Learning Permutation Distributions via Reflected Diffusion on Ranks

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.

Machine Learning (cs.LG); Artificial Intelligence (cs.AI)

arXiv:2603.17353 [cs.LG]

(or arXiv:2603.17353v2 [cs.LG] for this version)

https://doi.org/10.48550/arXiv.2603.17353 Focus to learn more arXiv-issued DOI via DataCite

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