Dimension independent bounds for general shallow networks

Hrushikesh N. Mhaskar

REVIEW 2 major objections 4 minor 35 references

Dimension independent bounds for general shallow networks

T0 review · 2 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read A single abstract theorem gives dimension-independent approximation rates for shallow kernel networks.

desk verdict The abstract theorem is coherent and worth taking seriously; the headline 'deep networks give no advantage' goes beyond what the theorem proves, and one smoothness corollary needs a fix. read the letter →

arxiv 1908.09880 v2 pith:5SCW5RDG submitted 2019-08-26 cs.LG stat.ML

Hrushikesh N. Mhaskar This is my paper · ORCID

classification cs.LGstat.ML MSC 41A2541A4665D3268T05

keywords shallownetworksdimensionindependentboundscurseofdimensionalityG-networksintegralrepresentationmanifoldlearningout-of-sampleextensiontractabilityintegration

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

This paper proves one abstract theorem that covers when a shallow network — a linear combination of a fixed kernel $G$ — can approximate a target function without the curse of dimensionality. The target must be an integral superposition $f(x)=\int G(x,y)\,d\tau(y)$ with $\tau$ a measure of bounded total variation supported on a $q$-dimensional subset of a compact metric space; the error bound for an $N$-term $G$-network then depends on $q$, not on the ambient dimension. The same theorem yields, as corollaries, rates for ReLU networks on the sphere, radial-basis-function networks on the cube, and out-of-sample extension in manifold learning. A by-product is the existence of quadrature formulas for high-dimensional integration with no tensor-product structure. The paper also argues that, without requiring robust parameter selection, deep networks hold no significant degree-of-approximation advantage over shallow ones.

What carries the argument

The machinery is a probabilistic quadrature construction on the metric measure space. A partition theorem (Theorem 5.1) tiles the support of $\tau$ into cells with controlled volume and density, using maximal distinguishable sets whose covering counts $H_\epsilon(A)$ define the notion of dimension. On each cell, Tchakaloff's theorem supplies a discrete probability measure with the same moments up to degree $R$ as the restricted target measure; the cell measures are then treated as independent random variables, and Hoeffding's inequality bounds the probability that the resulting network deviates from $f$ at any point. A net over the space transfers the pointwise bound to the uniform norm. The kernel's 'smoothness in the large' enters through the exceptional sets $E_x$, the growth function $F$, and the scale $\epsilon_n^*$ defined by $\widetilde F(t)=F(t)/t^{(q-s)/2}$, which balances the smoothness gain against the dimension loss.

What would settle it

Take $X=[-1,1]^Q$, $G(x,y)=e^{-|x-y|^2}$, and $\tau$ the uniform measure on the diagonal $\{(t,\dots,t):t\in[-1,1]\}$ ($q=1$). The theorem predicts that an $N$-term Gaussian network approximating the extension to the whole cube has uniform error $(\log N/N)^{1/2}$ times an arbitrarily high power $N^{-S}$. If the actual uniform error on the cube decays only like $N^{-c/Q}$ with $c$ independent of the smoothness, then the no-saturation claim for out-of-sample extension would be false.

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Extended reading notes

Core claim

The central claim is Theorem 3.1: if $\tau$ is $q$-admissible and $G$ belongs to the class $\mathcal{G}(\alpha,r,R,F)$ with $\{\operatorname{supp}(\tau)\cap E_x\}$ an $s$-dimensional family, then every $f(x)=\int G(x,y)\,d\tau(y)$ can be uniformly approximated by an $N$-term $G$-network whose centers lie within $1/n$ of $\operatorname{supp}(\tau)$, whose coefficients are $O(1/N)$, and whose error is at most $c\left((\log n-\log\epsilon_n^*)/(n^{q+2r}(\epsilon_n^*)^{s-q})\right)^{1/2}\|\tau\|_{TV}$, with $N\sim n^q$. The bound merges the dimension-independent $\sqrt{\log N/N}$ term with a smoothness-driven factor $N^{-r/q}$; for infinitely smooth kernels the approximation does not saturate. Specializing to ReLU activation on the sphere with $q=Q$ gives $O(N^{-(Q+3)/(2Q)}\sqrt{\log N})$, and the same theorem, with $\tau$ supported on a $q$-dimensional manifold, gives rates on the manifold with constants independent of the ambient space, plus rates for the out-of-sample Nyström extension to the ambient space. All of this is achieved without a robust parameter selector: the centers and coefficients are chosen depending on the target, which is what allows the improved rates.

