Principal subspace estimation for low-rank Toeplitz covariance matrices with binary sensing

In this paper, we consider the problem of estimating the principal subspace of data in decentralized sensing systems with resource constraints, where the sensors only transmit a single bit to the fusion center to minimize communication costs. In particular, the data covariance matrix is modeled as a low-rank Toeplitz positive semidefinite (PSD) matrix, which arises in applications such as array signal processing and power spectrum estimation in cognitive radios. Our algorithm is based on convex regularization using carefully designed one-bit measurements based on comparing the energy projections of the data seen at each sensor onto pairs of randomly selected Gaussian vectors. We provide finite sample performance guarantees for the accuracy of the subspace estimate in the rank-one case, even when a constant fraction of the 1-bit measurements are randomly flipped. Numerical experiments demonstrate the method continues to work well in the low-rank case, and is capable of separating close located modes from fewer bits in direction-of-arrival estimation, outperforming existing approaches that do not consider the Toeplitz structure.