Nonconvex Matrix Completion with Linearly Parameterized Factors
Nonconvex Matrix Completion with Linearly Parameterized Factors
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发表时间：
2020-03
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通讯作者：
Ji Chen;Xiaodong Li;Zongming Ma
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作者：
Ji Chen;Xiaodong Li;Zongming Ma
矩阵补全技术旨在通过一小部分已观测到的数据来填补数据矩阵中大量缺失的元素，在机器学习中有广泛应用，包括协同过滤、成对排序等。在实际应用中，通常会采用额外的结构来提高矩阵补全的准确性。例如，协同过滤中由辅助信息形成的子空间约束，以及成对排序中的斜对称。本文对具有线性参数化分解的非凸矩阵补全进行了统一分析，上述例子是其特殊情况。重要的是，在采样率满足由真实低秩矩阵的秩、条件数和不相干参数所确定的某些条件时，为所有局部极小值建立了估计误差的一致上界。数值模拟进一步说明了所提方法的经验效率。
Techniques of matrix completion aim to impute a large portion of missing entries in a data matrix through a small portion of observed ones, with broad machine learning applications including collaborative filtering, pairwise ranking, etc. In practice, additional structures are usually employed in order to improve the accuracy of matrix completion. Examples include subspace constraints formed by side information in collaborative filtering, and skew symmetry in pairwise ranking. This paper performs a unified analysis of nonconvex matrix completion with linearly parameterized factorization, which covers the aforementioned examples as special cases. Importantly, uniform upper bounds for estimation errors are established for all local minima, provided that the sampling rate satisfies certain conditions determined by the rank, condition number, and incoherence parameter of the ground-truth low rank matrix. Empirical efficiency of the proposed method is further illustrated by numerical simulations.