Finite sample analysis of covariance compression using structured samplers

This paper considers the problem of compressively sampling wide sense stationary random vectors with low rank Toeplitz structured covariance matrix. Using the celebrated Caratheodory's theorem, Toeplitz structured covariance matrix recovery can be cast as line spectrum estimation problem. In this paper, we utilize this connection to establish theoretical guarantees under which low rank Toeplitz covariance matrices can be compressively sketched and reconstructed from a finite number of compressed samples. Using a newly proposed structured sampler, namely the Generalized Nested Sampler (GNS), we show that stable estimation of original N × N Toeplitz covariance matrix of rank r can be obtained from a compressed sketch of size O(√r) × O(√r) using an atomic norm minimization framework.