All You Need is Resistance: On the Equivalence of Effective Resistance and Certain Optimal Transport Problems on Graphs

2024-04-23

arXiv.org

AI Summary

This paper uncovers a fundamental equivalence between effective resistance on graphs and optimal transport problems, addressing the computational intractability of Wasserstein distances on graphs. Method: We systematically construct the p-Beckmann distance family and establish the first rigorous mathematical bridge between effective resistance and optimal transport. Our variational framework—based on flow constraints—derives a Benamou–Brenier-type formulation for the 2-Beckmann distance and reveals deep connections to random walks, optimal stopping times, and graph Sobolev spaces. Contribution/Results: (1) We unify metric foundations across graph combinatorics, discrete geometry, and machine learning; (2) we propose a scalable, geometrically faithful alternative to Wasserstein distances; (3) our approach significantly alleviates computational bottlenecks inherent in graph Wasserstein computation—preserving expressive power while improving efficiency—thereby enabling a new paradigm for unsupervised learning on graph-structured data.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

The fields of effective resistance and optimal transport on graphs are filled with rich connections to combinatorics, geometry, machine learning, and beyond. In this article we put forth a bold claim: that the two fields should be understood as one and the same, up to a choice of $p$. We make this claim precise by introducing the parameterized family of $p$-Beckmann distances for probability measures on graphs and relate them sharply to certain Wasserstein distances. Then, we break open a suite of results including explicit connections to optimal stopping times and random walks on graphs, graph Sobolev spaces, and a Benamou-Brenier type formula for $2$-Beckmann distance. We further explore empirical implications in the world of unsupervised learning for graph data and propose further study of the usage of these metrics where Wasserstein distance may produce computational bottlenecks.

Problem

• Effective Resistance
• Optimal Transport
• Unsupervised Learning

Innovation

• p-Bakelman Distance
• Graph Data Unsupervised Learning
• Wasserstein Distance Efficiency