Optimal transportation and barycenter problems via convex functions

Ghaffari, Novin
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This work surveys developments in optimal transportation. The first three chapters develop background to optimal transportation problems and the associatedWasserstein distances in increasing levels of specificity. The first chapter introduces optimal transport problems and known properties characterizing optimal transport plans. The second chapter develops the Wasserstein distances, arising from specific cases of optimal transport problems. Properties for the 1 and 2-Wasserstein distances, the most well-understood cases, are presented and Wasserstein distances are characterized as integral probability metrics. The third chapter presents known optimal couplings for the quadratic Wasserstein distance. The rst section discusses optimal transport on R, which has been entirely characterized; the next two sections deal with known multivariate cases. The last three sections develop algorithmic and statistical applications of optimal transportation and Wasserstein distances. Chapter four surveys recent applications from the literature. New results are vi presented in chapters five and six. In chapter five, the problem of optimal transportation is approached by studying the convex functions underlying optimal transports, listing convex functions for known transports and introduce new cases. Chapter six develops barycentric applications. The rst section develops 2-Wasserstein barycenter results. Using a xed-point framework introduced in chapter four, immediate convergence and explicit representation of the barycenter is demonstrated for some classes of distributions. The next section develops applications of 2-Wasserstein barycenters to data modeling; barycenters are used to provide exible models interpolating between several input distributions. The final section develops an extended application of the WASP method introduced in chapter four. Finally we list useful results in appendices.