  A ridge is a function that is characterized by a one-dimensional profile
(activation) and a multidimensional direction vector. Ridges appear in the
theory of neural networks as functional descriptors of the effect of a neuron,
with the direction vector being encoded in the linear weights. In this paper,
we investigate properties of the Radon transform in relation to ridges and to
the characterization of neural networks. We introduce a broad category of
hyper-spherical Banach subspaces (including the relevant subspace of measures)
over which the back-projection operator is invertible. We also give conditions
under which the back-projection operator is extendable to the full parent space
with its null space being identifiable as a Banach complement. Starting from
first principles, we then characterize the sampling functionals that are in the
range of the filtered Radon transform. Next, we extend the definition of ridges
for any distributional profile and determine their (filtered) Radon transform
in full generality. Finally, we apply our formalism to clarify and simplify
some of the results and proofs on the optimality of ReLU networks that have
appeared in the literature.
