Open Access
Translator Disclaimer
2010 Some Remarks on Diffusion Distances
Maxim J. Goldberg, Seonja Kim
J. Appl. Math. 2010: 1-17 (2010). DOI: 10.1155/2010/464815


As a diffusion distance, we propose to use a metric (closely related to cosine similarity) which is defined as the ${L}^{2}$ distance between two ${L}^{2}$-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 the ${L}^{2}$ distance between certain ${L}^{1}$ 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.


Download Citation

Maxim J. Goldberg. Seonja Kim. "Some Remarks on Diffusion Distances." J. Appl. Math. 2010 1 - 17, 2010.


Published: 2010
First available in Project Euclid: 1 November 2010

zbMATH: 1218.60074
MathSciNet: MR2720540
Digital Object Identifier: 10.1155/2010/464815

Rights: Copyright © 2010 Hindawi


Vol.2010 • 2010
Back to Top