Markov Random Walk Representations with Continuous Distributions

TL;DR

This paper introduces a novel framework for Markov random walk representations that utilize continuous data distributions by employing a diffusion equation with density-dependent coefficients, enhancing clustering and classification tasks.

Contribution

It extends traditional discrete random walk representations to continuous distributions using a diffusion equation linked to a path integral, allowing better modeling of continuous data densities.

Findings

Framework effectively incorporates continuous data densities.

Diffusion coefficient inversely depends on data density.

Enhanced clustering and classification performance.

Abstract

Representations based on random walks can exploit discrete data distributions for clustering and classification. We extend such representations from discrete to continuous distributions. Transition probabilities are now calculated using a diffusion equation with a diffusion coefficient that inversely depends on the data density. We relate this diffusion equation to a path integral and derive the corresponding path probability measure. The framework is useful for incorporating continuous data densities and prior knowledge.