Low-Rank Covariance Matrix Recovery From Rank-One Measurements
An Analytical Solution
Peilan Wang (University of Electronic Science and Technology of China)
Jun Fang (University of Electronic Science and Technology of China)
Binyao Ma (University of Electronic Science and Technology of China)
Bin Wang (University of Electronic Science and Technology of China)
Geert Leus (TU Delft - Electrical Engineering, Mathematics and Computer Science)
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Abstract
In this letter, we propose an analytical solution for recovering a low-rank positive semi-definite (PSD) matrix from its rank-one measurements. We show that by utilizing a set of structured measurement vectors, we can analytically determine the null space of this low-rank PSD matrix. Based on the result, the PSD matrix can be efficiently recovered. Our analysis shows that the proposed method only requires (N - K)(2K + 1)+ K
2 measurements to guarantee exact recovery of the PSD matrix, where N and K respectively denote the dimension and the rank of the PSD matrix. Numerical results show that the proposed method achieves a considerable improvement over existing state-of-the-art methods in terms of both sample complexity and computational efficiency. Specifically, the proposed method helps improve the computational efficiency by an order of magnitude as compared with existing methods.