Barron Spaces: Dimension-Free Neural Approximation

• Barron spaces are Banach function spaces defined via weighted Fourier norms that allow shallow neural networks to approximate functions with dimension-independent error rates.
• Spectral Barron spaces extend this framework, incorporating weighted integrability conditions to capture varying levels of regularity and achieve refined approximation results.
• They play a pivotal role in high-dimensional PDE analysis and machine learning, facilitating operator learning and providing explicit network-size/accuracy trade-offs.

A Barron space is a Banach function space characterizing those functions that can be well-approximated by shallow (two-layer) neural networks, with the key property that this approximation is achieved at rates independent of the ambient dimension. Spectral Barron spaces generalize this notion to encode arbitrary levels of regularity, defined via weighted integrability conditions on the Fourier transform or, equivalently, via weighted moment conditions. These spaces serve as the mathematically natural habitat for the paper of function approximation, regularity theory, and operator theory in contexts ranging from high-dimensional PDE analysis to machine learning, including neural network expressivity, generalization, and inverse problems.

1. Definitions and Core Structure

The classical Barron space on $\mathbb{R}^d$ is defined as the set of functions whose Fourier transform $\widehat f$ satisfies

$$B^1(\mathbb{R}^d) = \left\{ f : \int_{\mathbb{R}^d} |\widehat f(\xi)| \, |\xi| d\xi < \infty \right\}.$$

This generalizes to the spectral Barron space $B^s(\mathbb{R}^d)$ for $s\in\mathbb{R}$: $$B^s(\mathbb{R}^d) = \left\{ f : \int_{\mathbb{R}^d} |\widehat f(\xi)| \, (1+|\xi|^2)^{s/2} d\xi < \infty \right\},$$ with norm $$\|f\|_{B^s} := \int_{\mathbb{R}^d} |\widehat f(\xi)| (1+|\xi|^2)^{s/2} d\xi$$. These are Banach spaces, satisfy density of the Schwartz class, and enjoy robust algebra and continuous embedding properties; for $s\geq0$, $B^s\subset L^\infty$ and the space is closed under multiplication and certain Fourier multipliers (Choulli et al., 9 Jul 2025, Chen et al., 2022).

On compact domains, Barron spaces can alternatively be defined by their infinite-width neural network representations. For the ReLU activation, the Barron norm is

$$\|f\|_B = \inf_{\mu} \int |a| (|w| + |b|) \, d|\mu|(a,w,b),$$

where the infimum is over Radon measures yielding the correct integral ridge function representation of $f$ (E et al., 2020, E et al., 2019).

2. Interpolation, Hierarchies, and Embeddings

Spectral Barron spaces $B^s$ form a continuous scale generated by the positive operator $L = I - \Delta$: $B^s = L^{-s/2} B^0$, with $B^0 = \{f: \widehat f \in L^1\}$ (Lu et al., 6 Feb 2025, Choulli et al., 9 Jul 2025). Real interpolation theory yields

$$\|f\|_{B^s} \leq \|f\|_{B^r}^\theta \|f\|_{B^t}^{1-\theta}, \quad \text{if } s = \theta r + (1-\theta) t,$$

and $B^s=(B^r,B^t)_{\theta,1}$ in the real interpolation sense. For $0<\theta<1$, $B^\theta$ embeds compactly into Hölder spaces $C_b^{0,\theta}$, i.e., $B^\theta\hookrightarrow C_b^{0,\gamma}$ for all $\gamma<\theta$ (Choulli et al., 9 Jul 2025).

Hierarchical structure is also visible in Barron spaces defined via higher-order activations, such as RePU$_s$, with explicit norm-continuous embeddings $B^{\text{RePU}_t}\hookrightarrow B^{\text{RePU}_s}$ for $0\leq s\leq t$ (Heeringa et al., 2023). In the context of Fourier-analytic and integral Barron norms, tight embedding inequalities parametrize the relationship between integral Barron spaces $\mathcal{B}_s$ and spectral Barron spaces $\mathcal{F}_s$, with

$$\delta \|f\|_{\mathcal{F}_{s-\delta}(\Omega)} \lesssim_s \|f\|_{\mathcal{B}_s(\Omega)} \lesssim_s \|f\|_{\mathcal{F}_{s+1}(\Omega)},$$

uniformly in the ambient dimension (Wu, 2023).

3. Approximation Theory and Dimension-Free Rates

A fundamental property of Barron and spectral Barron spaces is the dimension-independent rate of function approximation by shallow neural networks. If $f\in B^1$, then for every $m$ there exists a two-layer network $f_m$ such that

$\|f-f_m\|_{L^2} \lesssim \frac{\|f\|_B}{\sqrt{m}},$

with the leading exponent $1/2$ independent of $d$ (Choulli et al., 9 Jul 2025, Chen et al., 2022, Chen et al., 2021, E et al., 2019).

