Intelligent Initialization and Adaptive Thresholding for Iterative Matrix Completion: Some Statistical and Algorithmic Theory for <i>Adaptive-Impute</i>

Over the past decade, various matrix completion algorithms have been developed. Thresholded singular value decomposition (SVD) is a popular technique in implementing many of them. A sizable number of studies have shown its theoretical and empirical excellence, but choosing the right threshold level still remains as a key empirical difficulty. This article proposes a novel matrix completion algorithm which iterates thresholded SVD with theoretically justified and data-dependent values of thresholding parameters. The estimate of the proposed algorithm enjoys the minimax error rate and shows outstanding empirical performances. The thresholding scheme that we use can be viewed as a solution to a nonconvex optimization problem, understanding of whose theoretical convergence guarantee is known to be limited. We investigate this problem by introducing a simpler algorithm, generalized- <i>softImpute</i>, analyzing its convergence behavior, and connecting it to the proposed algorithm.近十年来，各类矩阵补全算法相继被提出。阈值化奇异值分解（Singular Value Decomposition, SVD）是实现多数此类算法的主流技术。已有大量研究证实该方法在理论与实证层面的优异性能，但如何选取恰当的阈值仍是核心实践难题。本文提出一种全新的矩阵补全算法，该算法通过迭代阈值化SVD实现，其阈值参数的取值兼具理论合理性与数据依赖性。所提算法的估计结果满足极小极大误差率要求，且展现出卓越的实证性能。我们采用的阈值化方案可视为非凸优化问题的一种求解方式，但目前学界对其理论收敛性保证的认知仍较为有限。为此，本文引入一种更简洁的算法——广义softImpute，通过分析其收敛特性并将其与所提算法建立关联，以此对该问题展开研究。

搜集汇总

背景与挑战

背景概述

该数据集是关于Adaptive-Impute矩阵补全算法的学术研究补充材料，包括论文PDF和原始数据ZIP文件，旨在提出一种结合智能初始化和自适应阈值化的迭代算法，以解决阈值化SVD中的参数选择难题，并展示其理论最优性和卓越的实证性能。数据集覆盖多个学科领域，如空间科学、生物技术和数学科学。