Gridless Line Spectrum Estimation and Low-Rank Toeplitz Matrix Compression Using Structured Samplers: A Regularization-Free Approach

Abstract

This paper considers the problem of compressively sampling wide sense stationary random vectors with a low rank Toeplitz covariance matrix. Certain families of structured deterministic samplers are shown to efficiently compress a high-dimensional Toeplitz matrix of size N × N, producing a compressed sketch of size O(√r) × O(√r).The reconstruction problem can be cast as that of line spectrum estimation, whereby, in absence of noise, Toeplitz matrices of any size N can be exactly recovered from compressive sketches of size O(√r) × O(√r), no matter how large N is. In the presence of noise and finite data, the line spectrum estimation algorithm is combined with a novel denoising technique that only exploits a positive semidefinite (PSD) Toeplitz constraint to denoise the compressed sketch using a simple least-squares minimization framework. A major advantage of the algorithm is that it does not require any regularization parameter. It also enjoys lower computational complexity owing to its ability to predict the unobserved entries of the low-rank Toeplitz matrix. Explicit bounds on the reconstruction error are established and it is shown that the PSD constraint on the denoiser is sufficient to ensure stable reconstruction from a sketch of size O(√r) × O(√r). Extensive simulations demonstrate that the proposed algorithm provides better performance over random samplers and algorithms that use nuclear-norm-based regularizers