Papers › SymmetricDiffusers: Learning Discrete Diffusion on Finite Symmetric Groups

SymmetricDiffusers: Learning Discrete Diffusion on Finite Symmetric Groups

3 Oct 2024arXiv:2410.02942archive 2025-07-28

Yongxing Zhang, Donglin Yang, Renjie Liao

Finite symmetric groups Sₙ are essential in fields such as combinatorics, physics, and chemistry. However, learning a probability distribution over Sₙ poses significant challenges due to its intractable size and discrete nature. In this paper, we introduce SymmetricDiffusers, a novel discrete diffusion model that simplifies the task of learning a complicated distribution over Sₙ by decomposing it into learning simpler transitions of the reverse diffusion using deep neural networks. We identify the riffle shuffle as an effective forward transition and provide empirical guidelines for selecting the diffusion length based on the theory of random walks on finite groups. Additionally, we propose a generalized Plackett-Luce (PL) distribution for the reverse transition, which is provably more expressive than the PL distribution. We further introduce a theoretically grounded "denoising schedule" to improve sampling and learning efficiency. Extensive experiments show that our model achieves state-of-the-art or comparable performances on solving tasks including sorting 4-digit MNIST images, jigsaw puzzles, and traveling salesman problems. Our code is released at https://github.com/NickZhang53/SymmetricDiffusers.

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