Load-bearing premise

The target function must be representable as an integral superposition of the kernel with respect to a finite-variation measure supported on a $q$-dimensional set; without that representation, the construction has no starting point, and the dimension-independent rates do not apply.

Editorial extensions

If this is right

For functions whose representing measure lives on a $q$-dimensional manifold in $\mathbb{R}^Q$, shallow $G$-networks achieve rates governed by $q$ alone; the ambient dimension enters only through constants.
For ReLU networks on the sphere $S^Q$, allowing target-dependent centers and coefficients improves the guaranteed rate from $O(N^{-2/Q})$ to $O(N^{-(Q+3)/(2Q)}\sqrt{\log N})$.
For infinitely smooth kernels (e.g., the Gaussian), the bounds do not saturate: for every $S>0$ the error is $O(N^{-S})$ on the manifold, and the same extension rate holds in the ambient space, giving a degree-of-approximation estimate for the Nyström extension.
Corollary 3.1 yields $N$-point quadrature formulas for integrals over arbitrary compact metric measure spaces, with error $O(N^{-c})$ and no tensor-product structure on the domain or the measure.
Without robust parameter selection, the gap between deep and shallow ReLU degrees of approximation is much smaller than with it: on 1024 inputs, $O(N^{-1.25})$ versus $O(N^{-0.5015})$ without, versus $O(N^{-1})$ versus $O(N^{-0.002})$ with.

Reading between the lines

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

The proof is existential: it shows that good networks exist but does not give an efficient algorithm to find the centers and coefficients; finding them may require solving a non-convex problem, so the bounds describe expressive power, not trainability.
The notion of dimension via maximal distinguishable sets is essentially a covering (Assouad-type) dimension; the arguments are likely to extend to fractal sets with fractional $q$, and the $q$-admissibility condition could be relaxed or shown necessary for the concentration step.
The rates depend on the total variation norm $\|\tau\|_{TV}$; this suggests that sparsity of the representing measure is the operative complexity measure, connecting to sparse dictionary approximation and potentially guiding kernel choice so targets have sparse representations.
The no-saturation property for analytic kernels hints that uniform-norm approximation of kernel superpositions is limited only by representability, not smoothness; a testable extension would compare these bounds against kernel ridge regression on manifolds.

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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

Short version: Theorem 3.1 is a real theorem, and the proof is worth taking seriously. The paper gives a unified treatment of dimension-independent approximation for G-networks: Barron-type integral-representation classes, ReLU and RBF networks, out-of-sample extension on manifolds, and non-tensor-product cubature. What is new is mixing the Bourgain–Lindenstrauss probabilistic quadrature idea with local smoothness of the kernel and the dimension of the measure support, so the rate improves with kernel smoothness and depends on q rather than ambient Q. That is a solid contribution. The partition/Tchakaloff/Hoeffding/union-bound proof is assembled carefully, and the examples in Corollaries 4.1–4.4 are genuinely useful.

The main soft spot is the abstract and Conclusions claim that without robust parameter selection deep ReLU networks offer no significant advantage over shallow ones. Theorem 3.1 is an upper bound for functions f(x)=∫G(x,y)dτ(y) with q-admissible τ. It is not a lower bound for shallow networks on the same class, and there is no argument that the compositional targets for which [24] gets O(N^{-1.25}) lie in class (2.6) with q=Q=1024. The displayed comparison in Conclusions therefore compares bounds for possibly different function classes. This is an interpretive overreach; the theorem itself is not damaged.

