Barron Space in Neural Networks

• Barron space is the canonical function space capturing the approximation and generalization of infinite-width two-layer neural networks using ReLU and higher-order activations.
• It offers two closely linked formulations—expectation-based and Fourier-based—that enable dimension-free approximation rates and sparse function representations.
• Barron spaces facilitate rigorous analysis of high-dimensional PDEs, quantum systems, and graph neural networks through precise norm equivalences, embeddings, and spectral decay.

Barron space is the canonical function space governing the approximation and generalization properties of infinite-width two-layer (single-hidden-layer) neural networks, particularly with ReLU and higher-order activations. It admits two principal, tightly-linked formulations: the (expectation-based) parameter-space Barron class and the (Fourier-based) spectral Barron class. The infrastructure of Barron spaces provides a quantitative, dimension-free framework for understanding when and how neural networks break the curse of dimensionality through network representations, sparse approximations, and regularity results for high-dimensional PDEs and quantum wavefunctions.

1. Definitions and Fundamental Norms

Barron spaces are defined for functions $f: \Omega \to \mathbb{R}$ (with $\Omega \subset \mathbb{R}^d$ compact) and parameterized by a smoothness index $s \in [0, \infty)$. There are two core variants:

a) Spectral Barron space $\mathcal{F}_s(\Omega)$ (Fourier-based):

$$\|f\|_{\mathcal{F}_s(\Omega)} = \inf_{f_e|_{\Omega} = f} \int_{\mathbb{R}^d} (1+\|\xi\|_\Omega)^s |\hat{f}_e(\xi)|\,d\xi$$

where the infimum is over all tempered-distribution extensions $f_e : \mathbb{R}^d \to \mathbb{R}$, and the dual norm $\|v\|_\Omega = \sup_{x \in \Omega} |v \cdot x|$ controls directional frequency components (Wu, 2023).

b) Barron space $\mathcal{B}_s(\Omega)$ (expectation-based):

$$\|f\|_{\mathcal{B}_s(\Omega)} = \inf_{\rho \in A_f} \mathbb{E}_{(a,w,b)\sim\rho} \left[ |a| ( \|w\|_\Omega + |b| )^s \right]$$

where $$f_\rho(x) = \mathbb{E}_{(a,w,b)\sim\rho}[ a \cdot \sigma(w \cdot x + b) ]$$ for $\sigma(z) = \max(0, z)^s$ (i.e., ReLU$^s$), and $A_f$ is the set of all parameter measures representing $f$ on $\Omega$ (Wu, 2023, E et al., 2019, E et al., 2020).

Barron norm minimization admits numerous equivalent representations: parameter-space measures, spherical transforms, and as variation norms over single-neuron activation bases (E et al., 2020).

2. Embedding Theorems and Equivalence

A principal contribution of Barron-space theory is the dimension-free equivalence and embeddings between $\mathcal{B}_s$ and $\mathcal{F}_s$ up to a sharp shift in the smoothness parameter: $$\delta \|f\|_{\mathcal{F}_{s-\delta}(\Omega)} \lesssim_s \|f\|_{\mathcal{B}_s(\Omega)} \lesssim_s \|f\|_{\mathcal{F}_{s+1}(\Omega)}$$ for any $\delta \in (0, 1)$ and positive integer $s$, where the hidden constants depend only on $s$ and not on dimension $d$ (Wu, 2023).

• Upper bound sharpness: $B_s(\Omega) \hookrightarrow F_{s+1}(\Omega)$ is optimal; e.g., $F_s \subset B_1$ only if $s \geq 2$ [Caragea–Petersen–Voigtländer].
• Lower bound sharpness: No embedding $B_s(\Omega) \rightarrow F_s(\Omega)$ exists, even in one dimension (construct explicit functions whose $B_1$ norm is finite but $F_1$ infinite).

These results formalize that expectation-based and Fourier-based Barron spaces are "essentially the same" up to a shift of unit order in smoothness—a key technical fact for neural approximation (Wu, 2023, Caragea et al., 2020).

3. Function-Theoretic and Geometric Structure

Barron spaces admit rich structural properties, crucial for both approximation and regularity theory.

• Path-norm and total variation: The Barron norm is the tightest $L^1$ total variation of the measure representing $f$ as a (possibly infinite) superposition of neurons (E et al., 2020, E et al., 2019).
• Spherical/homogeneous decompositions: Every Barron function on $\mathbb{R}^d$ admits decomposition into a bounded part and a positively one-homogeneous component; only affine $C^1$ diffeomorphisms preserve Barron structure (E et al., 2020).
• Singular set geometry: The non-smooth locus of a Barron function is supported on countably many affine hyperplanes, ruling out functions with curved or fractal singular sets from Barron spaces (E et al., 2020).
• Sobolev embeddings: If $f \in H^s(\mathbb{R}^d)$ with $s > d/2 + 2$, then $f \in B(\mathbb{R}^d)$, and every Barron function is globally Lipschitz (E et al., 2020).
• Spectral decay: Membership in spectral Barron spaces is equivalent to Fourier integrability with weighted decay. For $B^s$, the requirement is $|\widehat{f}(\xi)| = O(|\xi|^{-s-\varepsilon})$ as $|\xi| \to \infty$ (Liao et al., 2023, Choulli et al., 9 Jul 2025).

