Robust Low-Rank Matrix Completion via an Alternating Manifold Proximal Gradient Continuation Method
Robust Low-Rank Matrix Completion via an Alternating Manifold Proximal Gradient Continuation Method
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DOI：
10.1109/tsp.2021.3073544
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发表时间：
2020-08
影响因子：
5.4
通讯作者：
Minhui Huang;Shiqian Ma;L. Lai
中科院分区：
文献类型：
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作者：
Minhui Huang;Shiqian Ma;L. Lai
鲁棒低秩矩阵完备（RMC），或部分观测数据的鲁棒主成分分析，已被广泛研究用于计算机视觉，信号处理和机器学习应用。该问题旨在将部分观测矩阵分解为低秩矩阵和稀疏矩阵的叠加，其中稀疏矩阵捕获矩阵中严重损坏的元素。一种广泛使用的解决RMC的方法是考虑凸公式，它最小化低秩矩阵的核范数（以促进低秩）加上稀疏矩阵的$\ell _1$范数（以促进稀疏性）。本文在低秩矩阵完备化和黎曼优化的基础上，将该问题转化为Grassmann流形上的非光滑黎曼优化问题。这种新的公式是可扩展的，因为低秩矩阵被分解为两个小得多的矩阵的乘法。然后，我们提出了一个交替流形近端梯度延拓方法来解决所提出的新配方。算法的收敛速度进行了严格的分析。模拟数据和真实的数据的监控视频背景提取的数值结果表明，所提出的新的配方和算法的优势，几个流行的现有方法。
Robust low-rank matrix completion (RMC), or robust principal component analysis with partially observed data, has been studied extensively for computer vision, signal processing and machine learning applications. This problem aims at decomposing a partially observed matrix into the superposition of a low-rank matrix and a sparse matrix, where the sparse matrix captures the grossly corrupted entries of the matrix. A widely used approach to tackle RMC is to consider a convex formulation, which minimizes the nuclear norm of the low-rank matrix (to promote low-rankness) plus the $\ell _1$ norm of the sparse matrix (to promote sparsity). In this paper, motivated by some recent works on low-rank matrix completion and Riemannian optimization, we formulate this problem as a nonsmooth Riemannian optimization problem over Grassmann manifold. This new formulation is scalable as the low-rank matrix is factorized to the multiplication of two much smaller matrices. We then propose an alternating manifold proximal gradient continuation method to solve the proposed new formulation. Convergence rate of the proposed algorithm is rigorously analyzed. Numerical results on both synthetic data and real data on background extraction from surveillance videos are reported to demonstrate the advantages of the proposed new formulation and algorithm over several popular existing approaches.