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Fixed-size networks with elementary activations can approximate Hölder functions to any accuracy, with an explicit bound on how large the weights must grow.

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2026-07-10 21:26 UTC pith:2FG3PLHD

load-bearing objection First explicit non-asymptotic parameter-magnitude bounds for fixed elementary super-expressive nets; CRT encoding is the real novelty, constants are huge but the scaling is clean.

arxiv 2607.06781 v1 pith:2FG3PLHD submitted 2026-07-07 cs.LG

On Explicit Super-Expressive Approximation for Neural Networks

Feng-Lei Fan , Ze-Yu Li , Chen-Yu Wang , Jian-Jun Wang This is my paper
classification cs.LG MSC 68T0741A2541A63
keywords super-expressive approximationfixed-architecture networksChinese Remainder TheoremHölder-smooth functionsparameter magnitude boundselementary activationsgridwise polynomials
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

Standard neural-network approximation theory lets network size grow as the error shrinks while keeping weights bounded. This paper treats the dual problem: freeze width and depth completely, and ask how large the weights must become. Using the Chinese Remainder Theorem to encode which grid cell a point occupies and what quantized value or local Taylor polynomial belongs there, the authors build a single network whose architecture depends only on dimension and smoothness order. For Lipschitz targets the network has width at most max{D,4} and depth 5; for Hölder-smooth targets the width is max{2D,D+5N+1} and the depth is r+9. In both cases they give a fully explicit, non-asymptotic upper bound on the largest parameter magnitude needed to reach error ε. The bound improves when the target is smoother, replacing the exponent 2D by 2D/(r+γ). The construction therefore supplies the missing quantitative half of the super-expressiveness story that earlier existence proofs left open.

Core claim

Any function in C^{r,γ}_A([0,1]^D) can be approximated to accuracy ε by a network of fixed width max{2D,D+5N+1} and fixed depth r+9 that uses only floor, RePU and reciprocal activations, and whose largest parameter satisfies log_{2} P = O(ε^{-2D/(r+γ)} log(1/ε)). The Lipschitz case (r=0,γ=1) reduces to width max{D,4} and depth 5 with the corresponding exponent −2D.

What carries the argument

Chinese Remainder Theorem encoding of grid indices and rational coefficients: pairwise-coprime moduli p_i = 1 + i(J+1)(M^D)! convert a finite table of quantized values (or local Taylor coefficients) into a single integer K that a fixed-size network recovers by modular arithmetic realized with floor, reciprocal and ReQU multiplications.

Load-bearing premise

The moduli are built from the classical factorial construction that guarantees pairwise coprimality; any substantially smaller family of coprime integers would immediately improve the stated weight bound, yet the paper treats this encoding as given.

What would settle it

Compute the CRT integer K for a concrete low-dimensional Lipschitz or Hölder function and a sequence of shrinking ε; if the measured bit length of K grows strictly slower than the claimed O(ε^{-2D/(r+γ)} log(1/ε)), the quantitative bound is false.

Watch this falsifier — get emailed when new claim-graph text bears on it.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 6 minor

Summary. The paper constructs fixed-architecture neural networks with elementary activations (floor, RePU, reciprocal) that approximate Lipschitz and Hölder-smooth functions on [0,1]^D to arbitrary accuracy ε, with fully explicit non-asymptotic bounds on parameter magnitude. For f ∈ Lip_A([0,1]^D), Theorem 1 gives a network of width max{D,4} and depth 5 whose largest parameter P satisfies log_{2} P ≤ O(ε^{-2D} log(1/ε)). For f ∈ C^{r,γ}_A([0,1]^D), Theorem 2 gives width max{2D, D+5N+1} and depth r+9 with log_{2} P = O(ε^{-2D/(r+γ)} log(1/ε)), N = binom(D+r,r). The mechanism is a uniform grid partition, quantization (or rationalized local Taylor polynomials), Gödel-style pairwise-coprime moduli (Lemma 8), CRT recovery of the grid labels/coefficients, and exact ReQU monomial evaluation (Lemma 14). Layer-wise width/depth accounting appears in Tables 1–2.

Significance. The work supplies the first quantitative, non-asymptotic parameter-magnitude bounds for fixed-architecture super-expressive approximation with elementary activations. Prior results either left the parameter growth uncharacterized or made the activation itself error-dependent. The dual viewpoint (architecture fixed, parameters grow with ε) cleanly complements the classical architecture-unbounded literature. Strengths include fully constructive proofs, explicit constants (M_ε, J_ε, F_ε, C_{D,r,γ,A,B}), and transparent layer accounting. The enormous size of the CRT integer K is acknowledged by the authors and only affects the leading constant inside the O-notation; the scaling exponent itself is new and useful.

minor comments (6)
  1. Abstract and Theorem 1: the displayed bound is written as an exact product over moduli, while the abstract claims an O-notation; a short sentence equating the two would avoid any impression of inconsistency.
  2. Section 3.1, Step 3 and Lemma 8: the factorial (M^D)! produces astronomically large moduli. A brief remark that any pairwise-coprime sequence larger than J would suffice (and that the present choice is only for explicitness) would help readers who worry about practicality.
  3. Figure 3: the axis labels appear as Unicode escape sequences rather than readable mathematics; regenerating the figure with proper LaTeX labels would improve clarity.
  4. Table 2, rows 8…7+r: the layer indexing is slightly ambiguous (does the monomial stage begin at layer 8?). A one-line clarification of the cumulative depth would remove any residual doubt.
  5. Definition 7 and Eq. (10): the reciprocal activation is defined as 2-t for t<1; a short note that this choice is only needed for numerical stability outside the relevant range [1,∞) would be useful.
  6. References: a few arXiv e-prints (e.g., Hon & Yang 2021) could be updated to their published versions if available.

