We build universal approximators of continuous maps between arbitrary Polish
metric spaces $\mathcal{X}$ and $\mathcal{Y}$ using universal approximators
between Euclidean spaces as building blocks. Earlier results assume that the
output space $\mathcal{Y}$ is a topological vector space. We overcome this
limitation by "randomization": our approximators output discrete probability
measures over $\mathcal{Y}$. When $\mathcal{X}$ and $\mathcal{Y}$ are Polish
without additional structure, we prove very general qualitative guarantees;
when they have suitable combinatorial structure, we prove quantitative
guarantees for H\"older-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 $\mathcal{X}$ and
$\mathcal{Y}$. For barycentric $\mathcal{Y}$, including Banach spaces,
$\mathbb{R}$-trees, Hadamard manifolds, or Wasserstein spaces on Polish metric
spaces, our approximators reduce to $\mathcal{Y}$-valued functions. When the
Euclidean approximators are neural networks, our constructions generalize
transformer networks, providing a new probabilistic viewpoint of geometric deep
learning.

本文提出了一种使用 Euclidean 空间作为基石建造弥合任意 Polish 度量空间之间连续映射的通用逼近方法，其中涉及到了离散概率、H"older-like (map) 以及深度学习 (transformer network) 等关键词。