Generalized nested sampling for compression and exact recovery of symmetric Toeplitz matrices

Abstract

This paper considers the problem of estimating the symmetric and Toeplitz covariance matrix of compressive samples of wide sense stationary random vectors. A new structured deterministic sampling method known as the "Generalized Nested Sampling" is introduced which enables compressive quadratic sampling of symmetric Toeplitz matrices., by fully exploiting the inherent redundancy in the Toeplitz matrix. For a Toeplitz matrix of size N ×N, this sampling scheme can attain a compression factor of O(√N) even without assuming sparsity and/or low rank, and allows exact recovery of the original Toeplitz matrix. When the matrix is sparse, a new hybrid sampling approach is proposed which efficiently combines Generalized Nested Sampling and Random Sampling to attain even greater compression rates, which, under suitable conditions can be as large as O(√N), using a novel observation formulated in this paper.