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。

精确低秩约束下的一比特矩阵补全

1-Bit Matrix Completion under Exact Low-Rank Constraint

This paper studies a class of exponential family models whose canonical
parameters are specified as linear functionals of an unknown
infinite-dimensional slope function. The optimal minimax rates of convergence
for slope function estimation are established. The estimators that achieve the
optimal rates are constructed by constrained maximum likelihood estimation with
parameters whose dimension grows with sample size. A change-of-measure
argument, inspired by Le Cam's theory of asymptotic equivalence, is used to
eliminate the bias caused by the non-linearity of exponential family models.

研究了一类指数族模型，其规范参数被指定为未知的无限维斜率函数的线性泛函。建立了斜率函数估计的最优最小值收敛速率。构造了实现最优速率的估计量，这些估计量通过带有样本大小增长的参数的受限最大似然估计来构造。通过类似于 Le Cam 渐近等价理论的测度变换论证，消除了指数族模型非线性引起的偏差。