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数学大讲堂

A Geometric Understanding of Deep Learning via Optimal Transportation Theory

  • 演讲者:顾险峰(纽约州立大学石溪分校)

  • 时间:2021-06-25 08:45-10:00

  • 地点:Zoom: 938 8876 7428 Passcode:265569

Abstract

This work introduces an optimal transportation (OT) view of generative adversarial networks (GANs). Natural datasets have intrinsic patterns, which can be summarized as the manifold distribution principle: the distribution of a class of data is close to a low-dimensional manifold. GANs mainly accomplish two tasks: manifold learning and probability distribution transformation. The latter can be carried out using the classical OT method. From the OT perspective, the generator computes the OT map, while the discriminator computes the Wasserstein distance between the generated data distribution and the real data distribution; both can be reduced to a convex geometric optimization process. Furthermore, OT theory discovers the intrinsic collaborative—instead of competitive—relation between the generator and the discriminator, and the fundamental reason for mode collapse. We also propose a novel generative model, which uses an autoencoder (AE) for manifold learning and OT map for probability distribution transformation. This AE–OT model improves the theoretical rigor and transparency, as well as the computational stability and efficiency; in particular, it eliminates the mode collapse. The experimental results validate our hypothesis, and demonstrate the advantages of our proposed model.


Biography
David Gu is an associate professor (with tenure) at the Department of Computer Science, Stony Brook University. He received his Ph.D degree from the Department of Computer Science, Harvard University in 2003 and B.S. degree from the Tsinghua University, Beijing, China in 1995. His research focuses on applying modern geometry in engineering and medical fields. With his collaborators, David systematically develops discrete theories and computational algorithms in the interdisciplinary field: Computational Conformal Geometry, and apply them for solving real problems, such as global surface parameterization in graphics, deformable shape registration in vision, manifold spline in geometric modeling, curvature convergence analysis in geometric processing, efficient routing in networking, brain mapping and virtual colonoscopy in medical imaging, and so on. He is a recipient of Morningside Gold Medal of Applied Mathematics, International Congress of Chinese Mathematician (ICCM)2013; the Research Excellence Award, Computer Science department, Stony Brook University, 2010; Gaheon Award: The Best Paper in International Journal of CAD/CAM of 2009; Best Paper Award: The 10th International Conference on Computer Aided Design and Computer Graphics, 2007; National Science Foundation (NSF) Faculty Early Career Award, 2005.