Yang-Mills Meets Data

TL;DR

This work develops a discrete gauge-theoretic framework for data on graphs, modeling node features as sections of a discrete vector bundle and employing gauge-invariant heat kernels to study data transformations. It introduces a gauged Laplacian $L_{\chi}$ and gauged heat kernel $K_{\chi}^t$ on voltage graphs $\chi=(\eta,\kappa,\omega)$, linking the kernel to synchronizability and the underlying graph topology. A discrete Yang-Mills analysis is built around holonomy and an extended energy $\mathcal{YM}_{e}$, enabling topological obstructions to synchronization to be detected via graph homology. The framework unifies smooth and discrete vector-bundle approaches, connects to existing gauge-invariant data methods, and points to ML applications where $\kappa$ and $\omega$ serve as learnable, gauge-consistent parameters, with YM energy acting as a geometric regularizer and topological diagnostic.

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.

Figures (13)

• Figure 1: Illustration of the gauge invariant feature data model employed in this paper. Left: Visualization of node feature data $x \in X =\mathbb{R}^{N \times d}$ over a graph $\eta$ with vertex set $\nu$ of cardinality $|\nu| = N$. Features are represented by matrices $x\in X$, with rows $x_{i}$ indexed by the vertices $i\in\nu$. Right: The scenario on the left changes when gauge invariant feature vectors$\phi$ over $\eta$ are considered. The feature space $X$ is extended to the space of discrete bundle sections$\Gamma$. A section $\phi\in\Gamma$ assigns to every node $i\in\nu$ a gauge invariant feature vector $\phi_i=[\xi_{i},x_{i}]$, where $x_{i} \in \mathbb{R}^{d}$ is a regular feature vector, and $\xi_{i}$ is an element of $\mathop{\mathrm{SO}}\nolimits(d)$; see Section \ref{['sec:bundles']} for details. Gauge invariant feature vectors $\phi_{i}$ interact with each other along the edges of the underlying graph, where the interaction is mediated in terms of graph voltages$\alpha \in \mathcal{A}$ (a matrix $\alpha_{jk} \in \mathop{\mathrm{SO}}\nolimits(d)$ for every edge $j\sim k$ of $\eta$) and associated gauged Laplacians$L_{\chi}$, as defined and studied in Section \ref{['sec:hks']}.
• Figure 2: Local rotations applied to an array of MNIST digits. The relevant graph here is a 8 $\times$ 8 grid graph with node feature vectors in $\mathbb{R}^{784}$ (visualized by MNIST images). Thus, the relevant feature space is given by $X = \mathbb{R}^{64 \times 784}$ and the local rotation is performed by acting with the group $\mathop{\mathrm{SO}}\nolimits(784)^{64}$, or more precisely, via a $\mathop{\mathrm{U}}\nolimits(1)^{64}$ subgroup of $\mathop{\mathrm{SO}}\nolimits(784)^{64}$.
• Figure 3: Examples of non-synchronizable voltages: a graph with $\dim \ker L_{\chi} = 0$ (left panel) and $\dim \ker L_{\chi} = 1$ (right panel) for the groups $\mathop{\mathrm{U}}\nolimits(1)$ and $\mathop{\mathrm{SO}}\nolimits(3)$ respectively.
• Figure 4: Visualization of heat kernel transformation $K^{t}_{\chi}$applied to the data shown by the left-most panel, for various time steps $t=\{0,0.5,2.5,25\}$, computed using a simple Euler scheme. The $\mathbb{R}^{3}$ node feature vectors are visualized in terms of colors. The given $\mathop{\mathrm{SO}}\nolimits(3)$-voltage $\alpha$ configuration is depicted in the right panel of Figure \ref{['fig:examples for graphs']}. The heat kernel transformation causes the projection of the initial configuration over time to the one-dimensional nullspace of $L_{\chi}$. This contrast with the typical long-time transformations of classical graph Laplacians to constant output and also highlights a non-dissipative tendency of the gauged heat kernel: the channel-wise means of the initial data are not preserved during diffusion.
• Figure 5: The edges $\varepsilon_1$ of the complement graph of a spanning tree $\eta_{0}$ of the grid graph $\eta$, marked with red.

Theorems & Definitions (100)

• Remark 2.2: $\Gamma$ versus $X$
• Proposition 2.3: gauge invariant transformations
• proof
• Corollary 2.4: characterization of gauge invariant transformations
• proof
• Remark 2.5: subgroup-gauge invariance
• Definition 2.6: discrete vector bundle
• Remark 2.8: local symmetry properties
• Definition 2.9: discrete bundle sections
• Remark 2.10: connection to the fiber bundle formalism