In this paper, we study the popularly dubbed matrix completion problem, where the task is to “fill in” the unobserved entries of a matrix from a small subset of observed entries, under the assumption that the underlying matrix is of low rank. Our contributions herein enhance our prior work on nuclear norm regularized problems for matrix completion (Mazumder et al. in J Mach Learn Res 1532(11):2287–2322, 2010) by incorporating a continuum of nonconvex penalty functions between the convex nuclear norm and nonconvex rank functions. Inspired by Soft-Impute (Mazumder et al. 2010; Hastie et al. in J Mach Learn Res, 2016), we propose NC-Impute—an EM-flavored algorithmic framework for computing a family of nonconvex penalized matrix completion problems with warm starts. We present a systematic study of the associated spectral thresholding operators, which play an important role in the overall algorithm. We study convergence properties of the algorithm. Using structured low-rank SVD computations, we demonstrate the computational scalability of our proposal for problems up to the Netflix size (approximately, a 500,000 \(\times \) 20,000 matrix with \(10^8\) observed entries). We demonstrate that on a wide range of synthetic and real data instances, our proposed nonconvex regularization framework leads to low-rank solutions with better predictive performance when compared to those obtained from nuclear norm problems. Implementations of algorithms proposed herein, written in the R language, are made available on github.

中文翻译：

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非凸正则化的矩阵完成：频谱算子和可扩展算法

在本文中，我们研究了流行的配音矩阵完成问题，其中的任务是在基本矩阵的秩较低的前提下，从一小部分观测到的条目中“填充”矩阵的未观测到的条目。我们在本文中的贡献通过在凸核范数和凸核范数之间引入连续的非凸罚函数，加强了我们对矩阵完成的核规范正则化问题的先前工作（Mazumder等人，J Mach Learn Res 1532（11）：2287–2322，2010）。非凸秩函数。受软性冲击的启发 （Mazumder等人2010; Hastie等人在J Mach Learn Res中发表，2016），我们提出了NC-Impute— EM风格的算法框架，用于计算带有热启动的一系列非凸惩罚矩阵完成问题。我们对相关的频谱阈值运算符进行了系统的研究，它们在整个算法中起着重要的作用。我们研究了该算法的收敛性。使用结构化的低秩SVD计算，我们演示了针对Netflix大小的问题（大约500,000 \（\ times \）  20,000矩阵和\（10 ^ 8 \）的问题）的建议的计算可伸缩性。 观察到的条目）。我们证明，与从核规范问题获得的结果相比，我们提出的非凸正则化框架可在各种合成和实际数据实例中提供具有更好预测性能的低秩解决方案。本文提出的用R语言编写的算法的实现在github上可用。