Permutation-Symmetrized Diffusion for Unconditional Molecular Generation

Permutation invariance is fundamental in molecular point-cloud generation, yet most diffusion models enforce it indirectly via permutation-equivariant networks on an ordered space. We propose to model diffusion directly on the quotient manifold $\tilde{\calX}=\sR^{d\times N}/S_N$, where all atom permutations are identified. We show that the heat kernel on $\tilde{\calX}$ admits an explicit expression as a sum of Euclidean heat kernels over permutations, which clarifies how diffusion on the quotient differs from ordered-particle diffusion. Training requires a permutation-symmetrized score involving an intractable sum over $S_N$; we derive an expectation form over a posterior on permutations and approximate it using MCMC in permutation space. We evaluate on unconditional 3D molecule generation on QM9 under the EQGAT-Diff protocol, using SemlaFlow-style backbone and treating all variables continuously. The results demonstrate that quotient-based permutation symmetrization is practical and yields competitive generation quality with improved efficiency.

Learning Infinitesimal Generators of Continuous Symmetries from Data

Exploiting symmetry inherent in data can significantly improve the sample efficiency of a learning procedure and the generalization of learned models. When data clearly reveals underlying symmetry, leveraging this symmetry can naturally inform the design of model architectures or learning strategies. Yet, in numerous real-world scenarios, identifying the specific symmetry within a given data distribution often proves ambiguous. To tackle this, some existing works learn symmetry in a data-driven manner, parameterizing and learning expected symmetry through data. However, these methods often rely on explicit knowledge, such as pre-defined Lie groups, which are typically restricted to linear or affine transformations. In this paper, we propose a novel symmetry learning algorithm based on transformations defined with one-parameter groups, continuously parameterized transformations flowing along the directions of vector fields called infinitesimal generators. Our method is built upon minimal inductive biases, encompassing not only commonly utilized symmetries rooted in Lie groups but also extending to symmetries derived from nonlinear generators. To learn these symmetries, we introduce a notion of a validity score that examine whether the transformed data is still valid for the given task. The validity score is designed to be fully differentiable and easily computable, enabling effective searches for transformations that achieve symmetries innate to the data. We apply our method mainly in two domains: image data and partial differential equations, and demonstrate its advantages. Our codes are available at \url{https://github.com/kogyeonghoon/learning-symmetry-from-scratch.git}.