Yang-Mills Meets Data

TL;DR

This paper introduces a geometric gauge theory framework for data analysis using discrete vector bundles, connecting gauge symmetry, heat kernels, and graph Laplacians to unify existing methods and provide new insights.

Contribution

It develops a discrete gauge theory approach for data representation, linking lattice gauge theory concepts with data analysis and unifying various vector bundle-based methods.

Findings

Connection between gauge symmetric heat kernels and graph Laplacians.

Use of discrete Yang-Mills energy to analyze heat kernel properties.

Insights into data transformation via the nullspace of gauged Laplacians.

Abstract

Gauge symmetric methods for data representation and analysis utilize tools from the differential geometry of vector bundles in order to achieve consistent data processing architectures with respect to local symmetry and equivariance. In this work, we elaborate concepts of geometric gauge theory for data science. Motivated by lattice gauge theory, we focus on discrete descriptions of vector bundles for data representation and analysis, with clear relations to the established mathematical bundle formalism. Our approach unifies various existing approaches to data processing via vector bundles, within the framework of gauge theory. We provide geometric insights into gauge symmetric heat kernel operators that are closely related to graph connection Laplacians, and into their data transformation properties in terms of the non-trivial nullspace of the corresponding gauged Laplacians. In particular, we utilize a discrete Yang-Mills energy in order to characterize significant properties of the heat kernel in terms of associated synchronization problems.