Dear all,

The next session of the One World Probability Seminar will be next week on Thursday (December 14) from 14:00 to 16:00 *UTC time*.The speakers of this session are Marcel Nutz (Columbia University) and Stephan Eckstein (ETH Zürich).

Title, abstract and the zoom link are below the signature and can be also found on the website https://www.owprobability.org/one-world-probability-seminar.

Kind regards, Julio Backhoff and Ilya Chevyrev.

Title: Entropic Selection in Optimal Transport

Abstract:The optimal transport problem can admit many solutions, for instance when the cost function is the Euclidean distance on R^d (Monge's problem). On the other hand, entropically regularized optimal transport (EOT) always has a unique optimizer. It is a longstanding open problem whether EOT selects some particular optimal transport coupling in the vanishing regularization limit, and what that coupling would look like. We provide a surprising answer for Monge's problem in dimension d > 1. (Joint work with Chenyang Zhong.)

Speaker 1 (14:00-15:00 UTC): Marcel Nutz (Columbia University).

Title: Exponential convergence of Sinkhorn's algorithm and Hilbert's projective metric for unbounded functions

Abstract: Entropic regularization of optimal transport has found numerous applications in various fields recently. A major reason for this surge in applications is the popularization of Sinkhorn's algorithm to efficiently solve the entropic optimal transport problem numerically. This talk studies the (rate of) convergence of Sinkhorn's algorithm. In bounded settings, it is known that Sinkhorn's algorithm converges with exponential rate, which is a consequence of applying a commonly used version of Hilbert's metric corresponding to the cone of all non-negative functions. This talk shows how to define versions of Hilbert's metric so that we can show exponential convergence of Sinkhorn's algorithm even in unbounded settings. This is done through the use of cones which are relaxations of the cone of all non-negative functions, in the sense that they include all functions having non-negative integral values when multiplied with certain test functions. Along the way, we establish that kernel integral operators are contractions with respect to suitable versions of Hilbert’s metric, even if the kernel functions are not bounded away from zero.

Speaker 2 (15:00-16:00 UTC): Stephan Eckstein (ETH Zürich).

Zoom-link: https://univienna.zoom.us/j/69978648939?pwd=WllwSmd4bW9EQlZJNlRqdlA3M0I1Zz09

Meeting ID: 699 7864 8939 Passcode: 713222