Solving low-rank matrix completion problems efficiently

D. Goldfarb, Shiqian Ma, Zaiwen Wen
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引用次数: 21

Abstract

We present several first-order algorithms for solving the low-rank matrix completion problem and the tightest convex relaxation of it obtained by minimizing the nuclear norm instead of the rank of the matrix. Our first algorithm is a fixed point continuation algorithm that incorporates an approximate singular value decomposition procedure (FPCA). FPCA can solve large matrix completion problems efficiently and attains high rates of recoverability. For example, FPCA can recover 1000 by 1000 matrices of rank 50 with a relative error of 10−5 in about 3 minutes by sampling only 20% of the elements. We know of no other method that achieves as good recoverability. Our second algorithm is a row by row method for solving a semidefinite programming reformulation of the nuclear norm matrix completion problem. This method produces highly accurate solutions to fairly large nuclear norm matrix completion problems efficiently. Finally, we introduce an alternating direction approach based on the augmented Lagrangian framework.
有效求解低秩矩阵补全问题
本文给出了求解低秩矩阵补全问题的几种一阶算法,并通过最小化核范数而不是最小化矩阵的秩得到了低秩矩阵补全问题的最紧凸松弛。我们的第一个算法是包含近似奇异值分解过程(FPCA)的不动点延拓算法。FPCA可以有效地解决大矩阵完井问题,并获得较高的采收率。例如,FPCA可以在大约3分钟内通过采样20%的元素恢复1000 × 1000秩为50的矩阵,相对误差为10−5。据我们所知,没有其他方法能达到如此好的可恢复性。我们的第二种算法是求解核范数矩阵补全问题的半定规划重表述的逐行方法。该方法可有效地对较大的核范数矩阵补全问题产生高精度的解。最后,我们介绍了一种基于增广拉格朗日框架的交替方向方法。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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