SymmetricDiffusers: Learning Discrete Diffusion on Finite Symmetric Groups

TL;DR

SymmetricDiffusers is a novel discrete diffusion model designed for learning distributions over finite symmetric groups, leveraging group theory, effective forward transitions, and expressive reverse distributions to improve sampling efficiency and performance on combinatorial tasks.

Contribution

The paper introduces SymmetricDiffusers, a new discrete diffusion framework for symmetric groups, with a generalized Plackett-Luce distribution and a theoretically grounded denoising schedule.

Findings

Achieves state-of-the-art performance on sorting 4-digit MNIST.

Effectively solves jigsaw puzzles and traveling salesman problems.

Provides empirical guidelines for diffusion length selection.

Abstract

Finite symmetric groups $S_n$ are essential in fields such as combinatorics, physics, and chemistry. However, learning a probability distribution over $S_n$ 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 $S_n$ 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/DSL-Lab/SymmetricDiffusers.