Some Remarks on Diffusion Distances

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 towhy the normalizationmakes diffusion distancesmoremeaningful. 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.