Some Remarks on Diffusion Distances. (13th September 2010)

• Record Type: Journal Article
• Authors: Goldberg, Maxim J.; Kim, Seonja
• Other Names: Pickering Andrew Academic Editor.
• Is Part Of: Journal of applied mathematics. Volume 2010(2010)
• Publication Date: 2010-09-13
• Subjects: Mathematics -- Periodicals; 519.05
• DOI: 10.1155/2010/464815
• Languages: English
• ISSNs: 1110-757X

Abstract

Abstract : As a diffusion distance, we propose to use a metric (closely related to cosine similarity) which is defined as theL 2 distance between twoL 2 -normalized vectors. We provide a mathematical explanation as to why the normalization makes diffusion distances more meaningful. Our proposal is in contrast to that made some years ago by R. Coifman which finds theL 2 distance between certainL 1 unit vectors. In the second part of the paper, we give two proofs that an extension of mean first passage time to mean first passage cost satisfies the triangle inequality; we do not assume that the underlying Markov matrix is diagonalizable. We conclude by exhibiting an interesting connection between the (normalized) mean first passage time and the discretized solution of a certain Dirichlet-Poisson problem and verify our result numerically for the simple case of the unit circle.