Open Access
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.


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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
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