# Embedding Inequalities for Barron-type Spaces

arXiv: https://arxiv.org/abs/2305.19082 · HTML: https://mlsanity.com/paper/2305.19082/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 26 references that are in the map.
- First posted on arXiv 30 May 2023.

## Builds on (most cited first, 15 of 15)

- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Norm-Based Capacity Control in Neural Networks](https://mlsanity.com/paper/1503.00036/) (2015)
- [Primer: Searching for Efficient Transformers for Language Modeling](https://mlsanity.com/paper/2109.08668/) (2021)
- [A Function Space View of Bounded Norm Infinite Width ReLU Nets: The Multivariate Case](https://mlsanity.com/paper/1910.01635/) (2019)
- [Sharp Bounds on the Approximation Rates, Metric Entropy, and n-widths of Shallow Neural Networks](https://mlsanity.com/paper/2101.12365/) (2021)
- [A Priori Estimates of the Population Risk for Two-layer Neural Networks](https://mlsanity.com/paper/1810.06397/) (2018)
- [Representation formulas and pointwise properties for Barron functions](https://mlsanity.com/paper/2006.05982/) (2020)
- [Characterization of the Variation Spaces Corresponding to Shallow Neural Networks](https://mlsanity.com/paper/2106.15002/) (2021)
- [Neural network approximation and estimation of classifiers with classification boundary in a Barron class](https://mlsanity.com/paper/2011.09363/) (2020)
- [Risk Bounds for High-dimensional Ridge Function Combinations Including Neural Networks](https://mlsanity.com/paper/1607.01434/) (2016)
- [On the Representation of Solutions to Elliptic PDEs in Barron Spaces](https://mlsanity.com/paper/2106.07539/) (2021)
- [Complexity Measures for Neural Networks with General Activation Functions Using Path-based Norms](https://mlsanity.com/paper/2009.06132/) (2020)
- [A Priori Generalization Error Analysis of Two-Layer Neural Networks for Solving High Dimensional Schr"odinger Eigenvalue Problems](https://mlsanity.com/paper/2105.01228/) (2021)
- [Some observations on high-dimensional partial differential equations with Barron data](https://mlsanity.com/paper/2012.01484/) (2020)
- [Tighter Sparse Approximation Bounds for ReLU Neural Networks](https://mlsanity.com/paper/2110.03673/) (2021)

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

- [Solving the Poisson Equation with Dirichlet data by shallow ReLUα-networks: A regularity and approximation perspective](https://mlsanity.com/paper/2412.07728/) (2024)
- [Fractional Parabolic Partial Differential Equations in Anisotropic Spectral Barron Spaces: Regularity and Neural Approximation](https://mlsanity.com/paper/2607.27781/) (2026)

## Abstract (arXiv)

An important problem in machine learning theory is to understand the approximation and generalization properties of two-layer neural networks in high dimensions. To this end, researchers have introduced the Barron space Bs(Omega) and the spectral Barron space Fs(Omega), where the index s∈ [0,∞) indicates the smoothness of functions within these spaces and Omegasubsetℝd denotes the input domain. However, the precise relationship between the two types of Barron spaces remains unclear. In this paper, we establish a continuous embedding between them as implied by the following inequality: for any δ∈ (0,1), s∈ mathbbN+ and f: Omega mapstoℝ, it holds that [ δ |f|_Fs-δ(Omega)lesssims |f|Bs(Omega)lesssims |f|_Fs+1(Omega). ] Importantly, the constants do not depend on the input dimension d, suggesting that the embedding is effective in high dimensions. Moreover, we also show that the lower and upper bound are both tight.