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Entropy numbers of classes defined by integral operators

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Pith's one-line read For $r>1$, worst-case entropy of integral-operator classes with mixed-smoothness kernels is $k^{-2r}$ up to logarithmic factors.

desk verdict A genuine extension of the Kolmogorov-width sup-over-kernels program to entropy numbers, with clean proofs and a few correctable blemishes; the main thing to check is the endpoint Littlewood-Paley citation. read the letter →

arxiv 2505.08572 v2 pith:73CKRXHD submitted 2025-05-13 math.NA cs.NAmath.FA

V. Temlyakov This is my paper · ORCID
classification math.NAcs.NAmath.FA MSC 41A4641A6342B35
keywords entropynumbersintegraloperatorsmixedsmoothnesshyperboliccrossLittlewood-PaleyblocksKolmogorovwidthsBernoullikernelmultivariateapproximation
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 moves from studying the entropy numbers of one smoothness class to studying them over a whole collection: for every kernel $K$ in a mixed-smoothness class $H^{r,2d}_{1,\infty}$, form the class $W^K_1$ of all functions $f(x)=\int K(x,y)\varphi(y)\,d\mu_2(y)$ with $\|\varphi\|_{L_1}\le 1$, and ask how many $k$-bit covers suffice in $L_\infty$ uniformly in $K$. The main theorem asserts that this worst-case entropy is at most $k^{-2r}(\log 2k)^{2r}$ when $d=1$ and $k^{-2r}(\log 2k)^{4r+5/2}$ when $d=2$, for every $r>1$. A lower bound from the Bernoulli kernel $K=F_{2r}(x-y)$ gives a matching main term $k^{-2r}$ in both cases, leaving only logarithmic gaps. The point is that smoothness information about an unknown kernel guarantees a concrete approximation rate for every function its integral operator can produce, extending to entropy numbers a program previously developed for Kolmogorov widths.

What carries the argument

The load-bearing object is the dyadic Littlewood-Paley decomposition of the kernel. With de la Vallée Poussin kernels one forms blocks $A_s(K)$, $s\in\mathbb{N}_0^{2d}$, and Theorem 2.1 converts $K\in H^{r,2d}_{1,\infty}$ into the block decay $\|A_s(K)\|_{1,\infty}\ll 2^{-r\|s\|_1}$. Lemma 3.1 then splits every $f\in W^K_1$ into two low-frequency pieces inside the hyperbolic-cross space $T(Q_u)$ (handled by finite-dimensional covering-number estimates) and a tail piece belonging to the weighted sequence class $\overline{W}^{a,b,b'}_{X,n_0}$ with $X=L_1(\mathbb{T}^d)$ and $X_n=T(Q_{n-u})$. A new abstract lemma, Theorem 2.3, converts the known entropy estimates for the subspaces $T(Q_n)$ in $L_\infty$ into the bound $k^{-2a}(\log k)^{2ac+b+\alpha}$ for such tails; substituting the $d=1$ and $d=2$ estimates for hyperbolic crosses yields (1.4) and (1.5).

What would settle it

The sharpest concrete check is the extremal kernel $K=F_{2r}(x-y)$, whose image class is the classical class $W^{2r}_1$: in $d=2$ the known entropy of $W^{2r}_1$ in $L_\infty$ has rate $k^{-2r}(\log k)^{2r+1}$, which stays below the theorem's bound, so a counterexample would have to be a different kernel in $H^{r,4}_{1,\infty}$. One can search for it by solving, for dyadic weights $c_s$ with $\|c_s\|_{1,\infty}\le 2^{-r\|s\|_1}$, whether the entropy of the resulting image class exceeds $k^{-2r}(\log 2k)^{4r+5/2}$; in $d=1$ the same search with exponent $2r$ would decide the conjecture that the logarithms are unnecessary.

Watch

Extended reading notes

Core claim

The central claim is that the worst-case entropy of the collection is governed by the kernel smoothness exponent $r$: the upper bounds (1.4) and (1.5) hold, and the paper proves Theorem 3.1 giving $k^{-2r}(\log 2k)^{4r+2}$ in $L_p$ for $p<\infty$ in $d=2$. On the lower side, Lemma 4.1 shows $F_{2r}(x-y)\in H^{r,2d}_{1,\infty}$, so the image class $W^{2r}_1$ is part of the collection and the known entropy of $W^{2r}_1$ yields $k^{-2r}$ in $d=1$ and $k^{-2r}(\log k)^{2r}$ in $d=2$; Section 5 refines these lower bounds using sharper known results. The paper's own reading is that the upper bounds are close to optimal and the remaining questions are logarithmic: it states as open problems whether the logarithms can be removed and whether the kernel class can be enlarged to $H^{r,2}_{1,1}$.

Load-bearing premise

The proof stands on the Littlewood-Paley equivalence claiming that a kernel lies in $H^{r,2d}_{1,\infty}$ exactly when its dyadic frequency blocks decay at rate $2^{-r\|s\|_1}$ in a mixed $L_1$-$L_\infty$ norm; if that equivalence fails at this endpoint, the kernel decomposition in Lemma 3.1 and both upper bounds collapse.