This Monte-Carlo-type rate is sharp in the sense that it cannot be improved without additional smoothness assumptions or coefficient constraints (Lu et al., 21 Oct 2025). For $B^s$-regularity, more refined rates are available depending on activation smoothness and Fourier decay. For example, with ReLU activation, if $$\int |\widehat f(\xi)| (1+|\xi|^2)^{s/2} d\xi < \infty$$ for some $s \geq 1$, the same $O(m^{-1/2})$ approximation rate holds, and, by embedding, applies to higher-order RePU activations (Heeringa et al., 2023).

For exponential spectral Barron spaces—defined as those with $\|\widehat f\|_{L^1(\exp(c|\xi|^\beta))}<\infty$, $0<\beta<1$—sub-exponential-in-$m$ rates are achieved in every Sobolev norm, i.e.,

$\|f-f_N\|_{H^m} \lesssim e^{-c'N^{\beta/d}},$

demonstrating even faster convergence for ultra-smooth Gevrey-type targets (Abdeljawad et al., 27 Dec 2024).

4. Spectral Barron Spaces in Analysis and PDEs

Spectral Barron spaces admit a complete functional-analytic and PDE-theoretic framework. Fractional powers of $I-\Delta$ identify $B^s$ with the domain of $(I-\Delta)^{s/2}$, and interpolation and multiplier theorems provide access to classical and quantum operator theory (Choulli et al., 9 Jul 2025, Mensah, 18 Sep 2025).

In the analysis of elliptic and Schrödinger-type equations, regularity results assert that if $f\in B^s$ and $V = \alpha + W$ with $W\in B^s$ for $\alpha>0$, then the solution $u$ of $-\Delta u + V u = f$ satisfies $u\in B^{s+2}$ and an explicit a priori bound in the $B^{s+2}$-norm (Chen et al., 2022). This enables dimension-independent two-layer neural network approximation of PDE solutions in high dimensions.

On bounded domains, spectral Barron spaces (and their sine/cosine/exponential analogs) control expansion coefficients with mixed-norm weights, and are directly compatible with classical and PDE-based functional analysis. Embedding into $H^t$ and $C^{k,\gamma}$ spaces holds when $t>s+d/2$ and $s=k+\gamma$ (Choulli et al., 9 Jul 2025, Lu et al., 6 Feb 2025).

5. Applications in Machine Learning and Operator Learning

Barron and spectral Barron spaces are foundational for the expressivity, generalization, and sample complexity theory of shallow neural networks. The path norm, which defines the Barron Banach space, tightly controls both the approximation error and the Rademacher complexity, ensuring that consistent learning can occur at sample complexity $O(1/\sqrt{n})$ irrespective of the input dimension (E et al., 2019, Spek et al., 2022, Chung et al., 2023).

These spaces also appear as the correct functional setting for graph convolutional neural networks (GCNNs), where a graph-Barron structure can be developed as a reproducing kernel Banach space decomposable into a union of neuron-kernel RKHSs, again allowing dimension-free approximation and generalization (Chung et al., 2023).

For operator learning, shallow networks approximating functionals such as PDE symbols achieve uniform approximation in Fréchet metrics provided the symbol lies in a suitable exponential-spectral Barron space, enabling explicit network-size/accuracy tradeoffs for learning operators rather than just scalar functions (Abdeljawad et al., 27 Dec 2024).

6. Extensions: Quantum, Anisotropic, and Nonclassical Generalizations

Spectral Barron constructions extend to quantum harmonic analysis, where spectral Barron spaces of operators (rather than scalar functions) are defined in terms of quantum Fourier transforms. These non-commutative analogs retain Banach-space structure, completeness, and embedding/interpolation properties, with direct applications to quantum-mechanical PDEs (Mensah, 18 Sep 2025).

Anisotropic weighted Fourier–Lebesgue Barron spaces allow for separate space-time regularity and control approximation in mixed (Bochner-Sobolev) norms, important for time-dependent PDEs and operator learning in infinite-dimensional settings (Abdeljawad et al., 2023).

Nonclassical smoothness hierarchies, such as ADZ spaces defined by Mellin multipliers instead of Fourier multipliers, explain the dimension-independence of Barron approximation as arising from nonclassical scale invariance rather than from traditional Sobolev-type smoothness (Schavemaker, 17 Aug 2025).

7. Limitations, Rigidity, and Open Problems

While Barron and spectral Barron spaces dramatically mitigate or circumvent the classical curse of dimensionality under appropriate regularity, they exclude functions with fractal, curved, or manifold-supported singular sets; the singular locus of a Barron function is always a countable union of affine subspaces (E et al., 2020). Diffeomorphisms acting on Barron spaces are rigid: only affine maps preserve the Barron structure. Additionally, in boundary-value or highly non-linear PDE contexts, Barron-space containment may fail, necessitating deeper network architectures or new function space frameworks (E et al., 2020, Chen et al., 2022). Optimal approximation rates in the Barron regime become unattainable with insufficient smoothness or further constraints on coefficient $\ell^1$-norms (Lu et al., 21 Oct 2025).

The extent to which more general function classes—e.g., those with higher-order or nonclassical regularity—admit dimension-free neural approximation, or how to precisely characterize the interpolation between Barron and Sobolev/Besov spaces, remains an open domain for analysis (Wu, 2023, Schavemaker, 17 Aug 2025).