1-Bit Matrix Completion
Mark A. Davenport, Yaniv Plan, Ewout van den Berg, Mary Wootters

TL;DR
This paper develops a theoretical framework for matrix completion using only 1-bit binary measurements, showing that accurate recovery is possible under certain conditions and demonstrating practical advantages over traditional methods.
Contribution
It introduces a convex optimization approach for 1-bit matrix completion and provides near-optimal bounds, extending matrix completion theory to extreme quantization scenarios.
Findings
Maximum likelihood estimates are accurate under bounded infinity norm and low rank.
Convex programs can recover matrices from 1-bit data with theoretical guarantees.
Binary data-based methods outperform standard matrix completion in experiments.
Abstract
In this paper we develop a theory of matrix completion for the extreme case of noisy 1-bit observations. Instead of observing a subset of the real-valued entries of a matrix M, we obtain a small number of binary (1-bit) measurements generated according to a probability distribution determined by the real-valued entries of M. The central question we ask is whether or not it is possible to obtain an accurate estimate of M from this data. In general this would seem impossible, but we show that the maximum likelihood estimate under a suitable constraint returns an accurate estimate of M when ||M||_{\infty} <= \alpha, and rank(M) <= r. If the log-likelihood is a concave function (e.g., the logistic or probit observation models), then we can obtain this maximum likelihood estimate by optimizing a convex program. In addition, we also show that if instead of recovering M we simply wish to…
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