Subspace Learning From Bits

Networked sensing, where the goal is to perform complex inference using a large number of inexpensive and decentralized sensors, has become an increasingly attractive research topic due to its applications in wireless sensor networks and internet-of-things. To reduce the communication, sensing, and storage complexity, this paper proposes a simple sensing and estimation framework to faithfully recover the principal subspace of high-dimensional data streams using a collection of binary measurements from distributed sensors, without transmitting the whole data. The binary measurements are designed to indicate comparison outcomes of aggregated energy projections of the data samples over pairs of randomly selected directions. When the covariance matrix is a low-rank matrix, we propose a spectral estimator that recovers the principal subspace of the covariance matrix as the subspace spanned by the top eigenvectors of a properly designed surrogate matrix, which is provably accurate as soon as the number of binary measurements is sufficiently large. An adaptive rank selection strategy based on soft thresholding is also presented. Furthermore, we propose a tailored spectral estimator when the covariance matrix is additionally Toeplitz, and show that the reliable estimation can be obtained from a substantially smaller number of binary measurements. Our results hold even when a constant fraction of the binary measurements is randomly flipped. Finally, we develop a low-complexity online algorithm to track the principal subspace when new measurements arrive sequentially. Numerical experiments are provided to validate the proposed approach.