Diffusion and Superposition Distances for Signals Supported on Networks

TL;DR

This paper introduces diffusion and superposition distances as new metrics for comparing signals on networks, demonstrating their theoretical validity, stability, and practical utility in classification tasks including cancer histology and digit recognition.

Contribution

It proposes two novel metrics for network-supported signals, proves their mathematical properties, and showcases their application in classification and feature transformation.

Findings

Both distances are valid metrics and stable to network perturbations.

They improve classification accuracy in gene mutation profiles and handwritten digits.

Numerical experiments validate their effectiveness in real-world and synthetic data.

Abstract

We introduce the diffusion and superposition distances as two metrics to compare signals supported in the nodes of a network. Both metrics consider the given vectors as initial temperature distributions and diffuse heat trough the edges of the graph. The similarity between the given vectors is determined by the similarity of the respective diffusion profiles. The superposition distance computes the instantaneous difference between the diffused signals and integrates the difference over time. The diffusion distance determines a distance between the integrals of the diffused signals. We prove that both distances define valid metrics and that they are stable to perturbations in the underlying network. We utilize numerical experiments to illustrate their utility in classifying signals in a synthetic network as well as in classifying ovarian cancer histologies using gene mutation profiles of different patients. We also reinterpret diffusion as a transformation of interrelated feature spaces and use it as preprocessing tool for learning. We use diffusion to increase the accuracy of handwritten digit classification.