# Shallow ReLUs Networks in Lp-Type and Sobolev Spaces: Approximation and Path-Norm Controlled Generalization

arXiv: https://arxiv.org/abs/2605.18468 · HTML: https://mlsanity.com/paper/2605.18468/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 44 references that are in the map.
- First posted on arXiv 18 May 2026.

## Builds on (most cited first, 19 of 19)

- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [Surprises in High-Dimensional Ridgeless Least Squares Interpolation](https://mlsanity.com/paper/1903.08560/) (2019)
- [Breaking the Curse of Dimensionality with Convex Neural Networks](https://mlsanity.com/paper/1412.8690/) (2014)
- [Norm-Based Capacity Control in Neural Networks](https://mlsanity.com/paper/1503.00036/) (2015)
- [Two models of double descent for weak features](https://mlsanity.com/paper/1903.07571/) (2019)
- [A Function Space View of Bounded Norm Infinite Width ReLU Nets: The Multivariate Case](https://mlsanity.com/paper/1910.01635/) (2019)
- [On the Power and Limitations of Random Features for Understanding Neural Networks](https://mlsanity.com/paper/1904.00687/) (2019)
- [Sharp Bounds on the Approximation Rates, Metric Entropy, and n-widths of Shallow Neural Networks](https://mlsanity.com/paper/2101.12365/) (2021)
- [An overview on deep learning-based approximation methods for partial differential equations](https://mlsanity.com/paper/2012.12348/) (2020)
- [Characterization of the Variation Spaces Corresponding to Shallow Neural Networks](https://mlsanity.com/paper/2106.15002/) (2021)
- [Near-Minimax Optimal Estimation With Shallow ReLU Neural Networks](https://mlsanity.com/paper/2109.08844/) (2021)
- [Nonparametric regression using over-parameterized shallow ReLU neural networks](https://mlsanity.com/paper/2306.08321/) (2023)
- [Deep Neural Networks Are Effective At Learning High-Dimensional Hilbert-Valued Functions From Limited Data](https://mlsanity.com/paper/2012.06081/) (2020)
- [Optimal Approximation of Zonoids and Uniform Approximation by Shallow Neural Networks](https://mlsanity.com/paper/2307.15285/) (2023)
- [Expressivity and Approximation Properties of Deep Neural Networks with ReLUk Activation](https://mlsanity.com/paper/2312.16483/) (2023)
- [Learning with Norm Constrained, Over-parameterized, Two-layer Neural Networks](https://mlsanity.com/paper/2404.18769/) (2024)
- [Reproducing Kernel Hilbert Spaces Cannot Contain all Continuous Functions on a Compact Metric Space](https://mlsanity.com/paper/2002.03171/) (2020)
- [Minimum complexity interpolation in random features models](https://mlsanity.com/paper/2103.15996/) (2021)
- [A duality framework for analyzing random feature and two-layer neural networks](https://mlsanity.com/paper/2305.05642/) (2023)

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

- [Error bounds in Sobolev norms for approximations with norm constrained ReLU neural networks](https://mlsanity.com/paper/2609.19937/) (2026)
- [Minimax rates for learning spectral Barron functions by deep ReLU neural networks](https://mlsanity.com/paper/2609.39020/) (2026)

## Abstract (arXiv)

This paper studies approximation by shallow ReLUs networks, σs(t)=max0,ts, together with their generalization behavior under ℓ1 path-norm control. For the Lp-type integral spaces widetildeFp,τd,s, 1≤ p≤2, spherical harmonic analysis yields approximation bounds for shallow networks. In particular, when τd is the uniform measure and 1≤ p<2, the approximation rate is O!left(m^-fracp(2s+2d+1)-2d2dpright) for 1≤ p≤ p^* and O!left(m^-fracp(4s+3d-1)-2d+24dpright) for p^*<p<2, where p^*=frac2d+2d+3. Approximation bounds for Sobolev spaces Wα,p, 1≤ p<2, are obtained through embeddings into spectral Barron spaces. For nonparametric regression with sub-Gaussian noise, path-norm-regularized shallow ReLUs networks achieve minimax-optimal rates O!left(n^-fracd+2s+12d+2s+1log nright) over mathscrBs and O!left(n^-frac2α2α+dlog nright) over Wα,∞, with matching lower bounds up to logarithmic factors.