  Motivated by the developing mathematics of deep learning, we build universal
functions approximators of continuous maps between arbitrary Polish metric
spaces $\mathcal{X}$ and $\mathcal{Y}$ using elementary functions between
Euclidean spaces as building blocks. Earlier results assume that the target
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\"{o}lder-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.

Machine Learning;Numerical Analysis;Neural and Evolutionary Computing;Numerical Analysis;Probability;Machine Learning