On the Dimension-Free Approximation of Deep Neural Networks for Symmetric Korobov Functions

Jinchao Xu; Tong Mao; Yahong Yang; Yulong Lu

REVIEW 4 major objections 3 minor 46 references

On the Dimension-Free Approximation of Deep Neural Networks for Symmetric Korobov Functions

T0 review · 4 major / 3 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read This paper proves that permutation symmetry lets deep networks approximate symmetric Korobov functions without the curse of dimensionality: m parameters give energy-norm error O(m^{-1}), with a prefactor that does not grow exponentially wit

desk verdict Clever Vandermonde symmetrization and a plausible approach, but the central counting lemma contains a false implication that the main dimension-free rate depends on; the proof as written does not support Theorem 1. read the letter →

arxiv 2511.12398 v2 pith:B6UCGH2S submitted 2025-11-16 cs.LG cs.NAmath.NA

Yulong Lu , Tong Mao , Jinchao Xu , Yahong Yang This is my paper

classification cs.LGcs.NAmath.NA MSC 41A2541A3065D1568T07

keywords symmetricKorobovspacesdeepneuralnetworkscurseofdimensionalitysparsegridssquaredReLUpermutationsymmetryapproximationtheorygeneralizationbounds

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper's aim is to show that permutation symmetry removes the curse of dimensionality when neural networks approximate symmetric Korobov functions. It constructs squared-ReLU networks that achieve energy-norm error O(m^{-1}) with m parameters, and it proves that the leading constant is uniformly bounded in the ambient dimension d, so accuracy does not require exponentially many parameters as d grows. It also proves a learning bound for recovering the gradient of such a function from M noisy derivative samples, with squared error O(((log M)^2/M)^{2/3}) and a constant that depends only polynomially on d. A sympathetic reader should care because symmetric high-dimensional functions, such as interatomic potentials or score functions of permutation-invariant densities, are exactly the targets where exponential constants have blocked previous guarantees.

What carries the argument

The central object is the symmetrized sparse-grid basis ψ_{l,i}(x) = Σ_{τ∈S_d} φ_{l_1,i_1}(x_{τ(1)})⋯φ_{l_d,i_d}(x_{τ(d)}), formed from tensor products of one-dimensional hat functions. The paper avoids summing over all d! permutations by proving a Vandermonde identity: for any univariate functions φ_1,…,φ_d, the symmetrized product equals a linear combination of at most d 2^{d-1} functions G_ξ(x), each a product of univariate linear combinations of the φ_j. Each G_ξ is then realized by a squared-ReLU network with a binary-tree product circuit, giving O(d^3 2^{d-1} log d) parameters per basis. Together with an energy-based sparse-grid index set whose symmetrized cardinality is bounded by C_s

What would settle it

Compute the exact cardinality of the symmetric energy-based sparse grid ∪_{l∈X_n∩N^d_ord} i_l for d=2,3,4 and n=10,20,30 and compare it with C_s 2^n e^{π√(5d/3)}. In particular, test the displayed implication Σ 4^{l_j} ≥ (4n+4d−4)/32^i ⇒ |l|∞ ≥ n−⌊2.5i⌋: an explicit counterexample with small d and i would invalidate the bound and hence the dimension-free prefactor.

Watch

Extended reading notes

Core claim

Let X^{2,2}_{sym}([0,1]^d) be the space of functions with square-integrable mixed derivatives up to second order in each coordinate, vanishing on the boundary, and invariant under coordinate permutations. The paper constructs a symmetric squared-ReLU network φ with m parameters such that for every f in that space, ||f−φ||_E ≤ C |f|_{2,2} m^{-1}, where C = O(d^4 (5/6)^d e^{π√(5d/3)} (log d + 2)) is uniformly bounded in d. Since m^{-1} is also the optimal rate among continuous approximators, this gives a dimension-free approximation guarantee. The construction symmetrizes sparse-grid hat-function bases and represents each symmetrized basis as a linear combination of at most d 2^{d-1} product-t

Load-bearing premise

The proof's count of symmetric sparse-grid basis functions rests on the claim that the energy-based index condition forces the maximum coordinate level to be at least n−⌊2.5i⌋; if that implication is false, the number of basis functions could exceed the stated 2^n e^{π√(5d/3)} bound and the dimension-free rate would not follow.

