ADZ Spaces in Neural Network Approximation

• ADZ Spaces are a family of smoothness spaces that capture nonclassical regularity in infinitely wide shallow neural networks through harmonic decomposition and Mellin analysis.
• They bridge classical Barron spaces and neural-network spaces by decomposing functions into spherical harmonics and modulating radial smoothness via specialized Mellin multipliers.
• Their framework reinterprets dimension-independent approximation rates by employing Abel summation and anisotropic differentiation, aligning neural network theory with nonclassical smoothness.

Searching arXiv for the specified paper to ground the article in the cited source. ADZ spaces are a family of smoothness spaces introduced by Olov Schavemaker to capture a nonclassical notion of regularity associated with infinitely wide shallow neural networks. In the setting of the paper “Does the Barron space really defy the curse of dimensionality?” (Schavemaker, 17 Aug 2025), they are motivated by the observation that Barron spaces appear to evade the curse of dimensionality when smoothness is measured classically, yet may instead exhibit high regularity under a different smoothness calculus built from spherical harmonic decomposition, Mellin multipliers, and Abel summation. The resulting framework places ADZ spaces between Barron spaces and a class $N^\alpha$ of functions representable as infinitely wide shallow neural networks, thereby recasting dimension-independent approximation phenomena in terms of nonclassical smoothness rather than anomalously favorable classical behavior (Schavemaker, 17 Aug 2025).

1. Origin in Barron-space approximation theory

Barron spaces occupy a central position in the theory of shallow neural networks because they consist of functions that are efficiently approximable by such networks. In the formulation cited in the paper, the classical Barron space $B^1$ is

$$B^1 = \left\{ F\{\phi\} : \phi, |\cdot|\phi \in \ell_1(\mathbb{R}^n) \right\},$$

where $F$ is the distributional Fourier transform. Functions in this space can be approximated by shallow neural networks of the form

$$x \mapsto \sum_j a_j \rho(\langle w_j, x \rangle - b_j)$$

at rate $m^{-1/2}$, independently of the ambient dimension $n$ (Schavemaker, 17 Aug 2025).

This dimension independence appears to conflict with the classical smoothness heuristic according to which approximation accuracy scales as $m^{-(\text{smoothness})/n}$ in dimension $n$. The paper therefore asks whether classical smoothness is the correct regularity notion for neural-network approximation. Its answer is to construct ADZ spaces as a smoothness framework tailored to infinitely wide shallow neural networks. This suggests that the relevant regularity is not isotropic derivative regularity in the classical Sobolev or Bessel-potential sense, but a form of regularity organized by harmonic content and radial Mellin analysis.

2. Harmonic decomposition and Mellin-analytic construction

The construction begins by decomposing a continuous function $f \colon \mathbb{R}^n \to \mathbb{C}$ into spherical harmonics. The paper writes

$B^1$0

where, for each degree $B^1$1 and $B^1$2,

$B^1$3

with $B^1$4 the degree-$B^1$5 zonal spherical harmonic kernel (Schavemaker, 17 Aug 2025).

Smoothness is then measured not by applying classical derivatives directly to $B^1$6, but by acting separately on each harmonic component through multiplier operators defined in Mellin space. Given $B^1$7, the Mellin transform is

$B^1$8

The distributional Mellin transform is implemented through the isomorphism

$B^1$9

For each degree $$B^1 = \left\{ F\{\phi\} : \phi, |\cdot|\phi \in \ell_1(\mathbb{R}^n) \right\},$$0 and order $$B^1 = \left\{ F\{\phi\} : \phi, |\cdot|\phi \in \ell_1(\mathbb{R}^n) \right\},$$1, with technical caveats for even and odd $$B^1 = \left\{ F\{\phi\} : \phi, |\cdot|\phi \in \ell_1(\mathbb{R}^n) \right\},$$2, the multiplier function $$B^1 = \left\{ F\{\phi\} : \phi, |\cdot|\phi \in \ell_1(\mathbb{R}^n) \right\},$$3 is given in the generic holomorphic case by

$$B^1 = \left\{ F\{\phi\} : \phi, |\cdot|\phi \in \ell_1(\mathbb{R}^n) \right\},$$4

Here, $$B^1 = \left\{ F\{\phi\} : \phi, |\cdot|\phi \in \ell_1(\mathbb{R}^n) \right\},$$5 is the gamma function and $$B^1 = \left\{ F\{\phi\} : \phi, |\cdot|\phi \in \ell_1(\mathbb{R}^n) \right\},$$6 the Pochhammer symbol. The dependence on $$B^1 = \left\{ F\{\phi\} : \phi, |\cdot|\phi \in \ell_1(\mathbb{R}^n) \right\},$$7 shows that ADZ smoothness is anisotropic in the sense used in the paper: different spherical harmonic sectors are differentiated by different Mellin multipliers rather than a single isotropic Fourier multiplier.

3. Differential operators and formal definition

The basic differential operators are defined componentwise on the harmonic decomposition. For $$B^1 = \left\{ F\{\phi\} : \phi, |\cdot|\phi \in \ell_1(\mathbb{R}^n) \right\},$$8, where $$B^1 = \left\{ F\{\phi\} : \phi, |\cdot|\phi \in \ell_1(\mathbb{R}^n) \right\},$$9 is the set of even degrees, the paper sets

$F$0

For $F$1, a modified operator $F$2 is used to account for subtleties at even degrees. The total differentiated family is then

$F$3

Recombination is performed through Abel summation. If $F$4 denotes the differentiated components, then for each $F$5,

$F$6

and

$F$7

where the Abel sum $F$8 is taken over sequences absolutely Abel summable into the Banach space $F$9 (Schavemaker, 17 Aug 2025).

