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

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.

References in corpus (1)

• Massive MIMO Unsourced Random Access

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