ISE Seminar Series

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ISE Seminar Series Fall 2021

Time: Tuesday, September 28, 2021, 11:00 a.m.

Speaker: John Wright, Columbia University

Title: Deep Networks and the Multiple Manifold Problems

Abstract:

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

Bio: