Learning Permutation Distributions via Reflected Diffusion on Ranks

2026-03-18

AI Summary

Learning probability distributions over the symmetric group \( S_n \) is challenged by combinatorial explosion and the discrete, non-Euclidean structure of permutations, particularly causing existing diffusion methods to suffer from inefficient denoising on long sequences. This work proposes the Soft-Rank Diffusion framework, which maps discrete permutations to continuous representations via soft-rank relaxation and constructs a structured forward diffusion process. To enhance the expressiveness of reverse-time generation, the authors introduce a contextual generalized Plackett–Luce (cGPL) denoiser. By abandoning conventional shuffling mechanisms and incorporating reflection-based diffusion with soft-rank representations, the method achieves substantial performance gains over current diffusion models in ranking and combinatorial optimization tasks, especially excelling in scenarios involving long sequences and strong sequential dependencies.

Technology Category

• Machine Learning: Learning Preferences or Rankings
• Computer Vision: Diffusion Models for Vision
• Planning, Routing, and Scheduling: Planning with Language Models

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs

Abstract

The finite symmetric group S_n provides a natural domain for permutations, yet learning probability distributions on S_n is challenging due to its factorially growing size and discrete, non-Euclidean structure. Recent permutation diffusion methods define forward noising via shuffle-based random walks (e.g., riffle shuffles) and learn reverse transitions with Plackett-Luce (PL) variants, but the resulting trajectories can be abrupt and increasingly hard to denoise as n grows. We propose Soft-Rank Diffusion, a discrete diffusion framework that replaces shuffle-based corruption with a structured soft-rank forward process: we lift permutations to a continuous latent representation of order by relaxing discrete ranks into soft ranks, yielding smoother and more tractable trajectories. For the reverse process, we introduce contextualized generalized Plackett-Luce (cGPL) denoisers that generalize prior PL-style parameterizations and improve expressivity for sequential decision structures. Experiments on sorting and combinatorial optimization benchmarks show that Soft-Rank Diffusion consistently outperforms prior diffusion baselines, with particularly strong gains in long-sequence and intrinsically sequential settings.

Problem

• permutation distributions
• symmetric group
• discrete diffusion
• Plackett-Luce
• combinatorial optimization

Innovation

• Soft-Rank Diffusion
• Permutation Distributions
• Reflected Diffusion
• Generalized Plackett-Luce
• Discrete Diffusion