Mumford-Shah functionals on graphs and their asymptotics
This work addresses theoretical convergence and computational methods for graph-based functionals, with potential applications in machine learning, but it appears incremental as it builds on existing discretizations and approximations.
The paper tackles the adaptation of Mumford-Shah functionals to graphs, establishing conditions for convergence of minimizers to a continuum functional and identifying the limiting functional, while also providing an efficient algorithm for approximate minimization.
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.