Optimal transport (OT) has a long history, which originates in the 18th century by Gaspard Monge’s physical considerations of mass transportation and his studies on optimal spatial allocation of resources. Since then it has undergone a flourishing mathematical development and has influenced and shaped various areas within mathematics including analysis, probability, statistics and optimization. In parallel, it also has been proven to be a remarkably rich and fruitful concept for various other disciplines, such as economic theory, logistics and financial mathematics, and more recently computer science, machine learning and statistical data analysis. Fundamental to this is the relaxation of OT as a probability mass transportation problem, which dates back to Leonid Kantorovich in the first half of the last century. This laid the foundations of linear programing and opened the door for many further developments. Since then, the computation of OT is a highly active field of research. Modern methods often exploit duality, the specific structure of the ground space and of the cost functional. In fact, due to such computational progress and the flexibility of OT, various concepts for OT data analysis (OTDA) are beginning to find its way into real world applications, including video processing, tomography, cell biology and geophysics, to mention a few.
Hence, despite its great conceptual appeal and certain computational progress, OTDA is still at its infancy. This in particular concerns the development of statistical methodology and theory.
In this talk, we will discuss some recent developments in OTDA at the cutting edge of statistical methodology and computation. This includes OT-barycenters, which are summary measures of data with complex geometric structure, as well as novel ways to measure statistical dependency. Mathematical tools are limit laws and risk bounds for empirical OT plans and distances. Proofs are based on a combination of sensitivity analysis from convex optimization and empirical process theory. From this, we obtain methods for statistical inference and fast randomized computation schemes of OTDA tasks in large scale data applications. The performance of OTDA is illustrated in various computer experiments and on examples from cell biology.
Personal Website of Axel Munk
The talk also can be joined online via our ZOOM MEETING
Meeting room opens at: October 03, 2022, 4.30 pm Vienna
Meeting ID: 685 0132 5109
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