Learning orthogonal projections for Isomap

Abstract

We propose a dimensionality reduction technique in this paper, named Orthogonal Isometric Projection (OIP). In contrast with Isomap, which learns the low-dimension embedding, and solves problem under the classic Multidimensional Scaling (MDS) framework, we consider an explicit linear projection by capturing the geodesic distance, which is able to handle new data straightforward, and leads to a standard eigenvalue problem. We consider the orthogonal projection, and analyze the properties of orthogonal projection, and demonstrate the benefits, in which Euclidean distance, and angle at each pair in high-dimensional space are equivalent to ones in low-dimension, such that both global and local geometric structure are preserved. Numerical experiments are reported to demonstrate the performance of OIP by comparing with a few competing methods over different datasets.