# Riemannian Patch Assignment Gradient Flows

arXiv: https://arxiv.org/abs/2504.13024 · HTML: https://mlsanity.com/paper/2504.13024/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 9 of its 23 references that are in the map.
- First posted on arXiv 17 Apr 2025.

## Builds on (most cited first, 9 of 9)

- [Image Labeling by Assignment](https://mlsanity.com/paper/1603.05285/) (2016)
- [A Bayesian Hyperprior Approach for Joint Image Denoising and Interpolation, with an Application to HDR Imaging](https://mlsanity.com/paper/1706.03261/) (2017)
- [Assignment Flows for Data Labeling on Graphs: Convergence and Stability](https://mlsanity.com/paper/2002.11571/) (2020)
- [A Geometric Embedding Approach to Multiple Games and Multiple Populations](https://mlsanity.com/paper/2401.05918/) (2024)
- [Generative Assignment Flows for Representing and Learning Joint Distributions of Discrete Data](https://mlsanity.com/paper/2406.04527/) (2024)
- [Continuous-Domain Assignment Flows](https://mlsanity.com/paper/1910.07287/) (2019)
- [Learning Adaptive Regularization for Image Labeling Using Geometric Assignment](https://mlsanity.com/paper/1910.09976/) (2019)
- [Learning Linearized Assignment Flows for Image Labeling](https://mlsanity.com/paper/2108.02571/) (2021)
- [Sigma Flows for Image and Data Labeling and Learning Structured Prediction](https://mlsanity.com/paper/2408.15946/) (2024)

## Abstract (arXiv)

This paper introduces patch assignment flows for metric data labeling on graphs. Labelings are determined by regularizing initial local labelings through the dynamic interaction of both labels and label assignments across the graph, entirely encoded by a dictionary of competing labeled patches and mediated by patch assignment variables. Maximal consistency of patch assignments is achieved by geometric numerical integration of a Riemannian ascent flow, as critical point of a Lagrangian action functional. Experiments illustrate properties of the approach, including uncertainty quantification of label assignments.