Manifold denoising by Nonlinear Robust Principal Component Analysis

Rongrong Wang, Ming Yan, He Lyu, Yuying Xie, Ningyu Sha, Shuyang Qin

Abstract: The paper extends the idea of Robust Principal Component Analysis to nonlinear manifolds. Suppose the data matrix contains a sparse component and a component drawn from some low dimensional manifold. Is it possible to separate the two by using the low dimensionality assumption of the manifold? Is there any benefit to treat the manifold as a whole as opposed to treating each local region individually? We answer these two questions affirmatively by proposing an optimization framework that separates the two components in noisy data. Theoretical guarantees are provided under the assumption that tangent spaces of the manifold satisfy certain incoherence condition. We also provide near optimal choices for tuning parameters in the optimization formulation with the help of the local curvature. The efficacy of the proposed method is demonstrated in both synthetic and real dataset.

Code Link: https://github.com/rrwng/NRPCA/

CMT Num: 7377