# A Gap Between the Gaussian RKHS and Neural Networks: An Infinite-Center Asymptotic Analysis

arXiv: https://arxiv.org/abs/2502.16331 · HTML: https://mlsanity.com/paper/2502.16331/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 35 references that are in the map.
- First posted on arXiv 22 Feb 2025.

## Builds on (most cited first, 24 of 24)

- [A Mean Field View of the Landscape of Two-Layers Neural Networks](https://mlsanity.com/paper/1804.06561/) (2018)
- [On the Global Convergence of Gradient Descent for Over-parameterized Models using Optimal Transport](https://mlsanity.com/paper/1805.09545/) (2018)
- [Breaking the Curse of Dimensionality with Convex Neural Networks](https://mlsanity.com/paper/1412.8690/) (2014)
- [Gaussian Processes and Kernel Methods: A Review on Connections and Equivalences](https://mlsanity.com/paper/1807.02582/) (2018)
- [Bayesian Deep Convolutional Networks with Many Channels are Gaussian Processes](https://mlsanity.com/paper/1810.05148/) (2018)
- [Linearized two-layers neural networks in high dimension](https://mlsanity.com/paper/1904.12191/) (2019)
- [When Do Neural Networks Outperform Kernel Methods?](https://mlsanity.com/paper/2006.13409/) (2020)
- [A Function Space View of Bounded Norm Infinite Width ReLU Nets: The Multivariate Case](https://mlsanity.com/paper/1910.01635/) (2019)
- [Trainability and Accuracy of Neural Networks: An Interacting Particle System Approach](https://mlsanity.com/paper/1805.00915/) (2018)
- [On the Power and Limitations of Random Features for Understanding Neural Networks](https://mlsanity.com/paper/1904.00687/) (2019)
- [What Can ResNet Learn Efficiently, Going Beyond Kernels?](https://mlsanity.com/paper/1905.10337/) (2019)
- [Limitations of Lazy Training of Two-layers Neural Networks](https://mlsanity.com/paper/1906.08899/) (2019)
- [Harnessing the Power of Infinitely Wide Deep Nets on Small-data Tasks](https://mlsanity.com/paper/1910.01663/) (2019)
- [The Landscape of Empirical Risk for Non-convex Losses](https://mlsanity.com/paper/1607.06534/) (2016)
- [Deep Equals Shallow for ReLU Networks in Kernel Regimes](https://mlsanity.com/paper/2009.14397/) (2020)
- [Deep Neural Tangent Kernel and Laplace Kernel Have the Same RKHS](https://mlsanity.com/paper/2009.10683/) (2020)
- [What Kinds of Functions do Deep Neural Networks Learn? Insights from Variational Spline Theory](https://mlsanity.com/paper/2105.03361/) (2021)
- [Neural Kernels Without Tangents](https://mlsanity.com/paper/2003.02237/) (2020)
- [Understanding neural networks with reproducing kernel Banach spaces](https://mlsanity.com/paper/2109.09710/) (2021)
- [Near-Minimax Optimal Estimation With Shallow ReLU Neural Networks](https://mlsanity.com/paper/2109.08844/) (2021)
- [Learning sparse features can lead to overfitting in neural networks](https://mlsanity.com/paper/2206.12314/) (2022)
- [Deep Learning Meets Sparse Regularization: A Signal Processing Perspective](https://mlsanity.com/paper/2301.09554/) (2023)
- [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)

## Built on by (most cited first, 2 of 2)

- [A Function-Space Dichotomy for Compositional Learning: Exponential Sub-Optimality of the Neural Tangent Kernel](https://mlsanity.com/paper/2607.06382/) (2026)
- [A Gap Between Decision Trees and Neural Networks](https://mlsanity.com/paper/2601.03919/) (2026)

## Abstract (arXiv)

Recent works have characterized the function-space inductive bias of infinite-width bounded-norm single-hidden-layer neural networks as a kind of bounded-variation-type space. This novel neural network Banach space encompasses many classical multivariate function spaces, including certain Sobolev spaces and the spectral Barron spaces. Notably, this Banach space also includes functions that exhibit less classical regularity, such as those that only vary in a few directions. On bounded domains, it is well-established that the Gaussian reproducing kernel Hilbert space (RKHS) strictly embeds into this Banach space, demonstrating a clear gap between the Gaussian RKHS and the neural network Banach space. It turns out that when investigating these spaces on unbounded domains, e.g., all of ℝd, the story is fundamentally different. We establish the following fundamental result: Certain functions that lie in the Gaussian RKHS have infinite norm in the neural network Banach space. This provides a nontrivial gap between kernel methods and neural networks by exhibiting functions that kernel methods easily represent, whereas neural networks cannot.