标题： 具有双聚类结构的矩阵的最优估计与补全 显示英文标题

标题： Optimal Estimation and Completion of Matrices with Biclustering Structures

摘要： 数据矩阵中的双聚类结构最早由约翰·哈特igan（1972年）在一篇开创性论文中正式提出，在这篇论文中，人们试图同时对案例和变量进行聚类。这种结构在网络块建模中也很常见。 在本文中，我们开发了一种统一的理论来估计和完成具有双聚类结构的矩阵，其中数据是一个部分观测且带有噪声污染的数据矩阵，并具有某种双聚类结构。特别是，我们证明了在许多最重要的情况下，受约束的最小二乘估计量可以达到最优的渐近最小风险性能。为此，我们为所有次高斯数据推导出统一的高概率上界，并在高斯和二元情况下提供匹配的渐近最小下界。 由于图论与随机块模型之间的密切联系，我们的通用结果的一个直接推论是稀疏图论的渐近最小风险最优估计器。 显示英文摘要

摘要： Biclustering structures in data matrices were first formalized in a seminal paper by John Hartigan (1972) where one seeks to cluster cases and variables simultaneously. Such structures are also prevalent in block modeling of networks. In this paper, we develop a unified theory for the estimation and completion of matrices with biclustering structures, where the data is a partially observed and noise contaminated data matrix with a certain biclustering structure. In particular, we show that a constrained least squares estimator achieves minimax rate-optimal performance in several of the most important scenarios. To this end, we derive unified high probability upper bounds for all sub-Gaussian data and also provide matching minimax lower bounds in both Gaussian and binary cases. Due to the close connection of graphon to stochastic block models, an immediate consequence of our general results is a minimax rate-optimal estimator for sparse graphons.