BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Nicolás García Trillos (University of Wisconsin-Madison\, US) DTSTART:20200615T130000Z DTEND:20200615T134500Z DTSTAMP:20260423T021146Z UID:OWMADS/7 DESCRIPTION:Title: R egularity theory and uniform convergence in the large data limit of graph Laplacian eigenvectors on random data clouds.\nby Nicolás García Tri llos (University of Wisconsin-Madison\, US) as part of One World seminar: Mathematical Methods for Arbitrary Data Sources (MADS)\n\n\nAbstract\nGrap h Laplacians are omnipresent objects in machine learning that have been us ed in supervised\, unsupervised and semi supervised settings due to their versatility in extracting local and global geometric information from data clouds. In this talk I will present an overview of how the mathematical t heory built around them has gotten deeper and deeper\, layer by layer\, si nce the appearance of the first results on pointwise consistency in the 20 00’s\, until the most recent developments\; this line of research has fo und strong connections between PDEs built on proximity graphs on data clou ds and PDEs on manifolds\, and has given a more precise mathematical meani ng to the task of “manifold learning”. In the first part of the talk I will highlight how ideas from optimal transport made some of the initial steps\, which provided L2 type error estimates between the spectra of gra ph Laplacians and Laplace-Beltrami operators\, possible. In the second par t of the talk\, which is based on recent work with Jeff Calder and Marta L ewicka\, I will present a newly developed regularity theory for graph Lapl acians which among other things allow us to bootstrap the L2 error estimat es developed through optimal transport and upgrade them to uniform converg ence and almost C^{0\,1} convergence rates. The talk can be seen as a tale of how a flow of ideas from optimal transport\, PDEs\, and in general\, a nalysis\, has made possible a finer understanding of concrete objects popu lar in data analysis and machine learning.\n LOCATION:https://researchseminars.org/talk/OWMADS/7/ END:VEVENT END:VCALENDAR