On Optimal Transport with Respect to Non-Traditional Costs

Explore optimal transport with respect to non-traditional costs in this 59-minute lecture by Shiri Artstein-Avidan from the University of Tel Aviv. Begin with a brief overview of optimal transport, introducing key concepts such as c-transform and c-subgradients. Delve into the complexities of transportation with costs that can reach infinite values, examining necessary and sufficient conditions for the existence of transport plans supported on c-subgradients (Brenier-type maps) between pairs of measures. Focus on the polar cost underlying the polarity transform for functions as a primary example. Cover topics including transport plans, quadratic cost, cyclic monotonicity, C-path boundedness, strongly C-compatible measures, Hall's Marriage Theorem, and dualities for sets. Gain insights from this joint work with Shay Sadovsky and Kasia Wyczesany, presented at the Hausdorff Center for Mathematics.

Intro
Transport plans
Optimal transport
Ctransform
Csub gradient
quadratic cost
cyclic monotonicity
polar cost
Cpath boundedness
Strongly C compatible
Halls Marriage Theorem
Dualities for Sets
Conclusion