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"></span><span style='font-family:"Microsoft YaHei Light",sans-serif;color:#073763;mso-fareast-language:JA'> <a href="https://statistics.yale.edu/" target="_blank" title=Home><span style='font-size:22.0pt;color:#0563C1'>Department of Statistics and Data Science </span></a></span><i><span style='font-size:22.0pt;font-family:"Microsoft YaHei Light",sans-serif;color:#073763;mso-fareast-language:JA'> </span></i><span style='font-family:"Microsoft YaHei Light",sans-serif;mso-fareast-language:JA'><o:p></o:p></span></p><p class=MsoNormal style='background:white'><u><span style='font-size:16.0pt;color:#073763;mso-fareast-language:JA'>You are invited to a scheduled Zoom Seminar</span></u><u><span style='mso-fareast-language:JA'><o:p></o:p></span></u></p><h1 style='mso-margin-top-alt:.1in;margin-right:0in;margin-bottom:0in;margin-left:0in;background:white'><span style='font-size:23.0pt;font-family:Mallory;color:#003C76;text-transform:uppercase;font-weight:normal'>JOHN N. WRIGHT</span><span style='font-size:13.0pt;font-family:Mallory;color:#222222'>, </span><span class=odd><span style='font-size:13.0pt;font-family:Mallory;color:#222222'>Columbia University</span></span><span style='font-size:23.0pt;font-family:Mallory;color:#003C76;text-transform:uppercase;font-weight:normal'><o:p></o:p></span></h1><p class=MsoNormal style='background:white'><!--[if gte vml 1]><v:shapetype id="_x0000_t75" coordsize="21600,21600" o:spt="75" o:preferrelative="t" path="m@4@5l@4@11@9@11@9@5xe" filled="f" stroked="f">
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</v:shape><![endif]--><![if !vml]><img width=139 height=167 style='width:1.4479in;height:1.7395in' src="cid:image003.jpg@01D7C58F.10995B40" align=left hspace=12 v:shapes="Picture_x0020_3"><![endif]><span class=date-display-single><span style='font-size:13.5pt;font-family:Mallory;color:#003C76'>Monday, October 25, 2021<o:p></o:p></span></span></p><p class=MsoNormal style='background:white'><span class=date-display-start><span style='font-size:13.5pt;font-family:Mallory;color:#003C76'>4:00PM</span></span><span class=date-display-range><span style='font-size:13.5pt;font-family:Mallory;color:#003C76'> to </span></span><span class=date-display-end><span style='font-size:13.5pt;font-family:Mallory;color:#003C76'>5:00PM</span></span><span style='font-size:13.0pt;font-family:Mallory;color:#222222'><o:p></o:p></span></p><p class=MsoNormal style='background:white'><span style='font-size:13.0pt;font-family:Mallory;color:#222222'>Zoom: <a href="https://yale.zoom.us/j/99169700816?pwd=SWEvWHI5d3dPNVdHMkZMZURMWWJPUT09">https://yale.zoom.us/j/99169700816?pwd=SWEvWHI5d3dPNVdHMkZMZURMWWJPUT09</a> <o:p></o:p></span></p><p class=MsoNormal style='background:white'><span style='font-size:13.0pt;font-family:Mallory;color:#222222'>Password: 24<o:p></o:p></span></p><p class=MsoNormal style='background:white'><span style='font-size:13.0pt;font-family:Mallory;color:#222222'><a href="https://www.ee.columbia.edu/content/john-n-wright"><span style='font-size:12.0pt;color:#003C76'>Website</span></a><o:p></o:p></span></p><p class=MsoNormal style='background:white'><b><span style='font-size:13.0pt;font-family:Mallory;color:#222222'><o:p> </o:p></span></b></p><p class=MsoNormal style='background:white'><b><span style='font-size:13.0pt;font-family:Mallory;color:#222222'><o:p> </o:p></span></b></p><p class=MsoNormal style='background:white'><b><span style='font-size:15.0pt;font-family:Mallory;color:#222222'><o:p> </o:p></span></b></p><p class=MsoNormal style='background:white'><b><span style='font-size:15.0pt;font-family:Mallory;color:#222222'>Title: Deep Networks and the Multiple Manifold Problem<o:p></o:p></span></b></p><p class=MsoNormal style='background:white'><b><span style='font-size:13.0pt;font-family:Mallory;color:#222222'><o:p> </o:p></span></b></p><p class=MsoNormal style='background:white'><b><span style='font-size:13.0pt;font-family:Mallory;color:#222222'>Information and Abstract: <o:p></o:p></span></b></p><p style='mso-margin-top-alt:0in;margin-right:0in;margin-bottom:12.0pt;margin-left:0in;background:white;box-sizing: inherit'><span style='font-size:13.0pt;font-family:Mallory;color:#222222'>Data with low-dimensional nonlinear structure are ubiquitous in engineering and scientific problems. We study a model problem with such structure—a binary classification task that uses a deep fully-connected neural network to classify data drawn from two disjoint smooth curves on the unit sphere. Aside from mild