A Geometric Understanding of Generative Model in Deep Learning

Speaker: Professor David Xianfeng GU
         SUNY Empire Innovation Professor
         Department of Computer Science
         Stony Brook University

Title:  "A Geometric Understanding of Generative Model in Deep Learning"

Date:   Thursday, 1 June 2023

Time:   3:00pm - 4:00pm

Venue:  Room 2463 (via lift 25/26), HKUST

Abstract:

Deep learning (DL) has achieved great successes, but understanding of DL
remains primitive. In this talk, we try to answer some fundamental
questions about DL through a geometric perspective: what does a DL system
really learn ? How does the system learn? Does it really learn or just
memorize the training data sets? How to improve the learning process?

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. DL systems mainly accomplish two
tasks: manifold learning and probability distribution transformation. The
latter can be carried out based on optimal transportation (OT) theory.
This work introduces a geometric view of optimal transportation, which
bridges statistics and differential geometry and is applied for generative
adversarial networks (GANs) and diffusion models. From the OT perspective,
in a GAN model, the generator computes the OT map, while the discriminator
computes the Wasserstein distance between the real data distribution and
the counterfeit; both can be reduced to a convex geometric optimization
process. The diffusion model computes a transportation map from the data
distribution to the Gaussian distribution by a heat diffusion, and focuses
on the inverse flow. Furthermore, the regularity theory of the
Monge-Ampere equation discovers the fundamental reason for mode
collapse. In order to eliminate the mode collapses, a novel generative
model based on the geometric OT theory is proposed, which improves the
theoretical rigor and interpretability, as well as the computational
stability and efficiency. The experimental results validate our
hypothesis, and demonstrate the advantages of our proposed model.


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Biography:

Dr. David Xianfeng Gu got his Bachelor degree in Computer Science from
Tsinghua University; master degree from Harvard University, supervised by
a Fields medalist, Prof. David Mumford; and PhD from Harvard University,
supervised by another Fields medalist, Prof. Shing-Tung Yau. Dr. Gu
currently is a SUNY Empire Innovation Professor at the Computer Science
Department in the State University of New York at Stony Brook. David is
also affiliated with the Applied Mathematics Department at the SUNY Stony
Brook, Yau Mathematical Science Center at Tsinghua University and the
Center of Mathematical Sciences and Applications at Harvard University.
David was also the director of the 3D scanning laboratory of SUNY Stony
Brook, the chief scientist of Beijing Advanced Innovation Center for
Imaging Technology.

By collaborating with Prof.Yau and other colleagues, David has founded an
interdisciplinary field, Computational Conformal Geometry, which combines
modern geometry and computer science and is applied for a broad range of
fields, including computer vision, computer graphics, medical imaging,
networking and digital geometry processing. Recently, with his
collaborators, David has developed discrete theories and computational
algorithms for optimal transportation, and applied them for understandable
AI. David's work on spherical optimal transportation solved a Yau's
conjecture open for about 30 years. His work on the regularity theory of
the Monge-Ampere equation in OT explains the intrinsic reason for mode
collapsing in generative models. David has published about 10 academic
monographs, 400 papers on the top conferences and journals, and 15
patents, some of them have been licensed to Simens, GE, Microsoft. David
has long term collaborations with Google, Cadence, HP, Toyota and other
industrial research labs.

David has been area chairs, program committees for many top conferences in
vision, graphics and AI. He has been the chief editor and associated
editor for several academic journals, including IEEE Transaction on
Visualization and Computer Graphics. He is the recipient of National
Science Foundation (NSF) Faculty early Career Award, National Science
Foundation of China Outstanding Overseas Young Scholars Award,
Morningside Gold Medal of Applied Mathematics, and several best paper
awards. He has been selected as a member of the National Academy of
Inventors of the United States in 2020.