# Learning Permutation Distributions via Reflected Diffusion on Ranks

arXiv: https://arxiv.org/abs/2603.17353 · HTML: https://mlsanity.com/paper/2603.17353/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 23 of its 37 references that are in the map.
- First posted on arXiv 18 Mar 2026.

## Builds on (most cited first, 23 of 23)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Denoising Diffusion Implicit Models](https://mlsanity.com/paper/2010.02502/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Structured Denoising Diffusion Models in Discrete State-Spaces](https://mlsanity.com/paper/2107.03006/) (2021)
- [Discrete Diffusion Modeling by Estimating the Ratios of the Data Distribution](https://mlsanity.com/paper/2310.16834/) (2023)
- [Simplified and Generalized Masked Diffusion for Discrete Data](https://mlsanity.com/paper/2406.04329/) (2024)
- [Efficiently Scaling Transformer Inference](https://mlsanity.com/paper/2211.05102/) (2022)
- [DiGress: Discrete Denoising diffusion for graph generation](https://mlsanity.com/paper/2209.14734/) (2022)
- [A Continuous Time Framework for Discrete Denoising Models](https://mlsanity.com/paper/2205.14987/) (2022)
- [Fast Differentiable Sorting and Ranking](https://mlsanity.com/paper/2002.08871/) (2020)
- [Denoising Diffusion Bridge Models](https://mlsanity.com/paper/2309.16948/) (2023)
- [Learning Latent Permutations with Gumbel-Sinkhorn Networks](https://mlsanity.com/paper/1802.08665/) (2018)
- [Score-based Continuous-time Discrete Diffusion Models](https://mlsanity.com/paper/2211.16750/) (2022)
- [Likelihood-Based Diffusion Language Models](https://mlsanity.com/paper/2305.18619/) (2023)
- [Differentiable Ranks and Sorting using Optimal Transport](https://mlsanity.com/paper/1905.11885/) (2019)
- [Stochastic Optimization of Sorting Networks via Continuous Relaxations](https://mlsanity.com/paper/1903.08850/) (2019)
- [Reflected Diffusion Models](https://mlsanity.com/paper/2304.04740/) (2023)
- [SoftSort: A Continuous Relaxation for the argsort Operator](https://mlsanity.com/paper/2006.16038/) (2020)
- [Monotonic Differentiable Sorting Networks](https://mlsanity.com/paper/2203.09630/) (2022)
- [Revisit Recommender System in the Permutation Prospective](https://mlsanity.com/paper/2102.12057/) (2021)
- [SymmetricDiffusers: Learning Discrete Diffusion on Finite Symmetric Groups](https://mlsanity.com/paper/2410.02942/) (2024)
- [Generalized Neural Sorting Networks with Error-Free Differentiable Swap Functions](https://mlsanity.com/paper/2310.07174/) (2023)

## Abstract (arXiv)

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.