提出高效算法解决非随机缺失矩阵补全问题
Computational Efficient and Minimax Optimal Nonignorable Matrix Completion
- 用核范数正则化行列表示损失函数处理非随机缺失
- 计算效率接近随机缺失方法，统计收敛率近最优
- 适合处理复杂缺失机制的实证研究与数据建模
尽管矩阵补全问题已受到广泛关注，但针对非随机缺失问题的研究仍寥寥无几，且现有方法存在局限。本文提出一种基于核范数正则化的行-列联合U统计量损失函数，适用于广义非随机缺失机制——这一灵活通用的缺失机制包含可忽略和不可忽略两种情形。所提方法在计算效率上与现有的随机缺失方法相当，同时在更一般的非随机缺失场景下实现了近极小极大最优的统计收敛速率。我们设计了一种加速近端梯度算法求解相关优化问题，并刻画了算法收敛与统计收敛之间的相互作用关系。模拟实验与真实数据分析进一步验证了该方法的实际有效性。
原文摘要 · Abstract (English)
While the matrix completion problem has attracted considerable attention over the decades, few works address the nonignorable missing issue and all have their limitations. In this article, we propose a nuclear norm regularized row- and column-wise matrix U-statistic loss function for the generalized nonignorable missing mechanism, a flexible and generally applicable missing mechanism which contains both ignorable and nonignorable missing mechanism assumptions. The proposed method achieves computational efficiency comparable to the existing missing-at-random approaches, while providing the near minimax optimal statistical convergence rate guarantees for the more general nonignorable missing case. We propose an accelerated proximal gradient algorithm to solve the associated optimization problem, and characterize the interaction between algorithmic and statistical convergence. Simulations and real data analyzes further support the practical utility of the proposed method.
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