Statistical and Structural Aspects of Unbalanced Optimal Transport Barycenters

by Florian Heinemann
Cumulative thesis
Date of Examination:2022-09-12
Date of issue:2022-10-20
Advisor:Prof. Dr. Axel Munk
Referee:Prof. Dr. Axel Munk
Referee:Prof. Dr. Bernhard Schmitzer
Persistent Address: http://dx.doi.org/10.53846/goediss-9488

 

 

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Abstract

English

Optimal transport (OT) has seen a stellar rise in interest and relevance in the past two decades. More recently, severe limitation of OT have started to surface. Two key factors prevent it from becoming a standard tool in general data science applications. The first one is the fact that the era of big data and steadily improving measurement techniques in the natural sciences produce large scale data which is still out of reach for even modern state-of-the-art OT solvers. The second limitation which prevents the reasonable application of OT in several areas is that vanilla OT is only defined between measures of equal total mass intensity (usually probability measures). At the heart of this thesis lies the goal to advance research on OT to allow it to become a standard tool in modern data analysis. To achieve this, this thesis provides contributions to the research on both aforementioned limitations. It provides non-asymptotic deviation bounds for OT barycenters when the underlying measures are estimated from data and uses this to justify a randomised algorithm to approximate OT barycenter while controlling the induced statistical error. Additionally, it considers a specific notion of Unbalanced OT (UOT) and provides a detailled structural and statistical analysis of the resulting (p,C)-Kantorovich-Rubinstein distance and its corresponding barycenters.
Keywords: Unbalanced Optimal Transport; Wasserstein; Linear Programming; Barycenters; Deviation Bounds
 
Statistik