4. Barron Spaces and Neural Network Approximation

The foundational role of Barron spaces in neural network theory is established via the following core results:

• Dimension-free approximation: For any $f \in \mathcal{B}_s(\Omega)$ or $\mathcal{F}_s(\Omega)$, there exists a width-$m$ two-layer network $f_m$ such that

$$\| f - f_m \|_{L^2(\Omega)} \leq C_s\, \|f\|\, m^{-1/2}$$

with $C_s$ independent of $d$ (Wu, 2023, E et al., 2019, E et al., 2020).

• Generalization rates: The Rademacher/Monte Carlo complexity of the class $\{ f : \|f\|_{\mathcal{B}_s} \leq R \}$ scales as $O(R / \sqrt{n})$ without a dimension factor, yielding statistical risk bounds of order $O(m^{-1/2}) + O(n^{-1/2})$ (E et al., 2019, Caragea et al., 2020).
• Network expressivity: The two-layer Barron framework quantifies the function classes on which shallow networks outperform classic Sobolev rates (where $O(m^{-s/d})$ rates incur the "curse of dimensionality") (Lu et al., 21 Oct 2025, Schavemaker, 17 Aug 2025).
• Sparse parameterizations: Recent variational and inverse scale space methods enable learning sparse Barron representations with monotone convergence and stability under discretization, measurement noise, and sampling bias (Heeringa et al., 2023).

5. Spectral Barron Spaces and Deep Architectures

Spectral Barron norms generalize to settings beyond Euclidean spaces, with significant consequences:

• High-dimensional PDEs: Solutions to elliptic and parabolic PDEs with Barron-type data remain in Barron/spectral Barron spaces, allowing two-layer nets to approximate solutions with complexity scaling at most polynomially in $d$ (not exponentially), provided the right-hand side, coefficients, and boundary terms have Barron regularity (Chen et al., 11 Aug 2025, Chen et al., 2021, E et al., 2020).
• Schrödinger eigenfunctions: Electronic and many-body quantum eigenfunctions with singular Coulomb (or general inverse-power) potentials are shown to belong to spectral Barron spaces $\mathcal{B}^s$ for $s < 1$ or $s < s'$ as dictated by sharp decay in the Fourier domain, thus admitting dimension-free neural approximation (Yserentant, 25 Feb 2025, Ming et al., 25 Aug 2025).
• Groups and manifolds: For compact groups $G$ and vector-valued functions, spectral Barron spaces are defined via weighted Schatten-class summability of matrix-valued Fourier coefficients. These spaces enjoy completeness, interpolation, duality, and embedding into Sobolev/continuous function spaces, making them natural contexts for neural architectures over manifolds and symmetry groups (Mensah et al., 13 Dec 2025).
• Graph structures: Analogues of Barron space for graph convolutional neural networks (GCNNs) characterize expressivity, path-norm control, and universal approximation in the non-Euclidean regime (Chung et al., 2023).

6. Hierarchies, Embeddings, and Activation Dependence

The approximation power and inclusivity of Barron spaces are highly activation-dependent:

• RePU (ReLU$^s$) Barron spaces: There exists a strict hierarchy among Barron spaces with polynomial activation order: $B_{\mathrm{RePU}_t} \subset B_{\mathrm{RePU}_s}$ for all $0 \leq s \leq t$, mirroring the Sobolev scales (Heeringa et al., 2023). Smooth activations can be embedded into higher-order RePU spaces via push-forward measures and Taylor expansions.
• Optimal rates and limitations: For generic $W^{s, p}$-smooth targets, best possible approximation rates by shallow networks remain $O(m^{-s/d})$. In contrast, if $f$ lies in a (spectral) Barron class, the $O(m^{-1/2})$ rate is attained, dimension-free (Lu et al., 21 Oct 2025). However, imposing strong $\ell^1$ coefficient bounds may preclude achieving optimal exponents, and insufficient smoothness or nonclassical regularity incurs unavoidable dimension dependence (Lu et al., 21 Oct 2025, Schavemaker, 17 Aug 2025).
• Comparison to nonclassical ADZ spaces: Recent analyses using Mellin transforms show that Barron spaces’ claimed “dimension-independent” rates are explained by endowing functions with large “nonclassical” smoothness, quantified by symmetry and Mellin-analytic structure (Schavemaker, 17 Aug 2025).

7. Symmetry, Antisymmetry, and Quantum Applications

Special structure in function classes allows further efficiency gains:

• Antisymmetric Barron spaces: For problems requiring fully antisymmetric functions (e.g., electronic wavefunctions obeying Pauli statistics), explicit constructions show that sums of $m$ Slater determinants approximate any antisymmetric Barron function with an error bound scaling as $C / \sqrt{m}$ and only polynomial (not factorial) dependence in $n$ (number of particles). Determinant-based neural architectures thus acquire a rigorous theoretical justification as optimal in the Barron class (Abrahamsen et al., 2023).
• Factorial vs. polynomial complexity: Encoding antisymmetry in function representation yields a factorial improvement in sample complexity compared to naive symmetrization of generic Barron networks (Abrahamsen et al., 2023).

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The Barron space framework unifies infinite-width neural approximation theory, Fourier and functional analysis, and the study of high-dimensional PDEs and quantum systems. Its precise embedding inequalities, universality properties, and structural insights are central to both theoretical and practical advances in deep learning, scientific computing, and the mathematics of high-dimensional function spaces (Wu, 2023, E et al., 2019, Liao et al., 2023, E et al., 2020, Chen et al., 11 Aug 2025, Chen et al., 2021, Mensah et al., 13 Dec 2025, Yserentant, 25 Feb 2025, Ming et al., 25 Aug 2025, Lu et al., 21 Oct 2025, E et al., 2020).