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arxiv: 2607.01266 · v1 · pith:7DZ4SGXMnew · submitted 2026-06-29 · 🧮 math.LO · cs.LG· math.FA

Fast approximation and learning of binary classification tasks in o-minimal structures using ReLU neural networks

Clemens Kinn , Philipp Petersen This is my paper

Pith reviewed 2026-07-03 22:20 UTC · model grok-4.3

classification 🧮 math.LO cs.LGmath.FA
keywords o-minimal structuresReLU neural networksapproximation ratesbinary classificationempirical risk minimizationdefinable setstraceable setslearning theory
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The pith

ReLU neural networks approximate characteristic functions of traceable sets in o-minimal structures with size O(ε^{-p(n-1)/m}) and fixed depth.

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

The paper establishes that characteristic functions of traceable subsets of the unit cube can be approximated in the L^p norm to accuracy ε by ReLU networks whose total size scales as a power of 1/ε, while depth remains independent of ε and weights grow only polynomially. Traceable sets act as a classical stand-in for definable sets arising in o-minimal expansions of the reals, which include many algebraic, semi-algebraic, and exponential geometries. The same approximation rates apply to a subclass of definable real-valued maps, and feeding the networks into empirical risk minimization with hinge loss produces classifiers whose expected misclassification error on N uniform samples decays as N^{-m/(m+pn-p)}.

Core claim

Under uniform bounds on the number of connected components and suitable C^m extensions for boundary functions, the characteristic functions of traceable subsets of [-1/2,1/2]^n can be approximated in L^p to accuracy ε>0 by ReLU neural networks of size O(ε^{-p(n-1)/m}), with depth independent of ε and polynomially bounded weights. The same rates hold for a subclass of definable maps from the cube to the reals. Combining the approximation result with entropy estimates for ReLU network classes shows that empirical risk minimization with hinge loss achieves expected misclassification error of order N^{-m/(m+pn-p)} for N uniformly distributed samples.

What carries the argument

Traceable sets, defined via uniform bounds on connected components and C^m boundary extensions, serving as proxies for definable sets via cell decomposition in o-minimal structures.

Load-bearing premise

Traceable sets have boundaries that admit C^m extensions and the number of connected components stays uniformly bounded independent of the particular set.

What would settle it

A concrete traceable set in dimension n=2 whose characteristic function requires ReLU network size larger than order ε^{-p/m} to reach L^p approximation error ε.

Figures

Figures reproduced from arXiv: 2607.01266 by Clemens Kinn, Philipp Petersen.

Figure 1
Figure 1. Figure 1: The upper boundary of A approaches any x3 ∈ (0, 1) as (x1, x2) → (0, 0) x y z [PITH_FULL_IMAGE:figures/full_fig_p019_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The set A approaches {(0, x2, x3) : x 2 2 + x 2 3 = 1}, rather than {(0, x2, x3) : x 2 2 + x 2 3 ≤ 1}. 19 [PITH_FULL_IMAGE:figures/full_fig_p019_2.png] view at source ↗
read the original abstract

We study binary classification problems whose decision sets are given by definable sets in o-minimal expansions of the real field. Motivated by cell decomposition of definable sets, we introduce traceable sets as a classical proxy for definable decision regions and analyze their approximation by ReLU neural networks. Under uniform bounds on the number of connected components and suitable $C^m$ extensions for the boundary functions, we prove that characteristic functions of traceable subsets of $[-1/2,1/2]^n$ can be approximated in $L^p$ to accuracy $\varepsilon>0$ by ReLU neural networks of size $\mathcal{O}(\varepsilon^{-p(n-1)/m})$, with depth independent of $\varepsilon$ and polynomially bounded weights. This establishes quantitative approximation rates for certain definable collections in o-minimal structures using ReLU neural networks. The same approach also yields the stated approximation rates for a subclass of definable maps $[-1/2,1/2]^n \to \mathbb{R}$. We then combine the approximation capabilities with entropy estimates for ReLU neural network classes to obtain statistical learning rates for empirical risk minimization with hinge loss. For $N$ uniformly distributed samples, the resulting classifiers achieve expected misclassification error of order $N^{-m/(m+pn-p)}$ up to an arbitrarily small polynomial loss.

