- 控制理论和非线性滤波讨论班

An Approximation Theory for Metric Space-Valued Functions with A View Towards Deep Learning

组织者

丘成栋

演讲者

康家熠

时间

2023年07月31日 14:30 至 15:00

地点

数学系理科楼A-203

摘要

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.