Abstract
This article studies the approximate recovery of low-rank matrices acquired through binary measurements. Two types of recovery algorithms are considered, one based on hard singular value thresholding and the other one based on semidefinite programming. In case no thresholds are introduced before binary quantization, it is first shown that the direction of the low-rank matrices can be well approximated. Then, in case nonadaptive thresholds are incorporated, it is shown that both the direction and the magnitude can be well approximated. Finally, by allowing the thresholds to be chosen adaptively, we exhibit a recovery procedure for which low-rank matrices are fully approximated with error decaying exponentially with the number of binary measurements. In all cases, the procedures are robust to prequantization error. The underlying arguments are essentially deterministic: they rely only on an unusual restricted isometry property of the measurement process, which is established once and for all for Gaussian measurement processes.
Highlights
Low-rank matrices appear throughout science and engineering in topics as diverse as recommender systems, system identification, quantum-state tomography, etc
We aim to provide some theoretical justification for easy-to-implement algorithms to recover low-rank matrices from few binary measurements
Y = sgn(AX − τ ) = [sgn( Ai, X F − τi)]m i=1 ∈ {±1}m. This scenario of low-rank matrix recovery from binary measurements is a natural extension to the scenario of the sparse vector recovery from binary measurements and the techniques used here are adapted from the ones used there
Summary
Low-rank matrices appear throughout science and engineering in topics as diverse as recommender systems, system identification, quantum-state tomography, etc. Low-rank recovery, one-bit compressive sensing, quantization, hard singular value thresholding, semidefinite programming, adaptivity. Y = sgn(AX − τ ) = [sgn( Ai, X F − τi)]m i=1 ∈ {±1}m This scenario of low-rank matrix recovery from binary measurements is a natural extension to the scenario of the sparse vector recovery from binary measurements (and of one-bit compressive sensing) and the techniques used here are adapted from the ones used there.
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