Editorial extensions

If this is right

A single architecture with O(log d) depth and O(d^3 2^{d-1}) width per basis achieves the optimal continuous-approximation rate m^{-1} for the whole symmetric Korobov class, so no faster rate in m is possible.
For gradient-based learning from M samples, the expected energy error decays like M^{-2/3} up to logarithms, with no exponential dependence on d in the constant; this nearly matches the minimax rate in M.
The usual (log m)^{d-1} factor from sparse-grid DNN approximations disappears, so the dimension enters only through the prefactor.
Standard ReLU networks do not inherit the dimension-free constant: the same construction needs an extra factor of about d 2^{d-1} + log m in network size and yields (log m)/m error with an exponential-in-d constant, making squared ReLU essential.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Extension: The Vandermonde identity behind the symmetrization only requires univariate functions and does not use any special property of hat functions, so the strategy should transfer to other tensor-product local bases, such as B-splines or wavelets, by swapping the ReLU-hat approximant while keeping the O(d 2^{d-1}) term count.
Extension: The dimension-free guarantee is proved in the energy norm. A natural next test is whether the same symmetric sparse-grid construction yields L^2 or L^p rates with polynomial-in-d constants; the H^1 argument does not automatically imply those, and the lower bounds may differ.
Extension: In score-matching and force-learning settings, the target is a gradient field of a permutation-invariant potential. This theorem supplies a statistical rate for such networks, but only under uniform marginal and bounded-gradient assumptions, so extending it to non-uniform data distributions would be a concrete next step.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The genuinely new thing here is Lemma 5: the Vandermonde-based symmetrization that reduces the d! permutation sum to O(2^d) terms, together with the explicit squared-ReLU construction for each symmetrized sparse-grid basis. That is a real technical contribution, and the paper is honest about how it differs from ACE/polynomial approaches. If the main theorem were true, the paper would be an important step for symmetric function approximation and for gradient-learning guarantees in score-based and interatomic-potential settings. The energy-based sparse-grid starting point (Bungartz–Griebel) is standard but used thoughtfully.

However, the reader's stress-test hits the load-bearing step, and it is not a nitpick. Proposition 1 claims that the energy condition Σ 4^{l_j} ≥ (4n+4d−4)/32^i implies |l|∞ ≥ n−⌊2.5i⌋. That is false. For d=2, n=100, i=0, l=(50,51) satisfies the premise but |l|∞=51 < 100. The proof then replaces the index sum by one over the first d−1 components summing to ≤ d−1+1.5i, which undercounts badly: there are Θ(n^{d−1}) ordered multi-indices with |l|1 near n+d−1 and small |l|∞, each carrying about 2^{n−1} grid points. So the correct count is more like n^{d−1}2^n, not C_s 2^n e^{c√d}. Since Corollary 2 sets m to that smaller count and uses 2^{−n} = m^{−1}, the O(m^{−1}) rate collapses to O((log m)^{d−1}/m). The central claim is therefore unsupported as written. This is not a minor typo: it is the count that converts the sparse-grid error into a dimension-free network rate.

The secondary issues are less severe but real. Theorem 2's stated rate, ((log M)^2/M)^{2/3}, does not follow from the displayed bound in (25). Plugging M = O(m^3(log m)^2) gives an extra log M factor; the proof supports roughly (log M)^{7/3}/M^{2/3}. Also, the abstract claims the constant prefactor scales at most polynomially in d, but Theorem 1's C contains e^{π√(5d/3)}, which is not polynomial. That is an overclaim worth correcting.

On credit where due: the paper is self-contained enough to trace the error, the Vandermonde lemma is new and likely reusable, and the generalization framework via pseudo-dimension is reasonable. But the main result is not proved.