Editorial extensions

If this is right

  • For $d=1$, the theorem plus the Bernoulli-kernel lower bound gives $k^{-2r}\ll \sup_{K\in H^{r,2}_{1,\infty}}\varepsilon_k(W^K_1,L_\infty)\ll k^{-2r}(\log 2k)^{2r}$, so the main rate is fixed by $r$ alone.
  • For $d=2$, the corresponding two-sided bounds are $k^{-2r}(\log k)^{2r}$ and $k^{-2r}(\log 2k)^{4r+5/2}$; the remaining uncertainty is only the logarithmic exponent.
  • For target spaces $L_p$ with $p<\infty$, Theorem 3.1 improves the $d=2$ upper bound to $k^{-2r}(\log 2k)^{4r+2}$.
  • For $d=1$ and $2\le q,p\le\infty$, Theorem 5.3 yields the exact order $\sup_{K\in H^{r,2}_{1,w}}\varepsilon_k(W^K_q,L_p)\asymp k^{-2r}$ under $r>3/2$.

Reading between the lines

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

  • Beyond the paper: the same three-part kernel split should apply to other asymptotic characteristics of the collection, such as optimal sampling recovery and numerical integration, whenever the underlying subspaces satisfy an entropy bound of the form (2.6).
  • Beyond the paper: the extra logarithmic powers in $d=2$ appear to come from the $n^{1/2}$ factor in the hyperbolic-cross entropy estimate; improving that estimate for $T(Q_n)$ in $L_\infty$ would improve Theorem 1.2 without altering Lemma 3.1.
  • Beyond the paper: adapting the argument to nonperiodic domains would give worst-case guarantees for solution operators of elliptic and parabolic problems whose Green's functions lie in an appropriate mixed-smoothness class, at the price of new technical work near the boundary.
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Editorial analysis

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

Referee Report

2 major / 4 minor

Summary. The paper develops a general approach to entropy numbers of classes defined by integral operators whose kernels lie in a given smoothness class, extending the earlier program for Kolmogorov widths. The main result, Theorem 1.2, gives upper bounds for sup_{K∈H^{r,2}_{1,∞}} ε_k(W^K_1,L∞) ≪ k^{-2r}(log 2k)^{2r} in dimension d=1 and sup_{K∈H^{r,4}_{1,∞}} ε_k(W^K_1,L∞) ≪ k^{-2r}(log 2k)^{4r+5/2} in dimension d=2 for r>1. The proof decomposes a kernel K into low-frequency parts handled by finite-dimensional entropy estimates and a high-frequency tail handled by an abstract entropy theorem for classes of the form \bar W^{a,b,b'}_{X,n0}. Lower bounds are obtained by taking the Bernoulli kernel K=F_{2r}(x-y), which lies in the relevant kernel class, and applying known entropy lower bounds for classical mixed-smoothness classes. Section 5 compares the results with Kolmogorov-width bounds via Carl's inequality and states several open problems.

Significance. If the result is correct, it provides the first entropy-number analogues of the earlier Kolmogorov-width results for collections of integral-operator classes, showing that the worst-case entropy decays like k^{-2r} up to logarithmic factors uniformly over the kernel class. The paper is careful to separate upper and lower bounds, and the lower bounds in (4.2)-(4.3) and (5.6)-(5.9) show that the main exponent is sharp in many parameter regimes. The abstract tool Theorem 2.3 is useful beyond the specific application, and the paper explicitly identifies the remaining logarithmic-factor gap as an open problem. The argument is coherent and the exponents in the main theorem check out; the main caveats are the reliance on an endpoint Littlewood-Paley statement cited from the author's own earlier work and a correctable gap in the kernel-decomposition lemma.

