On the Restricted Isometry of the Columnwise Khatri-Rao Product

TL;DR

This paper analyzes the restricted isometry property (RIP) of columnwise Khatri-Rao product matrices, providing bounds and probabilistic guarantees that are useful for sparse recovery in linear inverse problems.

Contribution

It derives new upper bounds for the RIP constants of Khatri-Rao product matrices, including probabilistic bounds for random matrices with subgaussian entries.

Findings

Khatri-Rao product matrices satisfy RIP with high probability when dimensions are sufficiently large.

The Khatri-Rao product exhibits stronger RIP than its constituent matrices for the same order.

The bounds are useful for analyzing sample complexity in sparse recovery problems.

Abstract

The columnwise Khatri-Rao product of two matrices is an important matrix type, reprising its role as a structured sensing matrix in many fundamental linear inverse problems. Robust signal recovery in such inverse problems is often contingent on proving the restricted isometry property (RIP) of a certain system matrix expressible as a Khatri-Rao product of two matrices. In this work, we analyze the RIP of a generic columnwise Khatri-Rao product matrix by deriving two upper bounds for its $k^{\text{th}}$ order Restricted Isometry Constant ($k$-RIC) for different values of $k$. The first RIC bound is computed in terms of the individual RICs of the input matrices participating in the Khatri-Rao product. The second RIC bound is probabilistic, and is specified in terms of the input matrix dimensions. We show that the Khatri-Rao product of a pair of $m \times n$ sized random matrices comprising independent and identically distributed subgaussian entries satisfies $k$-RIP with arbitrarily high probability, provided $m$ exceeds $O(k \log n)$. Our RIC bounds confirm that the Khatri-Rao product exhibits stronger restricted isometry compared to its constituent matrices for the same RIP order. The proposed RIC bounds are potentially useful in the sample complexity analysis of several sparse recovery problems.