Regularising the ricci flow embedding

Abstract

This paper concerns the analysis of patterns that are specified in terms of non-Euclidean dissimilarity or proximity rather than ordinal values. In prior work we have reported a means of correcting or rectifying the similarities so that the non-Euclidean artifacts are minimized. This is achieved by representing the data using a graph, and evolving the manifold embedding of the graph using Ricci flow. Although the method provides encouraging results, it can prove to be unstable. In this paper we explore how this problem can be overcome using a graph regularisation technique. Specifically, by regularising the curvature of the manifold on which the graph is embedded, then we can improve both the stability and performance of the method. We demonstrate the utility of our method on the standard "Chicken pieces" dataset and show that we can transform the non-Euclidean distances into Euclidean space. © 2010 Springer-Verlag Berlin Heidelberg.