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 traceable sets as a proxy for definable sets arising from cell decomposition in o-minimal expansions of the real field. Under the explicit hypotheses of uniform bounds on the number of connected components and the existence of suitable C^m extensions of the boundary functions, it proves that the characteristic functions of traceable subsets of [-1/2,1/2]^n admit L^p approximation to accuracy ε by ReLU networks of size O(ε^{-p(n-1)/m}), with depth independent of ε and polynomially bounded weights. The same rates are obtained for a subclass of definable maps. These approximation results are then combined with standard entropy bounds on ReLU classes to derive that empirical risk minimization with hinge loss, on N uniform samples, yields expected misclassification error of order N^{-m/(m+pn-p)} (up to an arbitrarily small polynomial factor).

Significance. If the stated hypotheses suffice for the constructions, the manuscript supplies the first explicit quantitative approximation and learning rates that link o-minimal geometry with ReLU network approximation theory and statistical learning. The explicit dependence on dimension n, smoothness m, and accuracy ε, together with the use of cell-decomposition motivation followed by standard entropy estimates, constitutes a concrete contribution rather than an existence result. The paper correctly identifies the hypotheses as prerequisites and applies off-the-shelf techniques thereafter.

minor comments (3)
  1. [Abstract] Abstract, final sentence: the phrase 'up to an arbitrarily small polynomial loss' should be replaced by a precise statement of the loss factor (e.g., N^δ for any δ>0) already in the abstract, matching the main theorem.
  2. The definition of 'traceable sets' (presumably in §2 or §3) should include an explicit comparison, even if brief, to the cell-decomposition theorem that motivates it, so that the 'proxy' relation is immediately visible to readers outside o-minimality.
  3. Notation for the network size bound O(ε^{-p(n-1)/m}) should be accompanied by a short remark on whether the implicit constant depends on the uniform bound on connected components or only on n, m, p.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and positive assessment of the manuscript, including the recognition of its contribution in providing explicit quantitative rates linking o-minimal geometry to ReLU approximation and learning theory. The recommendation of minor revision is appreciated; we will address any editorial or clarity improvements in the revised version.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The derivation chain starts from explicit hypotheses (uniform bounds on connected components of traceable sets and existence of suitable C^m extensions of boundary functions) and proceeds via cell decomposition to construct ReLU networks whose size bound O(ε^{-p(n-1)/m}) and depth independence follow from standard entropy estimates for ReLU classes. The subsequent learning rate N^{-m/(m+pn-p)} is obtained by combining these approximation rates with hinge-loss ERM analysis. None of these steps reduce by the paper's own equations to fitted parameters, self-definitions, or load-bearing self-citations; the central claims remain independent of the target rates.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claims rest on the existence of cell decompositions in o-minimal structures and on standard entropy estimates for ReLU networks; traceable sets are introduced as an auxiliary class.

axioms (2)
  • domain assumption Every definable set in an o-minimal expansion of the real field admits a cell decomposition with finitely many cells.
    Invoked to motivate traceable sets as proxies and to control the number of connected components.
  • standard math ReLU network classes admit entropy bounds polynomial in the number of parameters.
    Used to convert approximation rates into statistical learning rates via empirical risk minimization.
invented entities (1)
  • traceable sets no independent evidence
    purpose: Classical proxy for definable decision regions that admit uniform component bounds and C^m boundary extensions.
    Introduced to obtain concrete approximation rates while remaining inside the o-minimal setting.

pith-pipeline@v0.9.1-grok · 5772 in / 1571 out tokens · 29564 ms · 2026-07-03T22:20:14.936994+00:00 · methodology

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