Some Remarks on Diffusion Distances

Abstract

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.