regularity conditions, we place no restrictions on the configuration of the curves. We prove that when (i) the network depth is large relative to certain geometric properties that set the difficulty of the problem and (ii) the network width and number of samples is polynomial in the depth, randomly-initialized gradient descent quickly learns to correctly classify all points on the two curves with high probability. To our knowledge, this is the first generalization guarantee for deep networks with nonlinear data that depends only on intrinsic data properties. Our analysis draws on ideas from harmonic analysis and martingale concentration for handling statistical dependencies in the initial (random) network. We sketch applications to invariant vision, and to gravitational wave astronomy, where leveraging low-dimensional structure leads to statistically optimal tests for identifying signals in noise. Joint work with Sam Buchanan, Dar Gilboa, Tim Wang, Jingkai Yan<o:p></o:p></span></p><p style='mso-margin-top-alt:0in;margin-right:0in;margin-bottom:12.0pt;margin-left:0in;background:white;box-sizing: inherit'><span style='font-size:13.0pt;font-family:Mallory;color:#222222'>Speaker Bio: John Wright is an associate professor in Electrical Engineering at Columbia University. He is also affiliated with the Department of Applied Physics and Applied Mathematics and Columbia’s Data Science Institute. He received his PhD in Electrical Engineering from the University of Illinois at Urbana Champaign in 2009. Before joining Columbia he was with Microsoft Research Asia from 2009-2011. His research interests include sparse and low-dimensional models for high-dimensional data, optimization (convex and otherwise), and applications in imaging and vision. <o:p></o:p></span></p><p style='margin:0in;background:white;box-sizing: inherit'><span style='font-size:13.0pt;font-family:Mallory;color:#222222'>You are invited to a scheduled Zoom meeting. Zoom is Yale’s audio and visual conferencing platform.<o:p></o:p></span></p><p style='margin:0in;background:white;box-sizing: inherit'><span style='font-size:13.0pt;font-family:Mallory;color:#222222'>Topic: Yale S&DS Department Seminar<o:p></o:p></span></p><p style='margin:0in;background:white;box-sizing: inherit'><span style='font-size:13.0pt;font-family:Mallory;color:#222222'>Time: 4:00pm - 5:00pm<o:p></o:p></span></p><p style='margin:0in;background:white;box-sizing: inherit'><span style='font-size:13.0pt;font-family:Mallory;color:#222222'>Join from PC, Mac, Linux, iOS or Android: <a href="https://yale.zoom.us/j/99169700816?pwd=SWEvWHI5d3dPNVdHMkZMZURMWWJPUT09" target="_blank"><span style='color:#286DC0'>https://yale.zoom.us/j/99169700816?pwd=SWEvWHI5d3dPNVdHMkZMZURMWWJPUT09</span></a><o:p></o:p></span></p><p style='margin:0in;background:white;box-sizing: inherit'><span style='font-size:13.0pt;font-family:Mallory;color:#222222'> Password: 24<o:p></o:p></span></p><p style='margin:0in;background:white;box-sizing: inherit'><span style='font-size:13.0pt;font-family:Mallory;color:#222222'> Or Telephone</span><span style='font-size:13.0pt;font-family:"MS Gothic";color:#222222'>：</span><span style='font-size:13.0pt;font-family:Mallory;color:#222222'>203-432-9666 (2-ZOOM if on-campus) or 646 568 7788<o:p></o:p></span></p><p style='margin:0in;background:white;box-sizing: inherit'><span style='font-size:13.0pt;font-family:Mallory;color:#222222'> Meeting ID: 991 6970 0816<o:p></o:p></span></p><p style='margin:0in;background:white;box-sizing: inherit'><span style='font-size:13.0pt;font-family:Mallory;color:#222222'> International numbers available: <a href="https://yale.zoom.us/u/acBOaD1ic6" target="_blank"><span style='color:#286DC0'>https://yale.zoom.us/u/acBOaD1ic6</span></a><o:p></o:p></span></p><p style='margin:0in;background:white;box-sizing: inherit'><span style='font-size:13.0pt;font-family:Mallory;color:#222222'>For H.323 and SIP information for video conferencing units please click here: <a href="https://yale.service-now.com/it?id=support_article&sys_id=434b72d3db9e8fc83514b1c0ef961924" target="_blank"><span style='color:#286DC0'>https://yale.service-now.com/it?id=support_article&sys_id=434b72d3db9e8fc83514b1c0ef961924</span></a><o:p></o:p></span></p><p class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto'><span style='font-size:12.0pt;font-family:"Arial",sans-serif;mso-fareast-language:JA'> <o:p></o:p></span></p><p class=MsoNormal><span style='mso-fareast-language:JA'><o:p> </o:p></span></p><p class=MsoNormal><o:p> </o:p></p></div></body></html>