  We study representations of data from an arbitrary metric space $\mathcal{X}$
in the space of univariate Gaussian mixtures with a transport metric (Delon and
Desolneux 2020). We derive embedding guarantees for feature maps implemented by
small neural networks called \emph{probabilistic transformers}. Our guarantees
are of memorization type: we prove that a probabilistic transformer of depth
about $n\log(n)$ and width about $n^2$ can bi-H\"{o}lder embed any $n$-point
dataset from $\mathcal{X}$ with low metric distortion, thus avoiding the curse
of dimensionality. We further derive probabilistic bi-Lipschitz guarantees,
which trade off the amount of distortion and the probability that a randomly
chosen pair of points embeds with that distortion. If $\mathcal{X}$'s geometry
is sufficiently regular, we obtain stronger, bi-Lipschitz guarantees for all
points in the dataset. As applications, we derive neural embedding guarantees
for datasets from Riemannian manifolds, metric trees, and certain types of
combinatorial graphs. When instead embedding into multivariate Gaussian
mixtures, we show that probabilistic transformers can compute bi-H\"{o}lder
embeddings with arbitrarily small distortion.

Machine Learning;Neural and Evolutionary Computing;Combinatorics;Metric Geometry;Machine Learning