Circularity Check

0 steps flagged

No significant circularity: pure constructive existence proofs with all constants defined from ε and known function norms

full rationale

Theorems 1 and 2 are fully constructive existence results. Every quantity used in the parameter bounds (M_ε = ⌈2A/ε⌉ or the Hölder analogue, J_ε, the CRT integer K bounded by the product of the Gödel-style moduli p_i = 1 + i(J+1)(M^D)!, the rational bit length F_ε from Lemma 11, and the Taylor remainder constant C_{D,r,γ}A) is defined directly from the target accuracy ε and the a-priori Lipschitz/Hölder constants of f; none is fitted to data or defined in terms of the final log_{2}P bound it is supposed to establish. The layer-wise width/depth accounting (Tables 1–2), the pairwise-coprimality argument (Lemma 8), the signed-residue recovery (Lemmas 12–13), and the exact monomial evaluation via Mult (Lemma 14) are self-contained elementary constructions that do not rely on self-citation chains, uniqueness theorems imported from the authors, or ansatzes smuggled via prior work. The dual scaling law is therefore an independent consequence of the CRT encoding plus standard approximation estimates, not a restatement of its inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The paper rests entirely on classical number theory (CRT, Gödel β-function style coprimality) and standard real analysis (Taylor with integral remainder, Hölder continuity). No free parameters are fitted; the only ‘invented’ objects are the concrete network modules built from already-published activations.

axioms (3)
  • standard math Chinese Remainder Theorem for pairwise coprime moduli (Lemma 9)
    Invoked to guarantee a single integer K that encodes all grid labels simultaneously.
  • standard math Taylor theorem with integral remainder for C^{r,γ} functions
    Used in the proof of Theorem 2 to bound the local polynomial error by C M^{-(r+γ)}.
  • domain assumption Existence of elementary activations floor, RePU_s and reciprocal that realize exact modular arithmetic and multiplication
    Taken from prior super-expressivity literature (Shen et al., Boullé et al.); the paper does not re-prove their algebraic identities.

pith-pipeline@v1.1.0-grok45 · 26801 in / 2131 out tokens · 26286 ms · 2026-07-10T21:26:30.254771+00:00 · methodology

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read the original abstract

In this work, we investigate the fixed-architecture neural network approximation with explicit parameter bounds and elementary activations. While prior work demonstrated super-expressive approximation using fixed-size networks, they lack quantitative and non-asymptotic characterizations of parameter magnitude with respect to the approximation error. We resolve this issue by introducing the Chinese Remainder Theorem as a constructive encoding mechanism. For Lipschitz continuous functions on $[0,1]^D$, we construct a width-$\max\{D,4\}$, depth-$5$ network with explicit parameter-error trade-offs. For H\"older-smooth functions in $C^{r,\gamma}_A\left([0,1]^D\right)$, our fixed network of width $\max\{2D,\ D+5N+1\}$ and depth $r + 9$ achieves the parameter magnitude $\mathcal{P}$ bounded by $\log_2 \mathcal{P}=\mathcal{O}\bigl(\varepsilon^{-2D/(r+\gamma)}\log(1/\varepsilon)\bigr)$. This is the dual result compared to those in the parameter-bounded and architecture-unbounded paradigm.

Figures

Figures reproduced from arXiv: 2607.06781 by Chen-Yu Wang, Feng-lei Fan, Jian-Jun Wang, Ze-Yu Li.

Figure 1
Figure 1. Figure 1: Visualization of the floor activation ρfloor(t), the RePU activation ρ1(t), ρ2(t), and the reciprocal activation ρinv. work that used irrational winding and KST, we apply CRT to give an explicit con￾struction. For the first time, our theorem provides an explicit bound on the parameter magnitude in terms of the approximation error. Theorem 1 (Lipschitz Continuous Function). Let Ω = [0, 1]D and f ∈ LipA(Ω). … view at source ↗
Figure 2
Figure 2. Figure 2: Illustration of Theorem 1 pipeline. we have ¯f(x) := −B + jmδ (15) for ∀x ∈ Qm. Then, for every x ∈ Qm, | ¯f(x) − f(x)| ≤ | ¯f(x) − f(xm)| + |f(xm) − f(x)| ≤ δ + A/M. (16) See [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Scaling law of the parameter magnitude log2 P(Ψε) with respect to the target accuracy log2 (1/ε). 24 [PITH_FULL_IMAGE:figures/full_fig_p026_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Architecture of the target network realizing [PITH_FULL_IMAGE:figures/full_fig_p031_4.png] view at source ↗

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