Local Smoothing for Manifold Learning

We propose methods for outlier handling and noise reduction using 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 step to obtain a more accurate reconstruction of the underlying nonlinear manifolds. Weighted PCA is used as a building block for our methods and we suggest an iterative weight selection scheme for 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. Synthetic examples along with several image data sets are presented to show that manifold learning methods combined with weighted local linear smoothing give more accurate results.