Rectifying Non-Euclidean Similarity Data Using Ricci Flow Embedding

Similarity based pattern recognition is concerned with the analysis of patterns that are specified in terms of object dissimilarity or proximity rather than ordinal values. For many types of data and measures, these dissimilarities are not Euclidean. This hinders the use of many machine-learning techniques. In this paper, we provide a means of correcting or rectifying the similarities so that the non-Euclidean artifacts are minimized. We consider the data to be embedded as points on a curved manifold and then evolve the manifold so as to increase its flatness. Our work uses the idea of Ricci flow on the constant curvature Riemannian manifold to modify the Gaussian curvatures on the edges of a graph representing the non-Euclidean data. 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. Keywords-similarity; embedding; Ricci flow;

• Type: Journal
• Year: 2010
• Where: ICPR
• Authors: Weiping Xu, Edwin R. Hancock, Richard C. Wilson
• Keywords: Computer Vision, Curvature Riemannian Manifold, ICPR 2010, Non-Euclidean, Similarity Based Pattern