# Learning Unbiased Permutations via Flow Matching

arXiv: https://arxiv.org/abs/2605.16755 · HTML: https://mlsanity.com/paper/2605.16755/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Flow matching, time series and science, one of 14 directions in Diffusion models.
- Builds on 18 of its 21 references that are in the map.
- First posted on arXiv 16 May 2026.

## Placed in

- Diffusion models / [Flow matching, time series and science](https://mlsanity.com/diffusion/flow-matching-time-series-and-science/)

## Builds on (most cited first, 18 of 18)

- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Scaling Rectified Flow Transformers for High-Resolution Image Synthesis](https://mlsanity.com/paper/2403.03206/) (2024)
- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Improving and generalizing flow-based generative models with minibatch optimal transport](https://mlsanity.com/paper/2302.00482/) (2023)
- [Discrete Flow Matching](https://mlsanity.com/paper/2407.15595/) (2024)
- [Generative Flows on Discrete State-Spaces: Enabling Multimodal Flows with Applications to Protein Co-Design](https://mlsanity.com/paper/2402.04997/) (2024)
- [Flow Matching on General Geometries](https://mlsanity.com/paper/2302.03660/) (2023)
- [Learning Latent Permutations with Gumbel-Sinkhorn Networks](https://mlsanity.com/paper/1802.08665/) (2018)
- [Riemannian Continuous Normalizing Flows](https://mlsanity.com/paper/2006.10605/) (2020)
- [Stochastic Optimization of Sorting Networks via Continuous Relaxations](https://mlsanity.com/paper/1903.08850/) (2019)
- [Neural Manifold Ordinary Differential Equations](https://mlsanity.com/paper/2006.10254/) (2020)
- [Optimal Flow Matching: Learning Straight Trajectories in Just One Step](https://mlsanity.com/paper/2403.13117/) (2024)
- [FlowLLM: Flow Matching for Material Generation with Large Language Models as Base Distributions](https://mlsanity.com/paper/2410.23405/) (2024)
- [Debiased Sinkhorn barycenters](https://mlsanity.com/paper/2006.02575/) (2020)
- [Neural Ordinary Differential Equations on Manifolds](https://mlsanity.com/paper/2006.06663/) (2020)
- [Learning Permutations with Sinkhorn Policy Gradient](https://mlsanity.com/paper/1805.07010/) (2018)
- [Probabilistic Permutation Synchronization using the Riemannian Structure of the Birkhoff Polytope](https://mlsanity.com/paper/1904.05814/) (2019)
- [Reparameterizing the Birkhoff Polytope for Variational Permutation Inference](https://mlsanity.com/paper/1710.09508/) (2017)

## Abstract (arXiv)

Learning permutations is fundamental to sorting, ranking, and matching, but existing differentiable methods based on entropy-regularized Sinkhorn produce a single softened solution and collapse under ambiguity. We present PermFlow, a conditional flow matching framework that operates directly on the affine subspace of matrices with unit row and column sums. A closed-form tangent-space projector preserves these constraints exactly along every trajectory, by construction rather than through iterative correction, and a nearest-target coupling routes distinct noisy initializations toward distinct valid permutations. The result is a model that captures multimodal permutation distributions rather than collapsing them to a single mode. On a visual sorting task with blended-digit ambiguity and a symmetric linear assignment problem, PermFlow achieves high accuracy on unambiguous inputs and recovers both valid permutations under ambiguity, where Sinkhorn-based baselines structurally fail.