# Rethinking Diffusion Models with Symmetries through Canonicalization with Applications to Molecular Graph Generation arXiv: https://arxiv.org/abs/2602.15022 · HTML: https://mlsanity.com/paper/2602.15022/ ## Key facts - Cited by 7 papers in the map (data through 14 Oct 2026). - Placed in Flow matching, time series and science, one of 14 directions in Diffusion models. - Builds on 67 of its 81 references that are in the map. - First posted on arXiv 16 Feb 2026. ## Placed in - Diffusion models / [Flow matching, time series and science](https://mlsanity.com/diffusion/flow-matching-time-series-and-science/) ## Builds on (most cited first, 30 of 67) - [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020) - [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021) - [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020) - [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021) - [Classifier-Free Diffusion Guidance](https://mlsanity.com/paper/2207.12598/) (2022) - [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022) - [How Powerful are Graph Neural Networks?](https://mlsanity.com/paper/1810.00826/) (2018) - [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019) - [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022) - [Video Diffusion Models](https://mlsanity.com/paper/2204.03458/) (2022) - [Make-A-Video: Text-to-Video Generation without Text-Video Data](https://mlsanity.com/paper/2209.14792/) (2022) - [Diffusion-LM Improves Controllable Text Generation](https://mlsanity.com/paper/2205.14217/) (2022) - [E(n) Equivariant Graph Neural Networks](https://mlsanity.com/paper/2102.09844/) (2021) - [Stochastic Interpolants: A Unifying Framework for Flows and Diffusions](https://mlsanity.com/paper/2303.08797/) (2023) - [Improving and generalizing flow-based generative models with minibatch optimal transport](https://mlsanity.com/paper/2302.00482/) (2023) - [Recipe for a General, Powerful, Scalable Graph Transformer](https://mlsanity.com/paper/2205.12454/) (2022) - [Equivariant Diffusion for Molecule Generation in 3D](https://mlsanity.com/paper/2203.17003/) (2022) - [Junction Tree Variational Autoencoder for Molecular Graph Generation](https://mlsanity.com/paper/1802.04364/) (2018) - [DiGress: Discrete Denoising diffusion for graph generation](https://mlsanity.com/paper/2209.14734/) (2022) - [GeoDiff: a Geometric Diffusion Model for Molecular Conformation Generation](https://mlsanity.com/paper/2203.02923/) (2022) - [DiffuSeq: Sequence to Sequence Text Generation with Diffusion Models](https://mlsanity.com/paper/2210.08933/) (2022) - [Provably Powerful Graph Networks](https://mlsanity.com/paper/1905.11136/) (2019) - [Graph Neural Networks with Learnable Structural and Positional Representations](https://mlsanity.com/paper/2110.07875/) (2021) - [DiffDock: Diffusion Steps, Twists, and Turns for Molecular Docking](https://mlsanity.com/paper/2210.01776/) (2022) - [GraphAF: a Flow-based Autoregressive Model for Molecular Graph Generation](https://mlsanity.com/paper/2001.09382/) (2020) - [GEOM: Energy-annotated molecular conformations for property prediction and molecular generation](https://mlsanity.com/paper/2006.05531/) (2020) - [Identity-aware Graph Neural Networks](https://mlsanity.com/paper/2101.10320/) (2021) - [Pure Transformers are Powerful Graph Learners](https://mlsanity.com/paper/2207.02505/) (2022) - [Geometric Latent Diffusion Models for 3D Molecule Generation](https://mlsanity.com/paper/2305.01140/) (2023) - [GraphDF: A Discrete Flow Model for Molecular Graph Generation](https://mlsanity.com/paper/2102.01189/) (2021) ## Built on by (most cited first, 7 of 7) - [Toward Better Geometric Representations for Molecule Generative Models](https://mlsanity.com/paper/2605.07693/) (2026) - [Fixed-Dimensional Latent Flow for Generating Variable-Size 3D Molecules](https://mlsanity.com/paper/2609.08333/) (2026) - [DiffATS: Diffusion in Aligned Tensor Space](https://mlsanity.com/paper/2605.09275/) (2026) - [When and How to Canonize: A Generalization Perspective](https://mlsanity.com/paper/2605.11008/) (2026) - [Terminal Symmetry as a Carrier of Asymmetric Process Knowledge: Statewise Refinement for Anytime Verified Construction](https://mlsanity.com/paper/2608.11318/) (2026) - [GLASS: Global Latent Aggregation with Slot-based Set Decoding for Scalable All-Atom Crystal Generation](https://mlsanity.com/paper/2609.37158/) (2026) - [Riemannian Flow Models with Reinforcement Learning for Molecular Crystal Structure Prediction](https://mlsanity.com/paper/2609.39773/) (2026) ## Abstract (arXiv) 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 Sn × 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.