These Limits Are Too Complicated for Calculus

Video Description

Thanks to my supporters on Patreon! Get early access to videos and more: https://www.patreon.com/EricRowland What numbers do you get when you iteratively scale a table? Approximations of them have been used since the 1930s to predict telephone traffic and in other applications. But mathematically, the exact values are extremely complicated! We discovered some interesting combinatorics behind Sinkhorn limits that didn't make it into this video. I talk about it here: https://youtu.be/qU-h4ogd3SQ?t=2240 Our paper: Eric Rowland and Jason Wu, The entries of the Sinkhorn limit of an m × n matrix (25 pages). https://arxiv.org/abs/2409.02789 ---------------- Other references: Marco Cuturi, Sinkhorn distances: lightspeed computation of optimal transportation distances, Advances in Neural Information Processing Systems 26 (2013) 2292-2300. https://papers.nips.cc/paper_files/paper/2013/hash/af21d0c97db2e27e13572cbf59eb343d-Abstract.html J. Kruithof, Telefoonverkeersrekening, De Ingenieur 52 (1937) E15-E25. English translation by Pieter-Tjerk de Boer: https://wwwhome.ewi.utwente.nl/~ptdeboer/misc/kruithof-1937-translation.html See Appendix 3d. Melvyn B. Nathanson, Alternate minimization and doubly stochastic matrices, Integers 20A (2020) Article A10 (17 pages). https://math.colgate.edu/~integers/uproc10/uproc10.pdf Robert D. Putnam, Bowling Alone: The Collapse and Revival of American Community (2000) Simon & Schuster. https://archive.org/details/bowlingalone00robe/page/166/mode/2up Richard Sinkhorn, A relationship between arbitrary positive matrices and doubly stochastic matrices, The Annals of Mathematical Statistics 35 (1964) 876-879. https://doi.org/10.1214/aoms/1177703591 ---------------- 0:00 Predicting telephone traffic 1:16 Kruithof's example 6:02 2x2 tables 8:35 3x3 tables 15:56 Rewriting the equation for 3x3 tables 22:20 Compact equation for 3x3 tables 24:47 Larger tables 27:40 Answer to Kruithof's example ---------------- Animated with Manim. https://www.manim.community Music by Callistio. Audio recorded at the Lawrence Herbert School of Communication at Hofstra University. https://www.hofstra.edu/communication/ Web site: https://ericrowland.github.io