Learning Permutation Distributions via Reflected Diffusion on Ranks
arXiv:2603.17353v1 Announce Type: new 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 expr
arXiv:2603.17353v1 Announce Type: new 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.
Executive Summary
This article proposes a novel approach to learning permutation distributions, named Soft-Rank Diffusion, which leverages a structured soft-rank forward process and contextualized generalized Plackett-Luce denoisers. The model addresses the challenges of permutation diffusion methods by generating smoother and more tractable trajectories, particularly in long-sequence and intrinsically sequential settings. Experiments on sorting and combinatorial optimization benchmarks demonstrate the superiority of Soft-Rank Diffusion over prior diffusion baselines. The proposed framework has the potential to improve the efficiency and accuracy of permutation learning, with implications for a range of applications, including machine learning and optimization.
Key Points
- ▸ Soft-Rank Diffusion is a discrete diffusion framework that replaces shuffle-based corruption with a structured soft-rank forward process.
- ▸ The model introduces contextualized generalized Plackett-Luce (cGPL) denoisers for the reverse process.
- ▸ Experiments show that Soft-Rank Diffusion outperforms prior diffusion baselines, particularly in long-sequence and intrinsically sequential settings.
Merits
Strength in Addressing Challenges
Soft-Rank Diffusion effectively addresses the challenges of permutation diffusion methods by generating smoother and more tractable trajectories, particularly in long-sequence and intrinsically sequential settings.
Improved Efficiency and Accuracy
The proposed framework has the potential to improve the efficiency and accuracy of permutation learning, with implications for a range of applications.
Demerits
Potential Overfitting
The introduction of contextualized generalized Plackett-Luce (cGPL) denoisers may increase the risk of overfitting, particularly if not properly regularized.
Scalability Limitations
The performance of Soft-Rank Diffusion may be limited in very large permutation spaces, requiring further investigation into scalability.
Expert Commentary
Soft-Rank Diffusion is a significant contribution to the field of permutation learning, addressing the challenges of permutation diffusion methods and providing a novel approach to learning permutation distributions. The introduction of contextualized generalized Plackett-Luce (cGPL) denoisers and the structured soft-rank forward process are key innovations that enable the generation of smoother and more tractable trajectories. However, further investigation into the scalability and potential overfitting of the model is required to fully realize its potential. The implications of Soft-Rank Diffusion are far-reaching, with potential applications in machine learning, optimization, and beyond.
Recommendations
- ✓ Further research is needed to investigate the scalability of Soft-Rank Diffusion and to develop more effective regularization techniques to mitigate the risk of overfitting.
- ✓ The proposed framework should be applied to a range of permutation-based machine learning and optimization tasks to fully realize its potential and to demonstrate its practical implications.
