On the Restricted Isometry of the Columnwise Khatri-Rao Product

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 paper, we analyze the RIP of a generic columnwise Khatri-Rao product matrix by deriving two upper bounds for its kth 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 real-valued 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 x n sized random matrices comprising independent and identically distributed sub-Gaussian entries satisfies k-RIP with arbitrarily high probability, provided m exceeds O(root k log(3/2) n). This is a substantially milder condition compared to O(k log n) rows needed to guarantee k-RIP of the input sub-Gaussian random matrices participating in the Khatri-Rao product. 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 obtaining improved performance guarantees in several sparse signal recovery and tensor decomposition problems.