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Spectral Diffusion for Sampling on ${\rm SU}(N)$

Gurtej Kanwar, Octavio Vega

TL;DR

This work extends score-based diffusion models to SU(N) group manifolds to address ensemble generation in lattice field theory. It introduces SU(N) specific constructions: (i) efficient sampling via the SU(N) heat kernel, with wrapped normal and Weyl character representations, and (ii) group-valued score matching and a reverse transport framework to realize unbiased sampling on the group. The authors demonstrate the method on toy SU(2) and SU(3) targets, showing accurate density reproduction and high effective sample sizes after reweighting. By providing explicit tooling for the SU(N) heat kernel and a stable training pipeline, this approach lays the groundwork for diffusion-based sampling in full SU(N) lattice gauge theories, including lattice QCD, offering a potentially scalable alternative to traditional Hybrid Monte Carlo methods.

Abstract

Although ensemble generation remains a central challenge in lattice field theory simulations, recent advances in generative modeling may offer a path to accelerated sampling in these contexts. In this work, we implement a framework for efficiently training diffusion models acting on ${\rm SU}(N)$ degrees of freedom, adapting the traditional score matching technique to the group manifold. We demonstrate that our models can effectively reproduce several target densities, resulting in precise unbiased expectation values. These results mark a step for diffusion models towards modeling full ${\rm SU}(N)$ lattice field theories, including lattice Quantum Chromodynamics.

Spectral Diffusion for Sampling on ${\rm SU}(N)$

TL;DR

This work extends score-based diffusion models to SU(N) group manifolds to address ensemble generation in lattice field theory. It introduces SU(N) specific constructions: (i) efficient sampling via the SU(N) heat kernel, with wrapped normal and Weyl character representations, and (ii) group-valued score matching and a reverse transport framework to realize unbiased sampling on the group. The authors demonstrate the method on toy SU(2) and SU(3) targets, showing accurate density reproduction and high effective sample sizes after reweighting. By providing explicit tooling for the SU(N) heat kernel and a stable training pipeline, this approach lays the groundwork for diffusion-based sampling in full SU(N) lattice gauge theories, including lattice QCD, offering a potentially scalable alternative to traditional Hybrid Monte Carlo methods.

Abstract

Although ensemble generation remains a central challenge in lattice field theory simulations, recent advances in generative modeling may offer a path to accelerated sampling in these contexts. In this work, we implement a framework for efficiently training diffusion models acting on degrees of freedom, adapting the traditional score matching technique to the group manifold. We demonstrate that our models can effectively reproduce several target densities, resulting in precise unbiased expectation values. These results mark a step for diffusion models towards modeling full lattice field theories, including lattice Quantum Chromodynamics.
Paper Structure (13 sections, 26 equations, 2 figures, 1 table)

This paper contains 13 sections, 26 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Diffusion and the Heat Equation
  3. Euclidean Diffusion
  4. Diffusion Processes on SU(N)
  5. The Heat Kernel over SU(N)
  6. Wrapped Normal Representation
  7. Dual Representation
  8. Sampling the Heat Kernel
  9. Spectral Diffusion
  10. Group-Valued Score Matching
  11. Reverse Transport
  12. Toy Example
  13. Outlook

Figures (2)

  • Figure 1: Evolution from left to right of the SU(2) spectral density on the first eigenangle under the forward diffusion process. The starting (target) density is progressively smoothed out by the heat kernel. After 1 unit of diffusion time, the diffused density closely matches the ${\rm SU}(2)$ Haar uniform density.
  • Figure 2: Reverse transport process for the toy ${\rm SU}(2)$ model with $\beta = 1.0$ and coefficient set $c^{(3)}$. The horizontal axis on each plot represents forward diffusion time, so the evolution should be interpreted as right to left. The KL divergence is seen to decrease while the ESS increases over time, signifying that the target density is being approximately reproduced by the model during sampling.