Second soft spot is the no-saturation statement in Remark 3.3 and Corollary 4.4. Setting E_x=X, s=q, R=r gives F constant, which conflicts with the assumption \tilde F(t)→∞ and makes the epsilon* definition degenerate. The no-saturation conclusion may be recoverable with a different F or a limiting argument, but as written it needs repair. The reader's report flagged this, and I think it is a legitimate issue, though minor compared with the deep/shallow framing.

The integral representation assumption (2.6) is heavy but standard in Barron-type work, and the paper is honest about using it. That is not a flaw if the claims are kept inside that class.

Who is this for? Approximation theorists and machine-learning theorists who care about when shallow networks escape the curse of dimensionality. It deserves a serious referee. I would send it out, with the expectation that the author narrows the deep/shallow conclusions and fixes the no-saturation corollary.

Referee Report

2 major / 4 minor

Summary. The paper proves an abstract approximation theorem for shallow G-networks on compact metric measure spaces. Assuming the target f has an integral representation f(x)=∫G(x,y)dτ(y) for a q-admissible measure τ, and assuming kernel smoothness and exceptional-set dimension conditions, the author constructs an N-term G-network with coefficients bounded by O(1/N) and an error bound of the form O((log N / N^{1+2r/q}(ε*)^{s-q})^{1/2}), specialized to rates such as O(N^{-(Q+3)/(2Q)}) for ReLU networks on the sphere. Applications include radial basis function networks, zonal function networks, manifold learning with out-of-sample extension, and non-tensor-product cubature. The paper also claims as a consequence that, absent robust parameter selection, deep ReLU networks offer no significant advantage over shallow networks in degree of approximation alone.

Significance. If taken as a theorem about the integral-representation class (2.6), the main result is a substantial and useful unification: it combines the probabilistic dimension-independent approach of Barron with a smoothness-driven improvement that does not saturate, and it applies in a genuinely general metric measure setting. The proof is self-contained and the construction via partitions, Tchakaloff quadrature, Hoeffding concentration, and a union bound is coherent. The applications to manifold out-of-sample extension and to cubature on non-tensor-product domains are well motivated. The paper does not provide code or machine-checked proofs, but the analytic proof is explicit. The main weakness is that the headline deep-versus-shallow comparison is not entailed by the theorem as stated, since there is no shallow lower bound and no proof that the relevant deep-network target functions lie in the class (2.6).

major comments (2)

[Abstract and Section 6 (Conclusions)] The headline conclusion that deep ReLU networks provide no significant advantage over shallow networks in degree of approximation alone is not a consequence of Theorem 3.1. The theorem is an upper bound for the class of functions satisfying the integral representation (2.6), and it contains no lower bound for shallow networks. Moreover, the paper does not prove that the compositional target functions for which [24] obtains deep rates such as O(N^{-1.25}) admit a representation (2.6) with q=Q=1024 and a q-admissible τ. The displayed comparison in Section 6 between O(N^{-1.25}) and O(N^{-0.5015}) therefore compares upper bounds for possibly different function classes. This claim should be removed or explicitly recast as conditional on the target lying in the integral-representation class, and the comparison with deep rates should be suppressed unless membership is established.
[Theorem 3.1, Remark 3.3, and Corollary 4.4] The 'no saturation' statement for infinitely smooth kernels with Ex=X, s=q, and F≡c is not covered by Theorem 3.1 as stated. The theorem requires that tilde-F(t)=F(t)/t^{(q-s)/2} tend to infinity as t tends to 0, but in this limiting case tilde-F is constant. The rate O((sqrt(log N)) N^{-S}) for every S>0 may be obtainable by a direct argument with ε*=1/n, but that argument is not present. Corollary 4.4 relies on this limiting case for its O(N^{-S}) claim, so the theorem statement or the corollary needs a separate proof for the case s=q with bounded F.

