Stable compressive low rank Toeplitz covariance estimation without regularization

This paper considers the problem of reconstructing a N × N low rank positive semidefinite Toeplitz matrix from a noisy compressed sketch of size O(√r) × O (√r) where r << N is the rank of the matrix. A novel algorithm is proposed which only exploits a positive semidefinite (PSD) constraint to denoise the compressed sketch using a simple least squares approach. A major advantage of our algorithm is that it does not require any regularization parameter. The PSD constraint, along with Vandermonde representation of PSD Toeplitz matrices are proved to be sufficient for stable reconstruction in presence of bounded noise.

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