  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 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 examples are
provided to validate the proposed approach.