minor comments (4)

[Section 5.3, Eq. (5.13)-(5.14)] The summation split in (5.14) uses the condition A∩Ex≠∅, whereas the bound (5.13) was derived under the condition A∩B(Ex,ε*_n)≠∅. The notation should be made consistent, since the counting argument depends on the ε*_n-neighborhood rather than on Ex itself.
[Proof of Theorem 5.2] There is a typo in the sentence beginning 'However, Thereom 5.3 applied with the system...'; 'Thereom' should be 'Theorem'.
[Section 2, Eq. (2.1)] The sentence 'where is X is some Banach space of functions on X' contains a duplicated 'is' and should be corrected.
[Title and Abstract] The phrase 'dimension independent bounds' is used for the rate, but the constants in Theorem 3.1 depend on the ambient dimension Q. This is stated in the body, but the abstract could be more precise by saying 'dimension independent rates' or 'bounds whose exponent is independent of dimension'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Theorem 3.1 is a genuine measure-discretization upper bound; the deep-vs-shallow conclusion is an overreach but not a circular reduction.

full rationale

The paper's central estimate is Theorem 3.1, which is conditional on the explicit representation f(x)=∫ G(x,y)dτ(y) (Eq. (2.6) and (3.13)). The approximant is an N-term G-network, i.e., a discrete-measure quadrature for the same kernel. This is the standard Barron/G-variation setup: the representation defines the target class, and the theorem proves a quantitative discretization rate; it does not assume the approximation error or the conclusion. The proof uses Tchakaloff's theorem (Theorem 5.3), Hoeffding's inequality (Lemma 5.3), and a Bourgain-Lindenstrauss-style partition argument (Section 5.1), all independent external tools. The q-admissibility and s-dimensionality conditions are hypotheses, not conclusions smuggled in. The ReLU corollary is a specialization of the general theorem, not a renaming of a known result. The self-citations (e.g., [23], [24], [25]) are contextual or used for the deep-network comparison in the Conclusions, not load-bearing for the proof of Theorem 3.1. The deep-vs-shallow comparative claim is an interpretive extrapolation that is not entailed by the theorem alone, since no matching shallow lower bound is proved and the deep-network target class is not shown to lie in class (2.6); however, an inference gap is not circularity. Therefore, no circular step is exhibited.

Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

The theorem's central inputs are explicit hypotheses: a q-admissible measure, a kernel with prescribed local and global smoothness, and an s-dimensional exceptional family. The proof uses standard tools from approximation theory and probability, and it introduces no new physical or mathematical entities such as a new force, conserved quantity, or algebraic structure. The only hand-chosen quantities are theoretical parameters beta and R in the corollaries, which do not come from data.

free parameters (2)

beta = arbitrary in (0,1)
In Corollaries 4.1-4.3, beta parameterizes the tradeoff between kernel smoothness and approximation rate. It is chosen by hand, not fitted to data, but the final displayed bound depends on it.
R = arbitrarily large, e.g. R > 2 gamma + 1 + beta(q-s)/(2-2 beta)
R is the large-scale smoothness parameter in Definition 3.3. The corollaries choose R sufficiently large to obtain the displayed rates; it is a proof-handpicked parameter, not a data-fitted constant.

