Some Remarks on Diffusion Distances

Maxim J. Goldberg and Seonja Kim

Journal of Applied Mathematics, 2010, vol. 2010, 1-17

Abstract:

As a diffusion distance, we propose to use a metric (closely related to cosine similarity) which is defined as the ð ¿ 2 distance between two ð ¿ 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 ð ¿ 2 distance between certain ð ¿ 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.

Date: 2010
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jnljam:464815

DOI: 10.1155/2010/464815

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