Categorical Trace Loop Networks for Gauge-Randomized Holonomy Regression

Yoshihiro Maruyama
Proceedings of GRaM: the Second Edition of the Workshop on Geometry-grounded Representation Learning and Generative Modeling, PMLR 326:398-415, 2026.

Abstract

Gauge ambiguity is a pervasive obstacle when learning from group-valued transport data: edge measurements depend on arbitrary local coordinate choices, while the quantities we care about are gauge-invariant. Inspired by lattice gauge theory, where meaningful observables are built from loop holonomies rather than individual edge variables, we study a {SO(3)} learning problem on a discrete torus with random vertex-wise gauges. We adopt a categorical viewpoint in which an edge connection defines holonomy functorially on edge-paths of the 1-skeleton; in the flat/noiseless regime, this holonomy descends to the fundamental groupoid of the torus cell complex. Gauge transformations act as natural isomorphisms. This viewpoint leads us to {Categorical Trace Loop Networks} ({CTLN}): a novel architecture that learns from loop- and face-based gauge invariants obtained by functorial holonomy composition and trace/angle scalarization. On gauge-randomized torus holonomy regression, {CTLN} achieves a test {MAE} of 0.1747, while a standard message passing network and a spectral connection-{Laplacian} baseline both remain near 0.81 {MAE}. These results show that in gauge-dominated regimes, learning on categorical invariants capturing global topology and higher-order consistency provides a highly effective method.

Cite this Paper

BibTeX
@InProceedings{pmlr-v326-maruyama26a, title = {Categorical Trace Loop Networks for Gauge-Randomized Holonomy Regression}, author = {Maruyama, Yoshihiro}, booktitle = {Proceedings of GRaM: the Second Edition of the Workshop on Geometry-grounded Representation Learning and Generative Modeling}, pages = {398--415}, year = {2026}, editor = {Pouplin, Alison and Vadgama, Sharvaree and Bekkers, Erik and Kaba, Sékou-Oumar and Lawrence, Hannah and Lecha, Manuel and Baker, Elizabeth and Suk, Julian and Walters, Robin and Tomczak, Jakub and Jegelka, Stefanie}, volume = {326}, series = {Proceedings of Machine Learning Research}, month = {26 Apr}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v326/main/assets/maruyama26a/maruyama26a.pdf}, url = {https://proceedings.mlr.press/v326/maruyama26a.html}, abstract = {Gauge ambiguity is a pervasive obstacle when learning from group-valued transport data: edge measurements depend on arbitrary local coordinate choices, while the quantities we care about are gauge-invariant. Inspired by lattice gauge theory, where meaningful observables are built from loop holonomies rather than individual edge variables, we study a {SO(3)} learning problem on a discrete torus with random vertex-wise gauges. We adopt a categorical viewpoint in which an edge connection defines holonomy functorially on edge-paths of the 1-skeleton; in the flat/noiseless regime, this holonomy descends to the fundamental groupoid of the torus cell complex. Gauge transformations act as natural isomorphisms. This viewpoint leads us to {Categorical Trace Loop Networks} ({CTLN}): a novel architecture that learns from loop- and face-based gauge invariants obtained by functorial holonomy composition and trace/angle scalarization. On gauge-randomized torus holonomy regression, {CTLN} achieves a test {MAE} of 0.1747, while a standard message passing network and a spectral connection-{Laplacian} baseline both remain near 0.81 {MAE}. These results show that in gauge-dominated regimes, learning on categorical invariants capturing global topology and higher-order consistency provides a highly effective method.} }
Endnote
%0 Conference Paper %T Categorical Trace Loop Networks for Gauge-Randomized Holonomy Regression %A Yoshihiro Maruyama %B Proceedings of GRaM: the Second Edition of the Workshop on Geometry-grounded Representation Learning and Generative Modeling %C Proceedings of Machine Learning Research %D 2026 %E Alison Pouplin %E Sharvaree Vadgama %E Erik Bekkers %E Sékou-Oumar Kaba %E Hannah Lawrence %E Manuel Lecha %E Elizabeth Baker %E Julian Suk %E Robin Walters %E Jakub Tomczak %E Stefanie Jegelka %F pmlr-v326-maruyama26a %I PMLR %P 398--415 %U https://proceedings.mlr.press/v326/maruyama26a.html %V 326 %X Gauge ambiguity is a pervasive obstacle when learning from group-valued transport data: edge measurements depend on arbitrary local coordinate choices, while the quantities we care about are gauge-invariant. Inspired by lattice gauge theory, where meaningful observables are built from loop holonomies rather than individual edge variables, we study a {SO(3)} learning problem on a discrete torus with random vertex-wise gauges. We adopt a categorical viewpoint in which an edge connection defines holonomy functorially on edge-paths of the 1-skeleton; in the flat/noiseless regime, this holonomy descends to the fundamental groupoid of the torus cell complex. Gauge transformations act as natural isomorphisms. This viewpoint leads us to {Categorical Trace Loop Networks} ({CTLN}): a novel architecture that learns from loop- and face-based gauge invariants obtained by functorial holonomy composition and trace/angle scalarization. On gauge-randomized torus holonomy regression, {CTLN} achieves a test {MAE} of 0.1747, while a standard message passing network and a spectral connection-{Laplacian} baseline both remain near 0.81 {MAE}. These results show that in gauge-dominated regimes, learning on categorical invariants capturing global topology and higher-order consistency provides a highly effective method.
APA
Maruyama, Y.. (2026). Categorical Trace Loop Networks for Gauge-Randomized Holonomy Regression. Proceedings of GRaM: the Second Edition of the Workshop on Geometry-grounded Representation Learning and Generative Modeling, in Proceedings of Machine Learning Research 326:398-415 Available from https://proceedings.mlr.press/v326/maruyama26a.html.

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