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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. [ABSTRACT FROM AUTHOR]

We investigate a family of regression problems in a semisupervised setting. The task is to assign real-valued labels to a set of n sample points provided a small training subset of N labeled points. A goal of semisupervised learning is to take advantage of the (geometric) structure provided by the large number of unlabeled data when assigning labels. We consider random geometric graphs, with connection radius ε(n), to represent the geometry of the data set. Functionals which model the task reward the regularity of the estimator function and impose or reward the agreement with the training data. Here we consider discrete p-Laplacian regularization. We investigate asymptotic behavior when the number of unlabeled points increases while the number of training points remains fixed. A delicate interplay between the regularizing nature of the functionals and the nonlocality inherent to the graph constructions is uncovered. Rigorous, almost optimal, ranges on the scaling of ε(n) for asymptotic consistency are obtained and in these admissible ranges it is shown that mini-mizers of the discrete functionals converge uniformly to the desired continuum limit. Furthermore, we discover that, for the standard model, there is a restrictive upper bound on how quickly ε(n) must converge to zero as n → ∞. A new model is introduced, which is as simple as the original model, but overcomes this restriction. [ABSTRACT FROM AUTHOR]

We consider point clouds obtained as random samples of a measure on a Euclidean domain. A graph representing the point cloud is obtained by assigning weights to edges based on the distance between the points they connect. Our goal is to develop mathematical tools needed to study the consistency, as the number of available data points increases, of graph-based machine learning algorithms for tasks such as clustering. In particular, we study when the cut capacity, and more generally total variation, on these graphs is a good approximation of the perimeter (total variation) in the continuum setting. We address this question in the setting of Γ-convergence. We obtain almost optimal conditions on the scaling, as the number of points increases, of the size of the neighborhood over which the points are connected by an edge for the Γ-convergence to hold. Taking of the limit is enabled by a transportation based metric which allows us to suitably compare functionals defined on different point clouds. [ABSTRACT FROM AUTHOR]

We present a family of first-order functionals which are displacement convex, that is convex along the geodesics induced by the quadratic transportation distance on the circle. The displacement convexity implies the existence and uniqueness of gradient flows of the given functionals. More precisely, we show the existence and uniqueness of gradient-flow solutions of a class of fourth-order degenerate parabolic equations with periodic boundary data. Moreover, positivity of the absolutely continuous part of the solutions is preserved along the flow. [ABSTRACT FROM AUTHOR]