# A Nonlocal Graph-PDE and Higher-Order Geometric Integration for Image Labeling

arXiv: https://arxiv.org/abs/2205.03991 · HTML: https://mlsanity.com/paper/2205.03991/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 14 of its 62 references that are in the map.
- First posted on arXiv 9 May 2022.

## Builds on (most cited first, 14 of 14)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [First Order Methods beyond Convexity and Lipschitz Gradient Continuity with Applications to Quadratic Inverse Problems](https://mlsanity.com/paper/1706.06461/) (2017)
- [Analysis of Optimization Algorithms via Integral Quadratic Constraints: Nonstrongly Convex Problems](https://mlsanity.com/paper/1705.03615/) (2017)
- [A Comparative Study of Modern Inference Techniques for Structured Discrete Energy Minimization Problems](https://mlsanity.com/paper/1404.0533/) (2014)
- [Image Labeling by Assignment](https://mlsanity.com/paper/1603.05285/) (2016)
- [Accelerating the DC algorithm for smooth functions](https://mlsanity.com/paper/1507.07375/) (2015)
- [Assignment Flows for Data Labeling on Graphs: Convergence and Stability](https://mlsanity.com/paper/2002.11571/) (2020)
- [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)
- [Unsupervised Assignment Flow: Label Learning on Feature Manifolds by Spatially Regularized Geometric Assignment](https://mlsanity.com/paper/1904.10863/) (2019)
- [Assignment Flow for Order-Constrained OCT Segmentation](https://mlsanity.com/paper/2009.04632/) (2020)
- [Learning Linearized Assignment Flows for Image Labeling](https://mlsanity.com/paper/2108.02571/) (2021)
- [Self-Assignment Flows for Unsupervised Data Labeling on Graphs](https://mlsanity.com/paper/1911.03472/) (2019)
- [PDE Evolutions for M-Smoothers in One, Two, and Three Dimensions](https://mlsanity.com/paper/2007.13191/) (2020)

## Abstract (arXiv)

This paper introduces a novel nonlocal partial difference equation (G-PDE) for labeling metric data on graphs. The G-PDE is derived as nonlocal reparametrization of the assignment flow approach that was introduced in textitJ.~Math.~Imaging & Vision 58(2), 2017. Due to this parameterization, solving the G-PDE numerically is shown to be equivalent to computing the Riemannian gradient flow with respect to a nonconvex potential. We devise an entropy-regularized difference-of-convex-functions (DC) decomposition of this potential and show that the basic geometric Euler scheme for integrating the assignment flow is equivalent to solving the G-PDE by an established DC programming scheme. Moreover, the viewpoint of geometric integration reveals a basic way to exploit higher-order information of the vector field that drives the assignment flow, in order to devise a novel accelerated DC programming scheme. A detailed convergence analysis of both numerical schemes is provided and illustrated by numerical experiments.