On anisotropic diffusion equations for label propagation

TL;DR

This paper introduces a microscopic ODE-based method for label propagation in data classification, deriving an anisotropic diffusion PDE that models label spreading with properties inherited from the data structure.

Contribution

It presents a novel microscopic ODE approach and derives a continuum anisotropic diffusion equation for label propagation, connecting agent-based dynamics with PDE models.

Findings

Continuum model inherits properties of the point cloud.

Microscopic and macroscopic simulations validate the approach.

Anisotropic diffusion with reaction term effectively propagates labels.

Abstract

In many problems in data classification one wishes to assign labels to points in a point cloud with a certain number of them being already correctly labeled. In this paper, we propose a microscopic ODE approach, in which information about correct labels is propagated to neighboring points. Its dynamics are based on alignment mechanisms, which are commonly used in large interacting agent systems in consensus formation. We derive the respective continuum description, which corresponds to an anisotropic diffusion equation with reaction term. Solutions of the continuum model on the bounded domain inherit certain properties of the underlying point cloud. We discuss these analytic properties and exemplify the results with micro- and macroscopic simulations.