The sequence space used in the seminorm is

$$x \mapsto \sum_j a_j \rho(\langle w_j, x \rangle - b_j)$$0

For $$x \mapsto \sum_j a_j \rho(\langle w_j, x \rangle - b_j)$$1, the ADZ space $$x \mapsto \sum_j a_j \rho(\langle w_j, x \rangle - b_j)$$2 is the set of all $$x \mapsto \sum_j a_j \rho(\langle w_j, x \rangle - b_j)$$3 such that:

1. for all $$x \mapsto \sum_j a_j \rho(\langle w_j, x \rangle - b_j)$$4,

$$x \mapsto \sum_j a_j \rho(\langle w_j, x \rangle - b_j)$$5

2. the differentiated family

$$x \mapsto \sum_j a_j \rho(\langle w_j, x \rangle - b_j)$$6

satisfies

$$x \mapsto \sum_j a_j \rho(\langle w_j, x \rangle - b_j)$$7

3. the seminorm is finite:

$$x \mapsto \sum_j a_j \rho(\langle w_j, x \rangle - b_j)$$8

The paper summarizes the construction by the formula

$$x \mapsto \sum_j a_j \rho(\langle w_j, x \rangle - b_j)$$9

4. Relation to Barron, Bessel potential, and neural-network spaces

The paper contrasts ADZ spaces with both Barron spaces and Bessel potential spaces. Barron smoothness is defined through Fourier-transform decay: $m^{-1/2}$0 iff $m^{-1/2}$1 with $m^{-1/2}$2. Bessel potential spaces $m^{-1/2}$3 are built via fractional powers of the Laplacian and encode the standard Sobolev notion that $m^{-1/2}$4 has $m^{-1/2}$5 derivatives in $m^{-1/2}$6, measured isotropically in all frequency directions (Schavemaker, 17 Aug 2025).

By contrast, ADZ spaces measure smoothness anisotropically. The decomposition into spherical harmonics separates angular structure from radial structure, and the radial regularity of each harmonic sector is analyzed through Mellin multipliers rather than ordinary derivatives or Fourier multipliers. Recombination through Abel summation is then essential for inverting the decomposition in the form used by the theory. The paper states that this organization is sensitive to the type of smoothness that appears in neural-network approximation and, more specifically, to functions realized by infinitely wide shallow neural networks.

A central embedding result is

$m^{-1/2}$7

where $m^{-1/2}$8 corresponds to functions representable as infinitely wide shallow neural networks. The paper further states the norm inequalities

$m^{-1/2}$9

These inclusions make ADZ spaces an intermediate class: broader than the Fourier-defined Barron space and narrower than the neural-network realizable class $n$0. A plausible implication is that ADZ regularity is intended as the intrinsic smoothness scale on which infinite-width shallow-network approximation is naturally measured.

5. Reinterpretation of the curse of dimensionality

A principal claim of the paper is that Barron spaces appear to defy the curse of dimensionality only when smoothness is assessed classically. In the classical picture, approximation rates of the form $n$1 reflect the interaction between smoothness $n$2 and ambient dimension $n$3. Barron-space approximation, by contrast, yields the dimension-independent rate $n$4 for $n$5, which can seem incompatible with that heuristic (Schavemaker, 17 Aug 2025).

The ADZ framework is introduced precisely to reinterpret this phenomenon. The paper argues that when smoothness is measured by the ADZ seminorm, Barron functions are highly regular in a nonclassical sense. It states that the Barron space’s regularity is “roughly $n$6 units” in the ADZ framework, and that this restores alignment with the standard curse-of-dimensionality folklore: improved approximation rates for Barron functions correlate not with low classical smoothness, but with high nonclassical ADZ regularity.

This does not negate the established approximation results for Barron functions. Rather, it relocates the explanation. The relevant regularity is no longer the isotropic smoothness measured by classical derivative counts or Fourier decay alone, but a Mellin-harmonic regularity tailored to the representation theory of infinitely wide shallow neural networks. This suggests that the apparent tension between Barron-space approximation and dimensionality heuristics is, in the paper’s framework, a consequence of using an unsuitable smoothness scale.

6. Examples, scope, and interpretive significance

The paper describes ADZ spaces as providing a mathematically precise and operationally meaningful notion of nonclassical smoothness relevant for neural-network function approximation. Their significance lies in the way they unify three ingredients: the spectral and angular decomposition of functions via spherical harmonics, nonclassical differentiation via Mellin multipliers, and the relation between high-dimensional infinite-width shallow networks and achievable approximation rates (Schavemaker, 17 Aug 2025).

The paper also emphasizes that classical and ADZ regularity can diverge sharply. In particular, examples in the appendix are said to show that Barron functions can be quite nonsmooth in the classical sense, “Lipschitz at best,” yet still belong to $n$7. This is used to support the claim that efficient neural-network approximation does not require high classical smoothness provided that the function possesses sufficient ADZ regularity. The paper further notes that it provides explicit realization formulas showing how Barron functions, when decomposed into harmonics and Mellin-regularized, yield ADZ regularity.

A common misconception addressed by this framework is that dimension-independent shallow-network approximation rates automatically imply a genuine escape from the curse of dimensionality in every meaningful sense. The paper’s position is narrower: Barron spaces do defy the curse of dimensionality from the point of view of classical smoothness, but they do not do so with respect to the nonclassical notion of smoothness encoded by ADZ spaces. In that sense, ADZ spaces function less as a replacement for Barron theory than as a reinterpretive layer that embeds Barron approximation into an alternative smoothness theory aligned with infinitely wide shallow neural networks.