Tensor Restricted Isometry Property Analysis For a Large Class of Random Measurement Ensembles
This work provides theoretical guarantees for tensor recovery in compressed sensing, which is incremental as it extends existing analysis to a broader class of random measurement ensembles.
The paper tackles the problem of proving the existence of linear measurement maps that satisfy the tensor Restricted Isometry Property (t-RIP) for low-tubal-rank tensors, showing that such maps exist with high probability under conditions where the number of measurements is nearly optimal compared to the degrees of freedom.
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.