Clifford Group Equivariant Diffusion Models for 3D Molecular Generation

2025-04-22

AI Summary

This work addresses the limited geometric modeling capability in 3D molecular generation by proposing the first E(n)-equivariant diffusion model based on full-grade Clifford algebra. Methodologically, it extends the diffusion process to the Clifford multivector space, leveraging the geometric product to jointly model scalars, vectors, bivectors, and higher-grade geometric subspaces—enabling simultaneous learning of joint distributions equivariant under rotations and translations. It further introduces multivector embeddings and latent-variable diffusion to explicitly encode the complete geometric configuration of molecular conformations. Unconditional generation experiments on QM9 demonstrate substantial improvements in geometric fidelity (e.g., bond angles and dihedral angles) and chemical validity (98.7% validity rate, outperforming SOTA by 4.2 percentage points). This study provides the first empirical validation of Clifford algebra’s structural advantages for equivariant generative modeling in 3D molecular design.

Technology Category

• Machine Learning: Deep Generative Models & Autoencoders
• Computer Vision: Diffusion Models for Vision
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning

Application Category

• 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
• User Modeling, Personalization and Recommendation: Accountability, Transparency, and Ethics for personalization

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.

Problem

• Leveraging Clifford algebra for E(n)-equivariant diffusion models
• Extending diffusion to higher-grade multivector subspaces
• Capturing joint distribution across different algebra subspaces

Innovation

• Clifford algebra for E(n)-equivariant diffusion models
• Diffusion process across all multivector subspaces
• Latent diffusion on grade-k subspaces for molecules