Ridges, Neural Networks, and the Radon Transform

TL;DR

This paper explores the mathematical properties of ridges and the Radon transform in neural networks, providing new theoretical insights and extending existing concepts to distributions, with implications for understanding ReLU network optimality.

Contribution

It introduces a broad class of Banach spaces for the Radon transform, characterizes sampling functionals, and extends ridge definitions to distributions, linking these to neural network theory.

Findings

Invertibility of the back-projection operator in certain Banach spaces

Characterization of sampling functionals in the Radon transform

Extension of ridge concepts to distributional profiles

Abstract

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.