We study the game theoretic p-Laplacian for semi-supervised learning on graphs, and show that it is well-posed in the limit of finite labeled data and infinite unlabeled data. In particular, we show that the continuum limit of graph-based semi-supervised learning with the game theoretic p-Laplacian is a weighted version of the continuous p-Laplace equation. Our proof uses the viscosity solution...

We consider a simple class of controlled queue length processes in the fluid rescaling, X(t) = 1 n X(nt). Consider the optimal control problem for X(t) which consists of minimizing the integral of an undiscounted positive running cost until the first time that X = 0. Our main result uses weak convergence ideas to show that the optimal value functions V n of the stochastic control problems for X...

We investigate the Dirichlet problem for the parabolic equation ut=Δum−buβ, m > 0, β > 0, b ∈ R, in a nonsmooth and closed domain Ω ⊂ RN+1, N ≥ 2, possibly formed with irregular surfaces and having a characteristic vertex point. Existence, boundary regularity, uniqueness, and comparison results are established. The main objective of the paper is to express the criteria for the well-posedness in...

Abstract Semi-discrete optimal transport problems, which evaluate the Wasserstein distance between a discrete and generic (possibly non-discrete) probability measure, are believed to be computationally hard. Even though such problems ubiquitous in statistics, machine learning computer vision, however, this perception has not yet received theoretical justification. To fill gap, we prove that com...