I would send this to peer review rather than desk reject, because the core idea is promising and the gap, though serious, may be repairable. The referee should be asked to verify the counting argument in Proposition 1 and the logarithmic powers in Theorem 2. I would not rely on the theorems in their current form, and I would not cite them this year.

Referee Report

4 major / 3 minor

Summary. The paper studies approximation and learning of symmetric Korobov functions by symmetric squared-ReLU networks. Theorem 1 claims that for any f in the symmetric Korobov space X^{2,2}_{sym}(Ω), there exists a symmetric squared-ReLU network with m parameters such that the energy-norm error is bounded by C |f|_{2,2} m^{-1}, where the constant C depends polynomially on the dimension d (with a decaying factor (5/6)^d). Theorem 2 claims a generalization error bound of order ((log M)^2/M)^{2/3} for gradient-based empirical risk minimization over such networks, with the constant depending polynomially on d. The proofs rely on an energy-based sparse-grid approximation, a symmetrization of the sparse-grid basis, a counting argument for the number of symmetric basis functions, and neural-network implementation of each basis. The central technical step is Proposition 1, which bounds the number of symmetric sparse-grid basis functions by C_s 2^n e^{π√(5d/3)}.

Significance. If the results were correct, they would constitute a substantial advance: a dimension-free approximation rate for a symmetric high-dimensional function class, with an explicit prefactor that does not grow exponentially in d, and a corresponding learning guarantee for gradient-fitting problems such as score matching and force-field learning. The idea of using a Vandermonde-type construction to reduce the d! symmetrization to O(2^d) terms is interesting and likely useful in other contexts. However, the manuscript’s central counting argument contains a false implication and an inconsistent lemma, and these errors are load-bearing: the claimed number of symmetric basis functions is too small by a polynomial-in-n factor, which reintroduces a dimension-dependent logarithmic term in the rate. The main theorems are therefore not established, and the flaws concern the core of the proof rather than presentation.

major comments (4)

[Section 4, Proposition 1] The displayed implication in the proof of Proposition 1 is false. The text asserts: if Σ_j 4^{l_j} ≥ (4n+4d−4)/32^i, then |l|_∞ ≥ n−⌊2.5i⌋. For d=2, n=100, i=0, take l=(50,51). Then |l|_1=101=n+d−1 and 4^50+4^51 is vastly larger than 4n+4d−4=404, so the premise holds; but |l|_∞=51<100, so the conclusion fails. This invalidates the subsequent replacement of the sum over l satisfying the energy condition by the partition sum over |l*|_1 ≤ d−1+1.5i. The actual number of sorted l with |l|_1=n+d−1−i and the energy condition is Θ(n^{d−2}) for each i (for fixed d), giving a total basis count Θ(n^{d−1}2^n), not O(2^n). Consequently Corollary 2's identification m=C_s 2^n e^{π√(5d/3)} is invalid, and the rate in terms of m becomes O((log m)^{d−1}/m), reintroducing a dimension-dependent logarithmic factor. Since Theorem 1's O(m^{-1}) rate is built on this count, the main approximation result is not
[Section 4, Lemma 3] Lemma 3 states that the energy-based index set X_n defined in (6) is a subset of {l∈N^d : |l|_1 ≤ n+d−1}. This is contradicted by the definition. For d=1 and n≥3, choose l=n+1. Then |l|_1−1/5 log2(Σ4^{l_j}) = (n+1) − (1/5)log2(4^{n+1}) = (3/5)(n+1), while the right-hand side is n − (1/5)log2(4n). The inequality (3/5)(n+1) ≤ n − (1/5)log2(4n) holds for all sufficiently large n, so l=n+1 belongs to X_n but has |l|_1=n+1 > n+d−1. Thus the subset claim is false. This is not a minor artifact: the proof of Proposition 1 sums only over indices with |l|_1=n+d−1−i for i≥0, thereby omitting valid elements of X_n and failing to produce the advertised upper bound on the number of basis functions.
[Section 5.2, Theorem 1] The proof of Theorem 1 combines Corollary 2 with the neural-network implementation of each basis function (Proposition 2). Since Corollary 2's bound on the number m of symmetric basis functions is based on Proposition 1 and Lemma 3, both of which contain the errors described above, the m^{-1} approximation rate is not established. The optimality remark invoking DeVore et al. (1989) is therefore also unsupported: the lower bound m^{-1} concerns the optimal rate, but the proposed construction does not achieve it. A correct counting of the energy-based symmetric sparse grid for the definition of X_n in (6) yields a Θ(n^{d−1}2^n) basis count, so the construction as presented cannot achieve the claimed dimension-free rate without a substantially different index set or error analysis.
[Section 6, Theorem 2] The generalization bound in Theorem 2 depends on the assumption that inf_{f∈F_{m,L}} ||f−fρ||_E attains the approximation rate of Theorem 1. Since Theorem 1 is not established, the learning-rate claim also fails. In addition, the proof conflates two uses of m: Proposition 4 bounds the pseudo-dimension of ∂_kΦ in terms of the number of subnetworks m, while the approximation error in the same argument (25) uses m as the total number of parameters via m^{-2}. The total parameter count is a factor of O(d^3 2^{d-1} log d) larger than the number of subnetworks, so the resulting bound in terms of the actual network size would carry additional d-dependent factors. Even granting the approximation theorem, the generalization argument needs a separate, carefully tracked trade-off between the number of subnetworks and the total parameter count.

