Optimal Transport - Entropic Regularisation

Math 707: Optimal Transport
Entropic Regularisation
October 30, 2019
This is a lecture on "Entropic Regularisation" given as a part of Brittany Hamfeldt's class Math 707: Optimal Transport at New Jersey Institute of Technology.
Syllabus: m.njit.edu/Gra...
This recorded lecture was supported by NSF DMS-1751996.

[@filippoelgorni3238]

Doctor Hamfeldt please be aware this lecture series, for free, on youtube, is an absolute godsend! I will forever be thankful

[@MuniSreenivasPydi]

Very nice lecture! I haven't seen the alternate characterization of the regularized OT in terms of minimizing KL with the Gibbs distribution before. Really cool!

[@AI_For_Scientists]

very nice lecture. thank you for sharing!

[@edisonmucllari7904]

Thank you so much for the lectures professor. I just needed to ask about the part at 20:42. How is \pi sparse in that case?

[@brittanyhamfeldt]

Keep in mind that \pi is defined on the product space X x Y. So in a simple example where X = Y = R, we have \pi defined on R^2. If the optimal map is a shift, say T(x) = x + a, then \pi is going to be supported on points of the form (x, x+a), which is a line. At any other points (x,y) in two-dimensions, no mass from x is transported to the point y, so \pi is zero around these points. Conclusion: all of the mass in \pi lives on a line in 2D, so \pi is sparse.

[@edisonmucllari7904]

@@brittanyhamfeldt Thank you so much. I got it.