# Mumford-Shah functionals on graphs and their asymptotics

arXiv: https://arxiv.org/abs/1906.09521 · HTML: https://mlsanity.com/paper/1906.09521/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 22 Jun 2019.

## Built on by (most cited first, 6 of 6)

- [On the existence of solutions to adversarial training in multiclass classification](https://mlsanity.com/paper/2305.00075/) (2023)
- [Manifold learning in Wasserstein space](https://mlsanity.com/paper/2311.08549/) (2023)
- [Rates of Convergence for Regression with the Graph Poly-Laplacian](https://mlsanity.com/paper/2209.02305/) (2022)
- [Continuum limit of p-biharmonic equations on graphs](https://mlsanity.com/paper/2404.19689/) (2024)
- [Higher-Order Regularization Learning on Hypergraphs](https://mlsanity.com/paper/2510.26533/) (2025)
- [Convergence of graph Dirichlet energies and graph Laplacians on intersecting manifolds of varying dimensions](https://mlsanity.com/paper/2509.24458/) (2025)

## Abstract (arXiv)

We consider adaptations of the Mumford-Shah functional to graphs. These are based on discretizations of nonlocal approximations to the Mumford-Shah functional. Motivated by applications in machine learning we study the random geometric graphs associated to random samples of a measure. We establish the conditions on the graph constructions under which the minimizers of graph Mumford-Shah functionals converge to a minimizer of a continuum Mumford-Shah functional. Furthermore we explicitly identify the limiting functional. Moreover we describe an efficient algorithm for computing the approximate minimizers of the graph Mumford-Shah functional.