# Sobolev Approximation of Deep ReLU Networks in Log-Barron Space

arXiv: https://arxiv.org/abs/2601.01295 · HTML: https://mlsanity.com/paper/2601.01295/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 26 references that are in the map.
- First posted on arXiv 3 Jan 2026.

## Builds on (most cited first, 11 of 11)

- [Error bounds for approximations with deep ReLU networks](https://mlsanity.com/paper/1610.01145/) (2016)
- [The Expressive Power of Neural Networks: A View from the Width](https://mlsanity.com/paper/1709.02540/) (2017)
- [Benefits of depth in neural networks](https://mlsanity.com/paper/1602.04485/) (2016)
- [Universal Approximation with Deep Narrow Networks](https://mlsanity.com/paper/1905.08539/) (2019)
- [Why and When Can Deep -- but Not Shallow -- Networks Avoid the Curse of Dimensionality: a Review](https://mlsanity.com/paper/1611.00740/) (2016)
- [Minimum Width for Universal Approximation](https://mlsanity.com/paper/2006.08859/) (2020)
- [Sharp Bounds on the Approximation Rates, Metric Entropy, and n-widths of Shallow Neural Networks](https://mlsanity.com/paper/2101.12365/) (2021)
- [Representation formulas and pointwise properties for Barron functions](https://mlsanity.com/paper/2006.05982/) (2020)
- [On the Representation of Solutions to Elliptic PDEs in Barron Spaces](https://mlsanity.com/paper/2106.07539/) (2021)
- [A general approximation lower bound in Lp norm, with applications to feed-forward neural networks](https://mlsanity.com/paper/2206.04360/) (2022)
- [Neural Hilbert Ladders: Multi-Layer Neural Networks in Function Space](https://mlsanity.com/paper/2307.01177/) (2023)

## Built on by (most cited first, 1 of 1)

- [Fractional Parabolic Partial Differential Equations in Anisotropic Spectral Barron Spaces: Regularity and Neural Approximation](https://mlsanity.com/paper/2607.27781/) (2026)

## Abstract (arXiv)

Universal approximation theorems show that neural networks can approximate any continuous function; however, the number of parameters may grow exponentially with the ambient dimension, so these results do not fully explain the practical success of deep models on high-dimensional data. Barron space theory addresses this: if a target function belongs to a Barron space, a two-layer network with n parameters achieves an O(n-1/2) approximation error in L2. Yet classical Barron spaces mathscrBs+1 still require stronger regularity than Sobolev spaces Hs, and existing depth-sensitive results often assume constraints such as sL ≤ 1/2. In this paper, we introduce a log-weighted Barron space mathscrBlog, which requires a strictly weaker assumption than mathscrBs for any s>0. For this new function space, we first study embedding properties and carry out a statistical analysis via the Rademacher complexity. Then we prove that functions in mathscrBlog can be approximated by deep ReLU networks with explicit depth dependence. We then define a family mathscrBs,log, establish approximation bounds in the H1 norm, and identify maximal depth scales under which these rates are preserved. Our results clarify how depth reduces regularity requirements for efficient representation, offering a more precise explanation for the performance of deep architectures beyond the classical Barron setting, and for their stable use in high-dimensional problems used today.