  This paper addresses the growing need to process non-Euclidean data, by
introducing a geometric deep learning (GDL) framework for building universal
feedforward-type models compatible with differentiable manifold geometries. We
show that our GDL models can approximate any continuous target function
uniformly on compact sets of a controlled maximum diameter. We obtain
curvature-dependent lower-bounds on this maximum diameter and upper-bounds on
the depth of our approximating GDL models. Conversely, we find that there is
always a continuous function between any two non-degenerate compact manifolds
that any "locally-defined" GDL model cannot uniformly approximate. Our last
main result identifies data-dependent conditions guaranteeing that the GDL
model implementing our approximation breaks "the curse of dimensionality." We
find that any "real-world" (i.e. finite) dataset always satisfies our condition
and, conversely, any dataset satisfies our requirement if the target function
is smooth. As applications, we confirm the universal approximation capabilities
of the following GDL models: Ganea et al. (2018)'s hyperbolic feedforward
networks, the architecture implementing Krishnan et al. (2015)'s deep
Kalman-Filter, and deep softmax classifiers. We build universal
extensions/variants of: the SPD-matrix regressor of Meyer et al. (2011), and
Fletcher (2003)'s Procrustean regressor. In the Euclidean setting, our results
imply a quantitative version of Kidger and Lyons (2020)'s approximation theorem
and a data-dependent version of Yarotsky and Zhevnerchuk (2019)'s uncursed
approximation rates.