assumptions (6)

domain assumption The ambient measure mu* satisfies the volume condition kappa1 delta^Q <= mu*(B(x,delta)) <= kappa2 delta^Q (Eq. 3.2).
This doubling-type condition is used throughout to bound net cardinalities, partition cell volumes, and covering numbers. It defines the ambient dimension Q.
domain assumption The target function admits an integral representation f(x)=integral G(x,y) dtau(y) with tau of bounded total variation (Eq. 2.6).
This is the defining hypothesis of the target class and the starting point for the probabilistic quadrature construction in the proof of Theorem 3.1.
standard math Tchakaloff's theorem provides positive quadrature for finitely many continuous moments (Theorem 5.3, cited as [29]).
Used in Theorem 5.2 to show that each piece of the partition can be represented by an atomic probability measure matching the moments of tau on Pi_R.
standard math Hoeffding's inequality for bounded zero-mean random variables (Lemma 5.3, cited as [28]).
Used to convert the summed variances of the random quadrature errors into an exponential concentration estimate.
standard math A compactness and barycenter argument, cited as Rudin [32, Theorem 3.28], lets the author pass from membership in the weak-star closed convex hull to a probability measure on the atomic approximation set.
This is the step that produces the random variables with a specified law in Theorem 5.2.
domain assumption Euclidean ReLU approximation is equivalent to approximation of even functions on the sphere by sums of |x dot y| (cited from [1,24]).
This external equivalence is used in Corollary 4.1 to convert the abstract G-network result into a statement about ReLU networks in Euclidean space.

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Pith. "Pith review of Dimension independent bounds for general shallow networks." pith.science (2026). https://pith.science/paper/5SCW5RDG

@misc{pith2026190809880,
  author       = {Pith},
  title        = {Pith review of: Dimension independent bounds for general shallow networks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5SCW5RDG}},
  note         = {Machine review of arXiv:1908.09880}
}

abstract

This paper proves an abstract theorem addressing in a unified manner two important problems in function approximation: avoiding curse of dimensionality and estimating the degree of approximation for out-of-sample extension in manifold learning. We consider an abstract (shallow) network that includes, for example, neural networks, radial basis function networks, and kernels on data defined manifolds used for function approximation in various settings. A deep network is obtained by a composition of the shallow networks according to a directed acyclic graph, representing the architecture of the deep network. In this paper, we prove dimension independent bounds for approximation by shallow networks in the very general setting of what we have called $G$-networks on a compact metric measure space, where the notion of dimension is defined in terms of the cardinality of maximal distinguishable sets, generalizing the notion of dimension of a cube or a manifold. Our techniques give bounds that improve without saturation with the smoothness of the kernel involved in an integral representation of the target function. In the context of manifold learning, our bounds provide estimates on the degree of approximation for an out-of-sample extension of the target function to the ambient space. One consequence of our theorem is that without the requirement of robust parameter selection, deep networks using a non-smooth activation function such as the ReLU, do not provide any significant advantage over shallow networks in terms of the degree of approximation alone.

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [24]

H. N. Mhaskar and T. Poggio. Deep vs. shallow networks: An app roximation theory perspective. Analysis and Applications, 14(06):829–848, 2016

2016

[2] [1]

F. Bach. Breaking the curse of dimensionality with convex neural networks. The Journal of Machine Learning Research, 18(1):629–681, 2017

work page 2017

[3] [2]

A. R. Barron. Neural net approximation. In Proc. 7th Yale Workshop on Adaptive and Learning Systems , volume 1, pages 69–72, 1992

1992

[4] [3]

A. R. Barron. Universal approximation bounds for superpositio ns of a sigmoidal function. Information Theory, IEEE Transactions on , 39(3):930–945, 1993

work page 1993

[5] [4]

Bourgain and J

J. Bourgain and J. Lindenstrauss. Distribution of points on sphe res and approximation by zonotopes. Israel Journal of Mathematics , 64(1):25–31, 1988

work page 1988

[6] [5]

C. K. Chui, F. Filbir, and H. N. Mhaskar. Representation of funct ions on big data: graphs and trees. Applied and Computational Harmonic Analysis, , 38(3):489–509, 2015

work page 2015

[7] [6]

G. David. Wavelets and singular integrals on curves and surfaces . Springer, 2006

work page 2006

[8] [7]

R. A. DeVore, R. Howard, and C. A. Micchelli. Optimal nonlinear app roximation. Manuscripta mathematica, 63(4):469–478, 1989

1989