  In this paper, we consider the matrix completion problem when the
observations are one-bit measurements of some underlying matrix M, and in
particular the observed samples consist only of ones and no zeros. This problem
is motivated by modern applications such as recommender systems and social
networks where only "likes" or "friendships" are observed. The problem of
learning from only positive and unlabeled examples, called PU
(positive-unlabeled) learning, has been studied in the context of binary
classification. We consider the PU matrix completion problem, where an
underlying real-valued matrix M is first quantized to generate one-bit
observations and then a subset of positive entries is revealed. Under the
assumption that M has bounded nuclear norm, we provide recovery guarantees for
two different observation models: 1) M parameterizes a distribution that
generates a binary matrix, 2) M is thresholded to obtain a binary matrix. For
the first case, we propose a "shifted matrix completion" method that recovers M
using only a subset of indices corresponding to ones, while for the second
case, we propose a "biased matrix completion" method that recovers the
(thresholded) binary matrix. Both methods yield strong error bounds --- if M is
n by n, the Frobenius error is bounded as O(1/((1-rho)n), where 1-rho denotes
the fraction of ones observed. This implies a sample complexity of O(n\log n)
ones to achieve a small error, when M is dense and n is large. We extend our
methods and guarantees to the inductive matrix completion problem, where rows
and columns of M have associated features. We provide efficient and scalable
optimization procedures for both the methods and demonstrate the effectiveness
of the proposed methods for link prediction (on real-world networks consisting
of over 2 million nodes and 90 million links) and semi-supervised clustering
tasks.
