# Learning Linearized Assignment Flows for Image Labeling

arXiv: https://arxiv.org/abs/2108.02571 · HTML: https://mlsanity.com/paper/2108.02571/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 5 of its 30 references that are in the map.
- First posted on arXiv 2 Aug 2021.

## Builds on (most cited first, 5 of 5)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [Image Labeling by Assignment](https://mlsanity.com/paper/1603.05285/) (2016)
- [Assignment Flows for Data Labeling on Graphs: Convergence and Stability](https://mlsanity.com/paper/2002.11571/) (2020)

## Built on by (most cited first, 2 of 2)

- [A Nonlocal Graph-PDE and Higher-Order Geometric Integration for Image Labeling](https://mlsanity.com/paper/2205.03991/) (2022)
- [Riemannian Patch Assignment Gradient Flows](https://mlsanity.com/paper/2504.13024/) (2025)

## Abstract (arXiv)

We introduce a novel algorithm for estimating optimal parameters of linearized assignment flows for image labeling. An exact formula is derived for the parameter gradient of any loss function that is constrained by the linear system of ODEs determining the linearized assignment flow. We show how to efficiently evaluate this formula using a Krylov subspace and a low-rank approximation. This enables us to perform parameter learning by Riemannian gradient descent in the parameter space, without the need to backpropagate errors or to solve an adjoint equation. Experiments demonstrate that our method performs as good as highly-tuned machine learning software using automatic differentiation. Unlike methods employing automatic differentiation, our approach yields a low-dimensional representation of internal parameters and their dynamics which helps to understand how assignment flows and more generally neural networks work and perform.