minor comments (3)

[Throughout] There are several typos and minor inconsistencies. For example, in the paragraph after Corollary 1, 'expoenentially' should be 'exponentially'; Theorem 1 states the width as C_s 6d^3(2^{d−1}−1)2^n e^{π√(5d/3)} while Corollary 3 states m=O(d 2^{n+d−1} e^{...}), omitting the d^3 log d factor from Theorem 1; these should be reconciled.
[Section 4, Lemma 3] The constant C_s in Proposition 1 and Corollary 2 is introduced without a precise definition; it is only said to be independent of d and n. Since the proof of Proposition 1 is the only place C_s arises, the constants R1 and R2 are not fully tracked. This matters because the final prefactor in Theorem 1 includes C_s.
[Section 5.1, Lemma 5] In the proof of Lemma 5, the notation for U(d) and the derivation of the carry process is dense and contains conflicting statements about the range of Λ(j). A cleaner presentation, perhaps with an example for small d, would improve readability. This point is not load-bearing for the central claim, as Lemma 5 appears plausible.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central derivation is assembled from external sparse-grid, partition-number, VC/covering-number, and lower-bound results plus in-paper lemmas; the self-citations are templates, not load-bearing. The Proposition 1 counting gap is a correctness risk, not a circular step.

full rationale

The derivation is self-contained rather than circular. Theorem 1 is built from Bungartz and Griebel (2004) energy-based sparse-grid approximation (Lemmas 2 and 3), the symmetrized sparse-grid error bound (Lemma 4), the symmetric basis count (Proposition 1), the Vandermonde symmetrization identity (Lemma 5), and the squared-ReLU implementation of each basis function (Lemma 7 and Proposition 2); the network size m is then set by the count times per-basis complexity, not by fitting the error. Theorem 2 uses the external covering-number bound of Györfi et al. (2002, Theorem 11.4), the Bartlett et al. polynomial-sign lemma, and an Appendix A proof of the pseudo-dimension bound of derivative networks; the citations to Yang et al. (2023) and Yang and He (2024) are used only as a template because the needed bound is re-proved in the paper. There is no fitted parameter renamed as a prediction, no imported uniqueness theorem, and no renaming of a known result. One non-circular correctness caveat should be flagged: in Proposition 1 the displayed implication "As n→∞, the condition Σ 4^{l_j} ≥ (4n+4d−4)/32^i implies |l|∞ ≥ n−⌊2.5i⌋" is not justified and appears false in general (e.g., d=2, n=100, i=0, l=(50,51) satisfies the premise but not the conclusion), which would threaten the basis count and hence the m^{-1} rate; however, this is a mathematical gap rather than a definitional or self-citational circularity, so it does not affect the circularity score.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central claim rests on standard external results (energy-based sparse grids, Hardy-Ramanujan, covering-number machinery) plus the paper's own proved lemmas. No free parameters are fitted; all constants are explicit or universal. The main unstated fragility is the index-set counting step and the unbounded Vandermonde coefficients.

