mlsanityData to 7 Oct 2026

Ridges, Neural Networks, and the Radon Transform

arXiv 2203.02543 ↗First posted 4 Mar 2022math.FA stat.MLcited by 18

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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.

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

Builds on

7 of 49 references.

References · 7

  1. 2014Breaking the Curse of Dimensionality with Convex Neural Networkscited by 329
  2. 2019A Function Space View of Bounded Norm Infinite Width ReLU Nets: The Multivariate Casecited by 113
  3. 2015Neural Network with Unbounded Activation Functions is Universal Approximatorcited by 63
  4. 2021Understanding neural networks with reproducing kernel Banach spacescited by 50
  5. 2018Sparsity of solutions for variational inverse problems with finite-dimensional datacited by 25
  6. 2019A unifying representer theorem for inverse problems and machine learningcited by 24
  7. 2021Convex Optimization in Sums of Banach Spacescited by 11

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2022: 3; 2023: 4; 2024: 7; 2025: 2; 2026: 2.

18 papers.

Sort:

  1. 2023Deep Learning Meets Sparse Regularization: A Signal Processing Perspectivecited by 21
  2. 2022Active Learning with Neural Networks: Insights from Nonparametric Statisticscited by 13
  3. 2023ReLU Neural Networks with Linear Layers are Biased Towards Single- and Multi-Index Modelscited by 7
  4. 2024Approximation Rates for Shallow ReLUk Neural Networks on Sobolev Spaces via the Radon Transformcited by 6
  5. 2023Function-Space Optimality of Neural Architectures with Multivariate Nonlinearitiescited by 5
  6. 2024Weighted Sobolev Approximation Rates for Neural Networks on Unbounded Domainscited by 5
  7. 2022From Kernel Methods to Neural Networks: A Unifying Variational Formulationcited by 4
  8. 2024Neural reproducing kernel Banach spaces and representer theorems for deep networkscited by 4
  9. 2024Random ReLU Neural Networks as Non-Gaussian Processescited by 3
  10. 2024A Low Rank Neural Representation of Entropy Solutionscited by 3
  11. 2024Depth Separation in Norm-Bounded Infinite-Width Neural Networkscited by 2
  12. 2024A Lipschitz spaces view of infinitely wide shallow neural networkscited by 2
6 more
  1. 2022Delaunay-Triangulation-Based Learning with Hessian Total-Variation Regularizationcited by 0
  2. 2023Outlier-robust neural network training: variation regularization meets trimmed loss to prevent functional breakdowncited by 0
  3. 2025The Spectral Bias of Shallow Neural Network Learning is Shaped by the Choice of Non-linearitycited by 0
  4. 2025Detection of Rail Line Track and Human Beings Near the Track to Avoid Accidentscited by 0
  5. 2026Random Neural Network Expressivity for Non-Linear Partial Differential Equationscited by 0
  6. 2026The Barron-Lipschitz Energy Gap and Depth Separation Phenomena in Scientific Machine Learningcited by 0