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This paper introduces a formal framework for diffusion modeling on quotient spaces, specifically addressing systems with inherent symmetries. By explicitly modeling the quotient space, the method avoids learning the group action component, simplifying the learning process compared to group-equivariant diffusion models. Experiments on molecular structure generation demonstrate that this quotient-space diffusion model outperforms existing symmetry treatments.
Quotient-space diffusion elegantly sidesteps the need to learn symmetry transformations, leading to more efficient and accurate generative models for systems with inherent symmetries.
Diffusion-based generative models have reformed generative AI, and have enabled new capabilities in the science domain, for example, generating 3D structures of molecules. Due to the intrinsic problem structure of certain tasks, there is often a symmetry in the system, which identifies objects that can be converted by a group action as equivalent, hence the target distribution is essentially defined on the quotient space with respect to the group. In this work, we establish a formal framework for diffusion modeling on a general quotient space, and apply it to molecular structure generation which follows the special Euclidean group $\text{SE}(3)$ symmetry. The framework reduces the necessity of learning the component corresponding to the group action, hence simplifies learning difficulty over conventional group-equivariant diffusion models, and the sampler guarantees recovering the target distribution, while heuristic alignment strategies lack proper samplers. The arguments are empirically validated on structure generation for small molecules and proteins, indicating that the principled quotient-space diffusion model provides a new framework that outperforms previous symmetry treatments.
