Tensor Restricted Isometry Property Analysis For a Large Class of Random Measurement Ensembles

TL;DR

This paper proves the existence of measurement maps satisfying the tensor Restricted Isometry Property (t-RIP) with high probability, enabling robust low-tubal-rank tensor recovery under nearly optimal measurement conditions.

Contribution

It establishes probabilistic conditions for measurement maps satisfying t-RIP, including a broad class of distributions, and verifies measurement bounds through numerical experiments.

Findings

Measurement maps satisfying t-RIP exist with high probability.

Minimal measurements are nearly optimal relative to tensor degrees of freedom.

Numerical experiments confirm the theoretical measurement bounds.

Abstract

In previous work, theoretical analysis based on the tensor Restricted Isometry Property (t-RIP) established the robust recovery guarantees of a low-tubal-rank tensor. The obtained sufficient conditions depend strongly on the assumption that the linear measurement maps satisfy the t-RIP. In this paper, by exploiting the probabilistic arguments, we prove that such linear measurement maps exist under suitable conditions on the number of measurements in terms of the tubal rank r and the size of third-order tensor n1, n2, n3. And the obtained minimal possible number of linear measurements is nearly optimal compared with the degrees of freedom of a tensor with tubal rank r. Specially, we consider a random sub-Gaussian distribution that includes Gaussian, Bernoulli and all bounded distributions and construct a large class of linear maps that satisfy a t-RIP with high probability. Moreover, the validity of the required number of measurements is verified by numerical experiments.