We consider the problem of noisy 1-bit matrix completion under an exact rank
constraint on the true underlying matrix $M^*$. Instead of observing a subset
of the noisy continuous-valued entries of a matrix $M^*$, we observe a subset
of noisy 1-bit (or binary) measurements generated according to a probabilistic
model. We consider constrained maximum likelihood estimation of $M^*$, under a
constraint on the entry-wise infinity-norm of $M^*$ and an exact rank
constraint. This is in contrast to previous work which has used convex
relaxations for the rank. We provide an upper bound on the matrix estimation
error under this model. Compared to the existing results, our bound has faster
convergence rate with matrix dimensions when the fraction of revealed 1-bit
observations is fixed, independent of the matrix dimensions. We also propose an
iterative algorithm for solving our nonconvex optimization with a certificate
of global optimality of the limiting point. This algorithm is based on low rank
factorization of $M^*$. We validate the method on synthetic and real data with
improved performance over existing methods.

本文提出利用低秩分解完成具有确切秩约束的嘈杂 1 位矩阵完成问题，并研究了在入口无限范数和确切秩约束下的最大似然估计，对于现有结果，本文提供了更快的收敛速率与矩阵维度无关的 1 位观察比例。