带信息性缺失的噪声矩阵补全

中文摘要 AI 辅助

噪声矩阵补全是统计学习中的一个基本问题，在过去二十年中引起了大量关注。在各种应用中，缺失模式具有高度信息性，但这一问题受到的关注相对较少。大多数现有方法要么忽视这一信息来源，要么过度简化其生成过程，很少有方法尝试对数据矩阵与其信息性缺失之间的关联进行建模。在本研究中，我们提出了一个通用建模框架，该框架使用一对由灵活共享线性结构耦合的低秩模型，联合对数据矩阵及其缺失模式进行建模。这种联合低秩方法利用信息性缺失来改进矩阵补全，并能灵活捕捉数据两种模式之间的各种关联。我们基于投影梯度下降为该框架开发了一种高效的联合估计程序，并建立了局部收敛保证，揭示了其计算和统计性质。我们进一步通过广泛的模拟研究和真实数据分析证明，我们提出的方法优于竞争方法，在缺失条目的预测准确性上取得了显著提升。

英文摘要

Noisy matrix completion is a fundamental problem in statistical learning and has attracted a substantial amount of interest over the last two decades. In a variety of applications, the missingness pattern is highly informative, yet it has received relatively less attention. Most existing methods either overlook this source of information or oversimplify its generating process, with few attempting to model the association between the data matrix and its informative missingness. In this study, we propose a general modeling framework that jointly models the data matrix and its missingness pattern using a pair of low-rank models coupled by a flexible shared linear structure. This joint low-rank approach incorporates the informative missingness to improve matrix completion and can flexibly capture various associations between the two modes of data. We develop an efficient joint estimation procedure for this framework based on projected gradient descent, and establish local convergence guarantees that unveil its computational and statistical properties. We further demonstrate, through extensive simulation studies and real-world data analysis, that our proposed approach outperforms competing methods, achieving substantial improvements in prediction accuracy on missing entries.

发表机构

• University of Michigan（密歇根大学）
• Pennsylvania State University（宾夕法尼亚州立大学）

机构由 AI 辅助整理，请以论文原文为准。