1-Bit matrix completion

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 realvalued 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 ∞ ≤ α 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 obtain an estimate of the distribution generating the 1-bit measurements, then we can eliminate the requirement that M ∞ ≤ α. For both cases, we provide lower bounds showing that these estimates are near-optimal. We conclude with a suite of experiments that both verify the implications of our theorems as well as illustrate some of the practical applications of 1-bit matrix completion. In particular, we compare our program to standard matrix completion methods on movie rating data in which users submit ratings from 1 to 5. In order to use our program, we quantize this data to a single bit, but we allow the standard matrix completion program to have access to the original ratings (from 1 to 5). Surprisingly, the approach based on binary data performs significantly better.

Quantized Massive MU-MIMO-OFDM Uplink

“…There, it was shown that amplitude information about the transmitted signal can be recovered even in the presence of a single-bit quantizer, provided that the number of receive antennas is sufficiently large and the signalto-noise ratio (SNR) is not too high. The observation that noise can help recovering magnitude information under 1-bit quantization has also been made in the compressive-sensing literature [9], [10].…”

“…In the above problem,ŷ w , w ∈ {Ω pilot ∪ Ω pilot }, are the frequency-domain entries of the matrices H w , w ∈ {Ω pilot ∪ Ω pilot } that correspond to the uth user and the bth BS antenna, which are estimated using the orthonormal pilot matrices T w (see Section IV-B). Although (MQ-CHE) has a closedform solution, 9 we will use a first-order method to reduce both computational complexity and memory requirements (see Section V). In practice, one can also obtain approximate solutions to (MQ-CHE) using more efficient methods (see e.g., [50]).…”

Individualized rank aggregation using nuclear norm regularization

“…Furthermore, the techniques outlined in the proof section our general and can be applied to a variety of sampling operators for matrix completion. For example, a simple modification of our proof yields a different class of results for the "one-bit" matrix completion problem [8].…”

Generalized Low Rank Models

“…For example, we might suppose that the entries of A are generated as noisy, 1-bit observations from an underlying low rank matrix XY . Surprisingly, it is possible to accurately estimate the underlying matrix with only a few observations |Ω| from the matrix by solving problem (18) (under a few mild technical conditions) with an appropriate loss function [DPBW12].…”