assumptions (4)

standard math Energy-based sparse-grid error bound for Korobov spaces (Bungartz and Griebel 2004, Theorem 3.10)
Used as Lemma 2 to get O(2^{-n}) error with the energy index set; the proof does not re-derive it.
standard math Hardy–Ramanujan asymptotic formula for integer partitions
Used in Proposition 1 to bound partition numbers by e^{π√(5d/3)}; standard analytic number theory background.
standard math Uniform covering number bound (Györfi et al. 2002, Theorem 11.4)
Used in Proposition 3 to convert covering numbers into a generalization error bound; external.
standard math VC-dimension and pseudo-dimension bounds for piecewise-polynomial neural networks (Bartlett et al. 2019, Anthony et al. 1999)
Used in Proposition 4's proof to bound the pseudo-dimension of derivative networks; external background.

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Cite this review

Pith. "Pith review of On the Dimension-Free Approximation of Deep Neural Networks for Symmetric Korobov Functions." pith.science (2026). https://pith.science/paper/B6UCGH2S

@misc{pith2026251112398,
  author       = {Pith},
  title        = {Pith review of: On the Dimension-Free Approximation of Deep Neural Networks for Symmetric Korobov Functions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/B6UCGH2S}},
  note         = {Machine review of arXiv:2511.12398}
}

read the original abstract

Deep neural networks have been widely used as universal approximators for functions with inherent physical structures, including permutation symmetry. In this paper, we construct symmetric deep neural networks to approximate symmetric Korobov functions and prove that both the convergence rate and the constant prefactor scale at most polynomially with respect to the ambient dimension. This represents a substantial improvement over prior approximation guarantees that suffer from the curse of dimensionality. Building on these approximation bounds, we further derive a generalization-error rate for learning symmetric Korobov functions whose leading factors likewise avoid the curse of dimensionality.

Figures

Figures reproduced from arXiv: 2511.12398 by the authors.

**Figure 1.** The difference between Xn and Vn for n = 5 and d = 2. similar construction is available for Korobov spaces with global Fourier bases; see Griebel and Hamaekers (2007); Shen and Wang (2010). However, Corollary 1 below reveals that—even with this refinement—the cardinality of the index set, and thus the leading constant in the approximation error, still grows exponentially with the dimension d. Combining Lemmas 2 and… view at source ↗

**Figure 2.** structure of Qd s=1 xs For the derivative term, we consider the partial derivative of s∗, we have that [PITH_FULL_IMAGE:figures/full_fig_p022_2.png] view at source ↗

**Figure 2.** Proposition 2. Let d ∈ N, and for any δ > 0 and l, i ∈ N d +, there exists a squared ReLU network with inputs x = (x1, . . . , xd) ⊤ and output ψl,i (x) that implements the basis ψl,i(x) defined in (7) with accuracy δ measured by H1 (Ω) norm with supp ψl,i ⊂ supp ψl,i . The neural network architecture has width 3d 3 [PITH_FULL_IMAGE:figures/full_fig_p023_2.png] view at source ↗

Figures from the paper (2 more)

**Figure 3.** Figure 3: 24 [PITH_FULL_IMAGE:figures/full_fig_p024_3.png]

**Figure 3.** Figure 3: Approximation of the hat function h by hδ with a smaller support. Hence, for any δ, ξ > 0, set φj as ϕlj ,ij in Lemma 5 and for each component of Gξ defined in (10), X d j=1 ξ 2 j ϕlj ,ij (xs), s = 1, . . . , d, (22) there exists a depth-1 squared ReLU network ψs of wi…

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Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.

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write newline

" write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION word.in bbl.in ":" * " " * FUNCTION f...

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