Exact and Stable Covariance Estimation from Quadratic Sampling via Convex Programming

Statistical inference and information processing of high-dimensional data often require efficient and accurate estimation of their second-order statistics. With rapidly changing data, limited processing power and storage at the sensor suite, it is desirable to extract the covariance structure from a single pass over the data stream and a small number of measurements. In this paper, we explore a quadratic random measurement model which imposes a minimal memory requirement and low computational complexity during the sampling process, and is shown to be optimal in preserving low-dimensional covariance structures. Specifically, four popular structural assumptions of covariance matrices, namely low rank, Toeplitz low rank, sparsity, jointly rank-one and sparse structure, are investigated. We show that a covariance matrix with either structure can be perfectly recovered from a near-optimal number of sub-Gaussian quadratic measurements, via efficient convex relaxation algorithms for the respective structure. The proposed algorithm has a variety of potential applications in streaming data processing, high-frequency wireless communication, phase space tomography in optics, non-coherent subspace detection, etc. Our method admits universally accurate covariance estimation in the absence of noise, as soon as the number of measurements exceeds the theoretic sampling limits. We also demonstrate the robustness of this approach against noise and imperfect structural assumptions. Our analysis is established upon a novel notion called the mixed-norm restricted isometry property (RIP-$\ell_{2}/\ell_{1}$), as well as the conventional RIP-$\ell_{2}/\ell_{2}$ for near-isotropic and bounded measurements. Besides, our results improve upon best-known phase retrieval (including both dense and sparse signals) guarantees using PhaseLift with a significantly simpler approach.