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

In this work we establish the Restricted Isometry Property (RIP) of the centered column-wise self Khatri-Rao (KR) products of n× 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× N-matrix which contains as columns the vectorized (self) outer products of the columns of the original n× 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(-cn) provided that s≲ n^2/^2(eN/n^2). Our result is applicable in various works on covariance matching like in activity detection and MIMO gain-estimation.

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