Lyu He, Sha Ningyu, Qin Shuyang, Yan Ming, Xie Yuying, Wang Rongrong
Department of Computational Mathematics, Science and Engineering Michigan State University.
Adv Neural Inf Process Syst. 2019 Dec;32.
This paper extends robust principal component analysis (RPCA) to nonlinear manifolds. Suppose that the observed data matrix is the sum of a sparse component and a component drawn from some low dimensional manifold. Is it possible to separate them by using similar ideas as RPCA? Is there any benefit in treating the manifold as a whole as opposed to treating each local region independently? We answer these two questions affirmatively by proposing and analyzing an optimization framework that separates the sparse component from the manifold under noisy data. Theoretical error bounds are provided when the tangent spaces of the manifold satisfy certain incoherence conditions. We also provide a near optimal choice of the tuning parameters for the proposed optimization formulation with the help of a new curvature estimation method. The efficacy of our method is demonstrated on both synthetic and real datasets.
本文将鲁棒主成分分析（RPCA）扩展到非线性流形。假设观测数据矩阵是一个稀疏分量与一个从某个低维流形中抽取的分量之和。是否有可能通过使用与RPCA类似的思路来将它们分离？将流形作为一个整体来处理而非独立地处理每个局部区域是否有任何益处？我们通过提出并分析一个在有噪声数据下将稀疏分量与流形分离的优化框架，对这两个问题给出了肯定的回答。当流形的切空间满足某些不相干条件时，给出了理论误差界。我们还借助一种新的曲率估计方法，为所提出的优化公式提供了调谐参数的近似最优选择。我们的方法在合成数据集和真实数据集上都证明了其有效性。
