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arxiv: 2112.14523 · v2 · submitted 2021-12-29 · 🧮 math.NA · cs.NA

Deep neural network approximation theory for high-dimensional functions

Pierfrancesco Beneventano , Patrick Cheridito , Robin Graeber , Arnulf Jentzen , Benno Kuckuck This is my paper

Pith reviewed 2026-05-24 12:34 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords deep neural networksapproximation theoryhigh-dimensional functionscurse of dimensionalityexpressive powerfunction compositionlocally Lipschitz functionsmaximum and product functions
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The pith

Deep neural networks can approximate high-dimensional functions composed of locally Lipschitz functions, maxima, and products using polynomially many parameters in the dimension and error.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a framework to analyze the approximation power of deep neural networks for high-dimensional functions. It introduces approximation spaces consisting of sequences of functions that can be built from locally Lipschitz continuous functions, maxima, and products through finite compositions. The authors prove that these spaces are closed under the relevant operations and that DNNs can approximate functions in these spaces with a number of parameters that grows at most polynomially in the input dimension and the reciprocal of the approximation error. This establishes that DNNs have sufficient expressive power to overcome the curse of dimensionality for this class of functions on compact sets.

Core claim

DNNs have sufficient expressive power to approximate, without the curse of dimensionality, certain sequences of functions which can be constructed by means of a finite number of compositions using locally Lipschitz continuous functions, maxima, and products. The number of parameters necessary to represent the approximating DNNs grows at most polynomially in 1/ε and in the input dimension d.

What carries the argument

Approximation spaces of function sequences that are closed under finite compositions with locally Lipschitz continuous functions, maxima, and products, allowing the combination of DNN approximation bounds for the individual operations.

If this is right

  • The parameter count for DNN approximations of such functions scales polynomially rather than exponentially with dimension.
  • Approximations hold on compact sets for any prescribed error ε > 0.
  • The result combines existing bounds for basic operations to cover their compositions.
  • Functions in these spaces can be approximated efficiently by DNNs without the curse of dimensionality.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This suggests a method to certify efficient approximability by checking if a given function can be expressed through such compositions.
  • It may be possible to enlarge the class by including other operations that admit similar polynomial bounds.
  • High-dimensional problems in applications could be addressed if their solutions fall into these approximation spaces.

Load-bearing premise

The functions belong to the closure of the basic operations under composition and the parameter bounds for the basic operations extend to the composed functions without introducing exponential factors in the number of compositions.

What would settle it

Constructing a sequence of functions using only the allowed operations for which the minimal number of DNN parameters needed to achieve error ε grows exponentially with the dimension d.

Figures

Figures reproduced from arXiv: 2112.14523 by Arnulf Jentzen, Benno Kuckuck, Patrick Cheridito, Pierfrancesco Beneventano, Robin Graeber.

Figure 1
Figure 1. Figure 1: Graphical illustration of an example neural network which h [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
read the original abstract

The purpose of this article is to develop a machinery to study the capacity of deep neural networks (DNNs) to approximate high-dimensional functions. In particular, we show that DNNs have the expressive power to overcome the curse of dimensionality in the approximation of a large class of functions. More precisely, we prove that these functions can be approximated by DNNs on compact sets such that the number of parameters necessary to represent the approximating DNNs grows at most polynomially in the reciprocal $1/\varepsilon$ of the prescribed approximation error $\varepsilon>0$ and in the input dimension $d\in\mathbb N$. To this end, we introduce certain approximation spaces, consisting of sequences of functions that can be efficiently approximated by DNNs. We then establish closure properties which we combine with known and new bounds on the number of parameters necessary to approximate locally Lipschitz continuous functions, maximum functions, and product functions by DNNs. The main result of this article demonstrates that DNNs have sufficient expressive power to approximate, without the curse of dimensionality, certain sequences of functions which can be constructed by means of a finite number of compositions using locally Lipschitz continuous functions, maxima, and products.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper introduces approximation spaces consisting of sequences of functions that can be constructed via a finite number of compositions of locally Lipschitz continuous functions, maxima, and products. It proves closure properties of these spaces under the listed operations and combines them with explicit bounds (new and previously published) on the number of DNN parameters required to approximate the basic operations, establishing that the resulting DNN approximants have parameter counts that scale at most polynomially in both the input dimension d and the reciprocal error 1/ε (with the polynomial degree permitted to depend on the fixed construction depth).

Significance. If the central claims hold, the work supplies a concrete, parameter-counting framework that identifies a broad class of high-dimensional functions approximable by DNNs without the curse of dimensionality. The explicit tracking of parameter counts through the closure operations, together with the combination of new and existing bounds on the elementary operations, constitutes a useful addition to the approximation theory of neural networks.

minor comments (3)
  1. The abstract states the main theorem but does not indicate where the explicit parameter bounds for the basic operations (locally Lipschitz, max, product) are proved or referenced; a pointer to the relevant lemmas or prior works in the introduction would improve readability.
  2. Notation for the approximation spaces (e.g., how the sequences are indexed and how the closure is formally defined) is introduced only in the body; a short definitional paragraph or diagram in §1 would help readers track the construction depth.
  3. The dependence of the polynomial degree on the (fixed) construction depth is stated in the abstract but should be restated explicitly when the main theorem is formulated, to avoid any ambiguity about uniformity in depth.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript, the clear summary of its contributions, and the recommendation for minor revision. No specific major comments appear in the provided report, so we have no individual points requiring rebuttal or clarification at this stage. We will proceed with preparing a revised version incorporating any minor editorial suggestions that may arise during the revision process.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The derivation introduces approximation spaces via explicit closure under locally Lipschitz functions, maxima, and products, then tracks parameter counts through these operations to obtain polynomial bounds in 1/ε and d. Bounds on the base operations are stated as either previously known or newly derived within the paper, with the closure lemmas providing an independent counting argument that does not reduce any final bound to a fitted quantity or to a self-referential definition. The central claim therefore rests on the explicit construction and parameter tracking rather than on any circular reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities are visible. The central claim rests on the existence of the approximation spaces and the validity of the cited bounds for basic operations.

pith-pipeline@v0.9.0 · 5751 in / 1083 out tokens · 19919 ms · 2026-05-24T12:34:05.874356+00:00 · methodology

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