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About
President
Governance
Partner Institutions
Visit
People
Management
Faculty
Postdocs
Visiting Scholars
Staff
Research
Research Groups
Courses
Seminars
Join Us
Faculty
Postdocs
Students
Events
Conferences
Workshops
Forum
Life @ BIMSA
Accommodation
Transportation
Facilities
Tour
News
News
Announcement
Downloads
Qiuzhen College, Tsinghua University
Yau Mathematical Sciences Center, Tsinghua University (YMSC)
Tsinghua Sanya International  Mathematics Forum (TSIMF)
Shanghai Institute for Mathematics and  Interdisciplinary Sciences (SIMIS)
BIMSA > Seminar on Control Theory and Nonlinear Filtering An Approximation Theory for Metric Space-Valued Functions with A View Towards Deep Learning
An Approximation Theory for Metric Space-Valued Functions with A View Towards Deep Learning
Organizer
Shing Toung Yau
Speaker
Jia Yi Kang
Time
Monday, July 31, 2023 2:30 PM - 3:00 PM
Venue
数学系理科楼A-203
Abstract
Motivated by the developing mathematics of deep learning, we build universal functions approximators of continuous maps between arbitrary Polish metric spaces X and Y using elementary functions between Euclidean spaces as building blocks. Earlier results assume that the target space Y is a topological vector space. We overcome this limitation by “randomization”: our approximators output discrete probability measures over Y. When X and Y are Polish without additional structure, we prove very general qualitative guarantees; when they have suitable combinatorial structure, we prove quantitative guarantees for Holder-like maps, including maps between finite graphs, solution operators to rough differential equations between certain Carnot groups, and continuous non-linear operators between Banach spaces arising in inverse problems. In particular, we show that the required number of Dirac measures is determined by the combinatorial structure of X and Y. For barycentric Y, including Banach spaces, R-trees, Hadamard manifolds, or Wasserstein spaces on Polish metric spaces, our approximators reduce to Y-valued functions. When the Euclidean approximators are neural networks, our constructions generalize transformer networks, providing a new probabilistic viewpoint of geometric deep learning.
Beijing Institute of Mathematical Sciences and Applications
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