Subspace Learning From Bits

This paper proposes a simple sensing and estimation framework to faithfully recover the principal subspace of high-dimensional datasets or data streams from a collection of one-bit measurements from distributed sensors based on comparing accumulated energy projections of their data samples of dimension n over pairs of randomly selected directions. By leveraging low-dimensional structures, the top eigenvectors of a properly designed surrogate matrix is shown to recover the principal subspace of rank $r$ as soon as the number of bit measurements exceeds the order of $nr^3 \log n$, which can be much smaller than the ambient dimension of the covariance matrix. The sample complexity to obtain reliable comparison outcomes is also obtained. Furthermore, we develop a low-complexity online algorithm to track the principal subspace that allows new bit measurements arrive sequentially. Numerical examples are provided to validate the proposed approach.