Orthogonal Neighborhood Preserving Projections: A Projection-Based Dimensionality Reduction Technique

This paper considers the problem of dimensionality reduction by orthogonal projection techniques. The main feature of the proposed techniques is that they attempt to preserve both the intrinsic neighborhood geometry of the data samples and the global geometry. In particular, we propose a method, named orthogonal neighborhood preserving projections, which works by first building an "affinity" graph for the data in a way that is similar to the method of locally linear embedding (LLE). However, in contrast with the standard LLE where the mapping between the input and the reduced spaces is implicit, ONPP employs an explicit linear mapping between the two. As a result, handling new data samples becomes straightforward, as this amounts to a simple linear transformation. We show how we can define kernel variants of ONPP, as well as how we can apply the method in a supervised setting. Numerical experiments are reported to illustrate the performance of ONPP and to compare it with a few competing methods.