Sparse Polynomial Mapping for Manifold Learning

Manifold learning is an approach for nonlinear dimensionality reduction and has been a hot research topic in the field of computer science. A disadvantage of manifold learning methods is, however, that there are no explicit mappings from the high-dimensional feature space to the low-dimensional representation space. It restricts the application of manifold learning methods in many practical problems such as target detection and classification. Previously, some methods have been proposed to provide linear or nonlinear mappings for manifold learning methods. However, a disadvantage of all these methods is that the learned projective functions are combinations of all the original features, thus it is often difficult to interpret the results. Moreover, the dense projection matrices of these approaches lead to a high cost of computation and storage. In this paper, a sparse polynomial mapping approach is proposed for manifold learning. We first get the low-dimensional representations of the high-dimensional input data by using a manifold learning method, and then a l1-based simplified polynomial regression is used to get a sparse polynomial mapping between the high-dimensional data and their low-dimensional representations. In particular, we apply this to the method of Laplacian eigenmap and derive a sparse nonlinear manifold learning algorithm, which is named sparse locality preserving polynomial embedding. Experimental results on real-world data show the effectiveness of our approach.