TY - JOUR
A2 - Pickering, Andrew
AU - Goldberg, Maxim J.
AU - Kim, Seonja
PY - 2010
DA - 2010/09/23
TI - Some Remarks on Diffusion Distances
SP - 464815
VL - 2010
AB - As a diffusion distance, we propose to use a metric (closely related to cosine similarity) which is defined as the L2 distance between two L2-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 L2 distance between certain L1 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.
SN - 1110-757X
UR - https://doi.org/10.1155/2010/464815
DO - 10.1155/2010/464815
JF - Journal of Applied Mathematics
PB - Hindawi Publishing Corporation
KW -
ER -