Local Linear Smoothing for Nonlinear Manifold Learning

In this paper, we develop methods for outlier removal and noise reduction based on weighted local linear smoothing for a set of noisy points sampled from a nonlinear manifold. The methods can be used by manifold learning methods such as Isomap, LLE and LTSA as a preprocessing procedure so as to obtain a more accurate reconstruction of the underlying nonlinear manifolds. Weighted principal component analysis is used as a building block of our methods and we develop an iterative weight selection scheme that leads to robust local linear fitting. We also develop an efficient and effective bias-reduction method to deal with the trim the peak and fill the valley phenomenon in local linear smoothing. Several illustrative examples are presented to show that nonlinear manifold learning methods combined with weighted local linear smoothing give more accurate reconstruction of the underlying nonlinear manifolds.