Incremental Manifold Learning Algorithm Using PCA on Overlapping Local Neighborhoods for Dimensionality Reduction

Abstract

A novel manifold learning algorithm called LPcaML is proposed in this paper. Based on the geometric intuition that d-dimensional manifold locally lies on or close to d-dimensional linear space, LPcaML first finds an α-TSLN of the whole high-dimensional input data set and then obtains the low-dimensional local coordinates of each neighborhood in the α-TSLN using classical PCA technique while preserving the local geometric and topological property of each neighborhood. At last LPcaML transforms each local coordinates to a unified global low-dimensional representation by processing each neighborhood in their order appeared in α-TSLN. And the transformation function of each neighborhood is obtained by solving a least square problem via the overlapped examples. By using the divide and conquer strategy, LPcaML can learn from incremental data and discover the underlying manifold efficiently even if the data set is large scale. Experiments on both synthetic data sets and real face data sets demonstrate the effectiveness of our LPcaML algorithm. Moreover the proposed LPcaML can discover the manifold from sparsely sampled data sets where other manifold learning algorithms can’t.