BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Hariharan Narayanan (Tata Institute for Fundamental Research) DTSTART:20210224T160000Z DTEND:20210224T170000Z DTSTAMP:20260414T173948Z UID:BUGeom/17 DESCRIPTION:Title: Testing the manifold hypothesis and fitting a manifold of large reach to n oisy data\nby Hariharan Narayanan (Tata Institute for Fundamental Rese arch) as part of Boston University Geometry/Physics Seminar\n\nLecture hel d in Zoom meeting ID: 974 5641 9902.\n\nAbstract\nThe hypothesis that high dimensional data tend to lie in the vicinity of a low dimensional manifol d is the basis of manifold learning. \nWe will discuss a joint work with C harles Fefferman and Sanjoy Mitter on testing the manifold hypothesis. We will outline an algorithm (with accompanying complexity guarantees) for fi tting a manifold to an unknown probability distribution supported in a sep arable Hilbert space\, only using i.i.d samples from that distribution.\nW e also give a solution based on joint work with Charles Fefferman\, Sergei Ivanov and Matti Lassas to the following question from manifold learning. \nSuppose data belonging to a high dimensional Euclidean space is sampled independently\, identically at random\, from a measure supported on a d di mensional twice differentiable embedded manifold M\, and corrupted by Gaus sian noise with small standard deviation sigma. How can we produce a manif old M_o whose Hausdorff distance to M is small and whose reach (normal inj ectivity radius) is not much smaller than the reach of M? We show how to p roduce a manifold within O(sigma^2) of M in Hausdorf distance\, whose reac h is smaller than that of M by a factor of no more than O(d^6).\n LOCATION:https://researchseminars.org/talk/BUGeom/17/ END:VEVENT END:VCALENDAR