Locally Adaptive Nonlinear Dimensionality Reduction

Abstract

Popular nonlinear dimensionality reduction algorithms, e.g., SIE and Isomap suffer a difficulty in common: global neighborhood parameters often fail in tackling data sets with high variation in local manifold. To improve the availability of nonlinear dimensionality reduction algorithms in the field of machine learning, an adaptive neighbors selection scheme based on locally principal direction reconstruction is proposed in this paper. Our method involves two main computation steps. First, it selects an appropriate neighbors set for each data points such that all neighbors in a neighbors set form a d-dimensional linear subspace approximately and computes locally principal directions for each neighbors set respectively. Secondly, it fits each neighbor by means of locally principal directions of corresponding neighbors set and deletes the neighbors whose fitting error exceeds a predefined threshold. The simulation shows that our proposal could deal with data set with high variation in local manifold effectively. Moreover, comparing with other adaptive neighbors selection strategy, our method could circumvent false connectivity induced by noise or high local curvature.

Supported by Science-Technology Development Project of Tianjin(04310941R) and Applied Basic Research Project of Tianjin (i05YFJMJC11700).