Permutation-Symmetrized Diffusion for Unconditional Molecular Generation

2026-03-24

AI Summary

This work addresses the challenge of accurately modeling the permutation-invariant structure of real molecules in diffusion-based 3D molecular generation. Existing approaches typically handle symmetry indirectly via permutation-equivariant networks in ordered spaces, which fails to fully respect molecular indistinguishability. In contrast, this study proposes the first diffusion process directly defined on the permutation quotient manifold $\tilde{\mathcal{X}} = \mathbb{R}^{d \times N}/S_N$, treating atomic permutations as equivalent. The authors derive an explicit, computable permutation-symmetrized score function using heat kernel theory on the quotient manifold and approximate the required posterior expectation via MCMC sampling over the permutation group for training. Integrated with the SemlaFlow backbone and evaluated under the EQGAT-Diff protocol, the method achieves competitive unconditional 3D molecular generation quality on QM9 while improving computational efficiency, demonstrating the effectiveness and practicality of quotient-space modeling.

Technology Category

• Machine Learning: Learning with Manifolds
• Computer Vision: Diffusion Models for Vision
• Natural Language Processing: Generation

Application Category

• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Economics, Online Markets and Human Computation: Economic ramifications for generative AI infrastructure and applications

Abstract

Permutation invariance is fundamental in molecular point-cloud generation, yet most diffusion models enforce it indirectly via permutation-equivariant networks on an ordered space. We propose to model diffusion directly on the quotient manifold $\tilde{\calX}=\sR^{d\times N}/S_N$, where all atom permutations are identified. We show that the heat kernel on $\tilde{\calX}$ admits an explicit expression as a sum of Euclidean heat kernels over permutations, which clarifies how diffusion on the quotient differs from ordered-particle diffusion. Training requires a permutation-symmetrized score involving an intractable sum over $S_N$; we derive an expectation form over a posterior on permutations and approximate it using MCMC in permutation space. We evaluate on unconditional 3D molecule generation on QM9 under the EQGAT-Diff protocol, using SemlaFlow-style backbone and treating all variables continuously. The results demonstrate that quotient-based permutation symmetrization is practical and yields competitive generation quality with improved efficiency.

Problem

• permutation invariance
• molecular generation
• diffusion models
• quotient manifold
• point-cloud

Innovation

• Permutation Symmetrization
• Quotient Manifold Diffusion
• Molecular Point Cloud Generation
• MCMC on Permutations
• Permutation-Invariant Diffusion