Rethinking Diffusion Models with Symmetries through Canonicalization with Applications to Molecular Graph Generation

TL;DR

This paper introduces a canonicalization approach for diffusion models that leverages symmetry properties, improving efficiency and performance in molecular graph generation by mapping samples to canonical forms before modeling.

Contribution

It provides a formal theory of canonical diffusion, demonstrating its universality and expressivity, and applies it to molecular graph generation with state-of-the-art results.

Findings

Canonical diffusion outperforms equivariant baselines in 3D molecule generation.

Canonicalization accelerates training and reduces complexity.

State-of-the-art performance achieved on GEOM-DRUG dataset.

Abstract

Many generative tasks in chemistry and science involve distributions invariant to group symmetries (e.g., permutation and rotation). A common strategy enforces invariance and equivariance through architectural constraints such as equivariant denoisers and invariant priors. In this paper, we challenge this tradition through the alternative canonicalization perspective: first map each sample to an orbit representative with a canonical pose or order, train an unconstrained (non-equivariant) diffusion or flow model on the canonical slice, and finally recover the invariant distribution by sampling a random symmetry transform at generation time. Building on a formal quotient-space perspective, our work provides a comprehensive theory of canonical diffusion by proving: (i) the correctness, universality and superior expressivity of canonical generative models over invariant targets; (ii) canonicalization accelerates training by removing diffusion score complexity induced by group mixtures and reducing conditional variance in flow matching. We then show that aligned priors and optimal transport act complementarily with canonicalization and further improves training efficiency. We instantiate the framework for molecular graph generation under $S_n \times SE(3)$ symmetries. By leveraging geometric spectra-based canonicalization and mild positional encodings, canonical diffusion significantly outperforms equivariant baselines in 3D molecule generation tasks, with similar or even less computation. Moreover, with a novel architecture Canon, CanonFlow achieves state-of-the-art performance on the challenging GEOM-DRUG dataset, and the advantage remains large in few-step generation.