# On the relationship between multivariate splines and infinitely-wide neural networks

arXiv: https://arxiv.org/abs/2302.03459 · HTML: https://mlsanity.com/paper/2302.03459/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 6 of its 26 references that are in the map.
- First posted on arXiv 7 Feb 2023.

## Builds on (most cited first, 6 of 6)

- [Breaking the Curse of Dimensionality with Convex Neural Networks](https://mlsanity.com/paper/1412.8690/) (2014)
- [Generalization Properties of Learning with Random Features](https://mlsanity.com/paper/1602.04474/) (2016)
- [On the Equivalence between Kernel Quadrature Rules and Random Feature Expansions](https://mlsanity.com/paper/1502.06800/) (2015)
- [Toward Deeper Understanding of Neural Networks: The Power of Initialization and a Dual View on Expressivity](https://mlsanity.com/paper/1602.05897/) (2016)
- [An Infinite-Feature Extension for Bayesian ReLU Nets That Fixes Their Asymptotic Overconfidence](https://mlsanity.com/paper/2010.02709/) (2020)
- [Relating Leverage Scores and Density using Regularized Christoffel Functions](https://mlsanity.com/paper/1805.07943/) (2018)

## Built on by (most cited first, 5 of 5)

- [Solving Nonlinear PDEs with Sparse Radial Basis Function Networks](https://mlsanity.com/paper/2505.07765/) (2025)
- [Nonuniform random feature models using derivative information](https://mlsanity.com/paper/2410.02132/) (2024)
- [A Convex Loss Function for Set Prediction with Optimal Trade-offs Between Size and Conditional Coverage](https://mlsanity.com/paper/2512.19142/) (2025)
- [Approximation by non-symmetric networks for cross-domain learning](https://mlsanity.com/paper/2305.03890/) (2023)
- [Sparse RBF Networks for PDEs and nonlocal equations: function space theory, operator calculus, and training algorithms](https://mlsanity.com/paper/2601.17562/) (2026)

## Abstract (arXiv)

We consider multivariate splines and show that they have a random feature expansion as infinitely wide neural networks with one-hidden layer and a homogeneous activation function which is the power of the rectified linear unit. We show that the associated function space is a Sobolev space on a Euclidean ball, with an explicit bound on the norms of derivatives. This link provides a new random feature expansion for multivariate splines that allow efficient algorithms. This random feature expansion is numerically better behaved than usual random Fourier features, both in theory and practice. In particular, in dimension one, we compare the associated leverage scores to compare the two random expansions and show a better scaling for the neural network expansion.