An Explicit Sparse Mapping for Nonlinear Dimensionality Reduction

A disadvantage of most nonlinear dimensionality reduction methods is that there are no explicit mappings to project high-dimensional features into low-dimensional representation space. Previously, some methods have been proposed to provide explicit mappings for nonlinear dimensionality reduction methods. Nevertheless, a disadvantage of these methods is that the learned mapping functions are combinations of all the original features, thus it is often difficult to interpret the results. In addition, the dense projection matrices of these approaches will cause a high cost of storage and computation. In this paper, a framework based on L1-norm regularization is presented to learn explicit sparse polynomial mappings for nonlinear dimensionality reduction. By using this framework and the method of locally linear embedding, we derive an explicit sparse nonlinear dimensionality reduction algorithm, which is named sparse neighborhood preserving polynomial embedding. Experimental results on real world classification and clustering problems demonstrate the effectiveness of our approach.