slight improvement over (Rudelson and Vershynin, Commun Pure Appl Math 61:1025–1045, 2008) in that the power of the logarithm is decreased by one unit.