Measuring Spatial Clustering via Metropolis-Hastings Diffusion Distance

What the math gives to ML

The paper introduces a graph-constrained notion of distributional discrepancy based on how rapidly a distribution f converges to a target distribution g under a Metropolis-Hastings chain. The transferable asset is the combination of a target-preserving Markov operator, multi-step convergence measurements, and spectral mixing behavior, which gives a tunable way to detect or penalize spatially concentrated probability mass. A practical neural-network adaptation is to regularize attention or routing distributions using multi-step MH diffusion distance rather than only one-hop smoothness or entropy. This can control locality and clustering while respecting arbitrary graph geometry and a nonuniform desired marginal.