major comments (2)
  1. [Section 2, Theorem 2.1; Section 3, Eq. (3.2)] The block estimate (3.2) is applied to kernels K∈H^{r,2d}_{1,∞}, that is, with vector q=(1,∞). Theorem 2.1 cites [35] for the vector case but does not state the precise result or indicate whether the endpoint values q_j=1 and q_j=∞ are covered. This is load-bearing: the kernel decomposition in Lemma 3.1 and both upper bounds (1.4)-(1.5) rest on (3.2). Please provide a precise reference (theorem and page) that covers the endpoint, or reproduce the endpoint argument. If [35] treats only 1<q_j<∞, then the proof of (3.2) must be supplied; the L∞ component is non-separable and the L1 component lacks unconditional structure, so this is not a routine limiting case.
  2. [Section 3, Lemma 3.1] The three kernels K^1_u, K^2_u, K^3_u do not form a partition of K. The block {(s1,s2): ∥s1∥1≤u, ∥s2∥1≤u} is counted in both K^1_u and K^2_u, so K^1_u+K^2_u+K^3_u equals K+I, where I is the double-counted block. Consequently the claimed representation f=f^1_u+f^2_u+f^3_u does not follow from the stated definitions. This is repairable by defining K^2_u with the additional restriction ∥s1∥1>u, or by absorbing the overlap into f^1_u; the estimates (3.4)-(3.5) are unaffected. The proof of the lemma should be corrected before publication.
minor comments (4)
  1. [Section 3, Lemma 3.1] The statement of Lemma 3.1 says X_n:=T(Q_n), but the proof and the application in Theorem 1.2 use X_n:=T(Q_{n−u}); the latter is the correct indexing because f_{n,u}∈T(Q_{n−u}). Please align the statement with the proof.
  2. [Section 2, Theorem 2.3] In (2.9), the notation k:=D_{2u} makes k a specific sequence rather than a free index. The proof is consistent with this, but it would help the reader if the statement explicitly said that the estimate holds for the sequence k=D_{2u}, with constants independent of u.
  3. [Section 3, proof of Lemma 3.1] There is a typo 'inequailities' for 'inequalities'. Also, the subspace S_u is not explicitly defined; it is clear from context that it is the span of the coefficient functions of the y-frequencies in Q_u, but a one-sentence definition would improve readability.
  4. [Section 5, Eq. (5.9)] The condition 'even r∈N' appears to be inherited from (5.5), but when (5.5) is applied with smoothness parameter 2r, the parity condition is automatically satisfied for integer r. Please clarify the intended condition.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the upper bounds follow from an independent Littlewood-Paley characterization and standard entropy estimates; heavy self-citation is present but is not circular.

full rationale

I walked the derivation chain for Theorem 1.2. The kernel class H^{r,2d}_{1,infty} is defined by mixed differences, and Lemma 3.1 decomposes K via the block bounds ||A_s(K)||_{1,infty} << 2^{-r||s||_1} supplied by Theorem 2.1, cited from [35] and [36]. This is a genuine external characterization theorem: its assumptions and conclusion concern smoothness classes and block decompositions, not entropy numbers of W^K_1, and it is not fitted to the target rates. The rest of the proof applies general entropy machinery (Theorem 2.3) to known entropy bounds for hyperbolic-cross polynomial classes (Propositions 3.2 and 3.3, cited from [42]); those are parameter-free published results whose assumptions do not include the desired bound. No quantity is fitted to a subset of data and later called a prediction, and the lower bounds come from independent known entropy estimates combined with a direct embedding check in Lemma 4.1. The frequent self-citations are load-bearing as mathematical references, but they are not circular in the sense of this review: none of them is equivalent by construction to the claimed entropy bound. The possible fragility at the endpoint vector norm q=(1,infty) is a correctness or robustness concern, not a circularity concern.

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

All inputs are standard published theorems in approximation theory; no data fitting, no invented objects, and no fitted parameters appear in the derivation.

assumptions (6)
  • domain assumption Theorem 2.1: membership in H^a_{q,l} is equivalent to Littlewood-Paley block bounds ∥A_s(f)∥_q ≪ 2^{-(a,s)}.
    Invoked in Lemma 3.1 to get norm bounds (3.2) and (3.3); cited from [36] and [35].
  • standard math Nikol'skii inequality for trigonometric polynomials in mixed L_{1,∞} norms.
    Used to pass from ∥A_s(K)∥_{1,∞} to ∥A_s(K)∥_∞ with loss 2^{∥s∥_1}, giving (3.3).
  • domain assumption Entropy bounds for hyperbolic cross polynomial classes, Propositions 3.2, 3.3, and 3.4.
    Supply condition (2.6) with α, β, γ for d=1 and d=2; cited from [42].
  • domain assumption Known entropy lower bounds for W^r_q classes, Theorem 4.1 and relations (5.3)-(5.5).
    Used with Lemma 4.1 to get lower bounds on the supremum; cited from [42], [38], and [14].
  • standard math Carl's inequality relating entropy numbers to Kolmogorov widths.
    Used in Section 5 to prove the sharp result in Theorem 5.3 for 2≤q,p≤∞.
  • standard math Finite-dimensional entropy bound ε_k(B_X,X)≤3·2^{-k/n} for an n-dimensional Banach space.
    Used in the proof of Theorem 1.2 for the low-frequency parts f^1_u and f^2_u.

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Pith. "Pith review of Entropy numbers of classes defined by integral operators." pith.science (2026). https://pith.science/paper/73CKRXHD

@misc{pith2026250508572,
  author       = {Pith},
  title        = {Pith review of: Entropy numbers of classes defined by integral operators},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/73CKRXHD}},
  note         = {Machine review of arXiv:2505.08572}
}
read the original abstract

In this paper we develop the following general approach. We study asymptotic behavior of the entropy numbers not for an individual smoothness class, how it is usually done, but for the collection of classes, which are defined by integral operators with kernels coming from a given class of functions. Earlier, such approach was realized for the Kolmogorov widths.

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Forward citations

Cited by 1 Pith paper

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