  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.
