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.

翻译：我们利用欧几里得空间之间的通用逼近器作为基本模块，构建了任意波兰度量空间 $\mathcal{X}$ 和 $\mathcal{Y}$ 之间连续映射的通用逼近器。先前的研究假设输出空间 $\mathcal{Y}$ 是拓扑向量空间。我们通过“随机化”克服了这一限制：我们的逼近器输出 $\mathcal{Y}$ 上的离散概率测度。当 $\mathcal{X}$ 和 $\mathcal{Y}$ 是波兰空间且无附加结构时，我们证明了非常广泛的定性保证；当它们具备适当的组合结构时，我们给出了 Hölder 类映射的定量保证，包括有限图之间的映射、某些 Carnot 群之间粗糙微分方程的解算子，以及反问题中 Banach 空间上的连续非线性算子。特别地，我们展示了所需 Dirac 测度的数量由 $\mathcal{X}$ 和 $\mathcal{Y}$ 的组合结构决定。对于重心型 $\mathcal{Y}$（包括 Banach 空间、$\mathbb{R}$-树、Hadamard 流形或波兰度量空间上的 Wasserstein 空间），我们的逼近器退化为取值于 $\mathcal{Y}$ 的函数。当欧几里得逼近器为神经网络时，我们的构造推广了 Transformer 网络，为几何深度学习提供了新的概率论视角。