Quaternion Gaussian matrices satisfy the RIP

Abstract

We prove that quaternion Gaussian random matrices satisfy the restricted isometry property (RIP) with overwhelming probability. We also explain why the restricted isometry random variables (RIV) approach is not appropriate for drawing conclusions on restricted isometry constants.

Compressed sensing in the quaternion algebra

“…The results of this article, together with aforementioned [2], form a theoretical background of the classical compressed sensing methods in the quaternion algebra. We extended the fundamental result of this theory to the full quaternion case, namely we proved that if a quaternion measurement matrix satisfies the RIP with a sufficiently small constant, then it is possible to reconstruct sparse quaternion signals from a small number of their measurements via ℓ 1 -norm minimization.…”

“…It has been believed that quaternion Gaussian random matrices satisfy RIP and, therefore, they have been widely used in numerical experiments [4,17,27] but there was a lack of theoretical justification of this conviction. In the subsequent article [2] we prove that this hypothesis is true, i.e. quaternion Gaussian matrices satisfy the RIP, and we provide estimates on matrix sizes that guarantee the RIP with overwhelming probability.…”

“…In view of the main results of this paper (Theorem 4.1, Corollary 5.1) and having in mind that quaternion Gaussian random matrices satisfy (with overwhelming probability) the restricted isometry property [2], we performed similar experiments for the case of quaternion matrix Φ ∈ H m×n -as in [27]. By a quaternion Gaussian random matrix we mean a matrix Φ = (φ ij ) ∈ H m×n whose entries φ ij are independent quaternion Gaussian random variables with distribution denoted by N H (0, σ 2 ), i.e.…”