Normalized Projection and Graph Embedding via Angular Decomposition

Abstract

Dimensionality reduction plays a vital role in pattern recognition. However, for normalized vector data, existing methods do not utilize the fact that the data is normalized. In this chapter, the authors propose to employ an Angular Decomposition of the normalized vector data which corresponds to embedding them on a unit surface. On graph data for similarity/kernel matrices with constant diagonal elements, the authors propose the Angular Decomposition of the similarity matrices which corresponds to embedding objects on a unit sphere. In these angular embeddings, the Euclidean distance is equivalent to the cosine similarity. Thus data structures best described in the cosine similarity and data structures best captured by the Euclidean distance can both be effectively detected in our angular embedding. The authors provide the theoretical analysis, derive the computational algorithm, and evaluate the angular embedding on several datasets. Experiments on data clustering demonstrate that the method can provide a more discriminative subspace.