Manifold Denoising by Nonlinear Robust Principal Component Analysis
This work addresses data denoising for applications where data resides on manifolds, offering a novel extension of RPCA, though it is incremental in building upon existing linear methods.
The paper tackles the problem of separating sparse noise from data lying on a nonlinear manifold, extending robust principal component analysis (RPCA) to handle such structures, and provides theoretical error bounds and a parameter tuning method, with efficacy shown on synthetic and real datasets.
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.