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Clifford Group Equivariant Diffusion Models for 3D Molecular Generation

Cong Liu, Sharvaree Vadgama, David Ruhe, Erik Bekkers, Patrick Forré

TL;DR

The paper tackles the challenge of generating 3D molecular structures in a way that respects $\,\mathrm{E}(3)$ symmetry by introducing Clifford Diffusion Models (CDMs) that operate on Clifford algebra multivectors. It proposes two diffusion schemes: one-vector diffusion, which diffuses grade-1 (vectors) via a Clifford-EGNN denoiser, and all-grade diffusion, which uses a latent Clifford encoder to diffuse across all grades and capture richer geometric information. Empirical results on QM9 show that CDMs are competitive with state-of-the-art $\,\mathrm{E}(3)$-equivariant diffusion methods, with all-grade diffusion often yielding higher-quality samples. The work highlights the potential of geometric-algebra–based representations to enhance generative modeling for molecular design and points to future exploration of higher-grade features in more complex scenarios.

Abstract

This paper explores leveraging the Clifford algebra's expressive power for $\E(n)$-equivariant diffusion models. We utilize the geometric products between Clifford multivectors and the rich geometric information encoded in Clifford subspaces in \emph{Clifford Diffusion Models} (CDMs). We extend the diffusion process beyond just Clifford one-vectors to incorporate all higher-grade multivector subspaces. The data is embedded in grade-$k$ subspaces, allowing us to apply latent diffusion across complete multivectors. This enables CDMs to capture the joint distribution across different subspaces of the algebra, incorporating richer geometric information through higher-order features. We provide empirical results for unconditional molecular generation on the QM9 dataset, showing that CDMs provide a promising avenue for generative modeling.

Clifford Group Equivariant Diffusion Models for 3D Molecular Generation

TL;DR

The paper tackles the challenge of generating 3D molecular structures in a way that respects symmetry by introducing Clifford Diffusion Models (CDMs) that operate on Clifford algebra multivectors. It proposes two diffusion schemes: one-vector diffusion, which diffuses grade-1 (vectors) via a Clifford-EGNN denoiser, and all-grade diffusion, which uses a latent Clifford encoder to diffuse across all grades and capture richer geometric information. Empirical results on QM9 show that CDMs are competitive with state-of-the-art -equivariant diffusion methods, with all-grade diffusion often yielding higher-quality samples. The work highlights the potential of geometric-algebra–based representations to enhance generative modeling for molecular design and points to future exploration of higher-grade features in more complex scenarios.

Abstract

This paper explores leveraging the Clifford algebra's expressive power for -equivariant diffusion models. We utilize the geometric products between Clifford multivectors and the rich geometric information encoded in Clifford subspaces in \emph{Clifford Diffusion Models} (CDMs). We extend the diffusion process beyond just Clifford one-vectors to incorporate all higher-grade multivector subspaces. The data is embedded in grade- subspaces, allowing us to apply latent diffusion across complete multivectors. This enables CDMs to capture the joint distribution across different subspaces of the algebra, incorporating richer geometric information through higher-order features. We provide empirical results for unconditional molecular generation on the QM9 dataset, showing that CDMs provide a promising avenue for generative modeling.

Paper Structure

This paper contains 11 sections, 5 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Related Work
  3. Background
  4. $\mathrm{E}(3)$ Equivariant Diffusion Models
  5. Clifford Group Equivariant Neural Networks
  6. Methods
  7. Clifford Group Equivariant One-Vector Diffusion Models
  8. Clifford Group Equivariant All-Grade Diffusion Models
  9. Experiments
  10. Conclusion
  11. Acknowledgements

Figures (1)

  • Figure 1: During sampling of Clifford all-grade diffusion models, subspace features are initialized from a Gaussian distribution, and CGENNs are used as the denoising model $\phi_\theta$ for each subspace. At time step zero, the molecular structure is read out by projecting a one-vector from Clifford space.