On the Restricted Isometry Property of Centered Self Khatri-Rao Products

TL;DR

This paper proves that centered self Khatri-Rao products of certain random matrices satisfy the Restricted Isometry Property, enabling reliable covariance estimation in applications like activity detection and MIMO systems.

Contribution

It establishes RIP conditions for centered self Khatri-Rao products of matrices with iid columns, extending understanding of their use in covariance matching tasks.

Findings

Centered self KR products have small RIP constants under certain conditions.

The results apply to matrices with iid sub-Gaussian or uniform sphere columns.

The findings support their use in covariance matching applications.

Abstract

In this work we establish the Restricted Isometry Property (RIP) of the centered column-wise self Khatri-Rao (KR) products of $n\times N$ matrix with iid columns drawn either uniformly from a sphere or with iid sub-Gaussian entries. The self KR product is an $n^2\times N$-matrix which contains as columns the vectorized (self) outer products of the columns of the original $n\times N$-matrix. Based on a result of Adamczak et al. we show that such a centered self KR product with independent heavy tailed columns has small RIP constants of order $s$ with probability at least $1-C\exp(-cn)$ provided that $s\lesssim n^2/\log^2(eN/n^2)$. Our result is applicable in various works on covariance matching like in activity detection and MIMO gain-estimation.