# Mumford-Shah functionals on graphs and their asymptotics

## TL;DR

This paper extends Mumford-Shah functionals to graphs, analyzing their asymptotic behavior and convergence to continuum functionals, with applications in machine learning and an efficient algorithm for minimization.

## Contribution

It introduces graph-based Mumford-Shah functionals, establishes convergence conditions to continuum limits, and provides an efficient algorithm for their minimization.

## Key findings

- Minimizers of graph Mumford-Shah functionals converge to continuum minimizers.
- Explicit identification of the limiting functional.
- Development of an efficient algorithm for approximate minimization.

## Abstract

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.