# Ridges, Neural Networks, and the Radon Transform

arXiv: https://arxiv.org/abs/2203.02543 · HTML: https://mlsanity.com/paper/2203.02543/

## Key facts

- Cited by 18 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 49 references that are in the map.
- First posted on arXiv 4 Mar 2022.

## Builds on (most cited first, 7 of 7)

- [Breaking the Curse of Dimensionality with Convex Neural Networks](https://mlsanity.com/paper/1412.8690/) (2014)
- [A Function Space View of Bounded Norm Infinite Width ReLU Nets: The Multivariate Case](https://mlsanity.com/paper/1910.01635/) (2019)
- [Neural Network with Unbounded Activation Functions is Universal Approximator](https://mlsanity.com/paper/1505.03654/) (2015)
- [Understanding neural networks with reproducing kernel Banach spaces](https://mlsanity.com/paper/2109.09710/) (2021)
- [Sparsity of solutions for variational inverse problems with finite-dimensional data](https://mlsanity.com/paper/1809.05045/) (2018)
- [A unifying representer theorem for inverse problems and machine learning](https://mlsanity.com/paper/1903.00687/) (2019)
- [Convex Optimization in Sums of Banach Spaces](https://mlsanity.com/paper/2104.13127/) (2021)

## Built on by (most cited first, 18 of 18)

- [Deep Learning Meets Sparse Regularization: A Signal Processing Perspective](https://mlsanity.com/paper/2301.09554/) (2023)
- [Active Learning with Neural Networks: Insights from Nonparametric Statistics](https://mlsanity.com/paper/2210.08367/) (2022)
- [ReLU Neural Networks with Linear Layers are Biased Towards Single- and Multi-Index Models](https://mlsanity.com/paper/2305.15598/) (2023)
- [Neural reproducing kernel Banach spaces and representer theorems for deep networks](https://mlsanity.com/paper/2403.08750/) (2024)
- [Approximation Rates for Shallow ReLUk Neural Networks on Sobolev Spaces via the Radon Transform](https://mlsanity.com/paper/2408.10996/) (2024)
- [Function-Space Optimality of Neural Architectures with Multivariate Nonlinearities](https://mlsanity.com/paper/2310.03696/) (2023)
- [Weighted Sobolev Approximation Rates for Neural Networks on Unbounded Domains](https://mlsanity.com/paper/2411.04108/) (2024)
- [From Kernel Methods to Neural Networks: A Unifying Variational Formulation](https://mlsanity.com/paper/2206.14625/) (2022)
- [Depth Separation in Norm-Bounded Infinite-Width Neural Networks](https://mlsanity.com/paper/2402.08808/) (2024)
- [Random ReLU Neural Networks as Non-Gaussian Processes](https://mlsanity.com/paper/2405.10229/) (2024)
- [A Low Rank Neural Representation of Entropy Solutions](https://mlsanity.com/paper/2406.05694/) (2024)
- [A Lipschitz spaces view of infinitely wide shallow neural networks](https://mlsanity.com/paper/2410.14591/) (2024)
- [Delaunay-Triangulation-Based Learning with Hessian Total-Variation Regularization](https://mlsanity.com/paper/2208.07787/) (2022)
- [Outlier-robust neural network training: variation regularization meets trimmed loss to prevent functional breakdown](https://mlsanity.com/paper/2308.02293/) (2023)
- [The Spectral Bias of Shallow Neural Network Learning is Shaped by the Choice of Non-linearity](https://mlsanity.com/paper/2503.10587/) (2025)
- [Detection of Rail Line Track and Human Beings Near the Track to Avoid Accidents](https://mlsanity.com/paper/2507.03040/) (2025)
- [Random Neural Network Expressivity for Non-Linear Partial Differential Equations](https://mlsanity.com/paper/2605.25057/) (2026)
- [The Barron-Lipschitz Energy Gap and Depth Separation Phenomena in Scientific Machine Learning](https://mlsanity.com/paper/2607.25905/) (2026)

## Abstract (arXiv)

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.