{"id":"5991d62a-bc61-49ea-95c1-8d301da0322c","arxiv_id":"2505.08572","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"For kernels with mixed smoothness r>1, the worst-case entropy numbers of the associated L1-to-L∞ integral operator classes decay like k^{-2r} up to logarithmic factors in dimensions one and two.","lead":"This paper proves new upper bounds on entropy numbers for whole families of function classes built from integral operators whose kernels have mixed smoothness. The bounds hold for all kernels in the class at once and match known lower bounds up to logarithmic factors, extending a program previously developed for Kolmogorov widths.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Upper bounds depend on the vector-q Littlewood-Paley equivalence at q=(1,∞); only a citation supports this endpoint, so Theorem 1.2 is conditional on that theorem.","rationale":"I read the proof carefully and found no arithmetic error in Theorem 2.3 or in its application inside Theorem 1.2. The indexing discrepancy between Lemma 3.1 (X_n=T(Q_n)) and its proof (X_n=T(Q_{n-u})) is absorbable because Q_{n-u}⊂Q_n and the proof only needs the smaller subspaces. The lower-bound construction in Lemma 4.1 is plausible and is not part of the central upper-bound claim. The genuinely load-bearing step for the upper bounds is the vector-q block characterization: Lemma 3.1 and (3.5) are just summation of (3.2), and (3.2) is exactly Theorem 2.1 at q=(1,∞). The paper's citation to [35] may well cover it, but no statement or proof is included in this manuscript at precisely the point where endpoint issues are most likely to appear. This is the same weakest assumption identified by the reader, so I agree with that identification. I do not move the verdict: the concern is external and plausible, not a demonstrated error, and the reader's conditional verdict already flags the proof-level caveats.","tokens_in":15168,"tokens_out":42099,"duration_ms":418003,"concrete_test":"Verify the two implications of Theorem 2.1 for q=(1,∞) by hand. Direct: show each A_s is a convolution with an L1 kernel, hence bounded on L1(dx)L∞(dy), and combine with the mixed-difference inequalities. Converse: let f=Σ A_s f and estimate ∥∆_t^l(e)f∥_{1,∞} by splitting over blocks, using only 2^{-r||s||1} decay; check whether the constants stay independent of f and s. If both directions close, the concern is resolved; if not, Lemma 3.1's (3.2)-(3.5) need an alternative proof and Theorem 1.2 is not established.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 1.2's two upper bounds are derived through Lemma 3.1, whose estimate (3.2) uses Theorem 2.1 in the vector mixed norm q=(1,∞). For q=(1,∞), q_1=1 and q_2=∞ are endpoint values: classical Littlewood-Paley characterizations for Besov/Nikol'skii spaces are usually stated for 1<q_j<∞, and endpoint cases require separate arguments (L∞ is non-separable and L1 lacks unconditional structure). The paper does not reproduce the proof of Theorem 2.1 for this vector q; it cites [35]. If the cited result does not cover q=(1,∞), or covers only the direct inequality (2.5), then Lemma 3.1's conclusion that f^3_u belongs to \\bar W^{r,2d-2,1}_{L1,n0} with the stated norm bounds does not follow. Consequently both estimates (1.4) and (1.5) lose their core kernel-decomposition step. This is a load-bearing dependence on an external endpoint theorem rather than an internal inconsistency.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":15136,"tokens_out":19404,"duration_ms":175307,"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":[{"comment":"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.","section":"Section 2, Theorem 2.1; Section 3, Eq. (3.2)"},{"comment":"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.","section":"Section 3, Lemma 3.1"}],"minor_comments":[{"comment":"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.","section":"Section 3, Lemma 3.1"},{"comment":"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.","section":"Section 2, Theorem 2.3"},{"comment":"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.","section":"Section 3, proof of Lemma 3.1"},{"comment":"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.","section":"Section 5, Eq. (5.9)"}],"recommendation":"major_revision","confidential_remarks":"The central claim is plausible and the main exponents check out, but the two major comments above both touch the proof of the main theorem: one concerns an unverified endpoint Littlewood-Paley statement in the author's own earlier work, and the other is a genuine gap in the decomposition lemma, though easily corrected. The heavy self-citation is not circular, but the editor may wish to ensure independent verification of the endpoint theorem in [35] before accepting the paper."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This paper does something actually new: instead of entropy numbers for one fixed smoothness class, it bounds the worst-case entropy numbers over all kernels from a mixed-smoothness class, for the induced integral-operator classes. The main result, Theorem 1.2, gives sup over K in H^{r,2d}_{1,∞} of ε_k(W^K_1, L∞) ≤ k^{-2r} times log factors, for d=1,2 and r>1. That is a real extension of the program Temlyakov developed for Kolmogorov widths, not a repackaging.\n\nThe proof is coherent and mostly elementary given the known ingredients: decompose the kernel into low- and high-frequency parts, handle the low part with finite-dimensional entropy estimates, and the high part with a weighted tail-sum theorem. The lower bounds via the Bernoulli kernel are legitimate, since F_{2r}(x-y) does lie in the required kernel class. The self-citation is heavy, but that is natural—the framework and most tools are the author's own. I see no circularity.\n\nSoft spots, in proportion. The one worth flagging before anyone builds on this: Lemma 3.1 uses Theorem 2.1 at the vector endpoint q=(1,∞), and that theorem is cited from [35] for the vector case. Endpoint Littlewood-Paley characterizations often need separate arguments, and L1 and L∞ are exactly the cases where standard references get quiet. If [35] does not actually cover q=(1,∞), the kernel decomposition in Lemma 3.1 loses its footing. This is an external citation check, not an internal contradiction, but it is load-bearing. The rest is minor: the proof of Theorem 2.3 misstates the relation between k and n0 (should be k ≍ 2^{n0/2} n0^c, not k ≍ 2^{n0} n0^c), and Lemma 3.1 says X_n = T(Q_n) while the proof of Theorem 1.2 uses X_n = T(Q_{n-u}). The final exponents are consistent with the corrected version, so these are typos, not mathematical errors.\n\nWho gets value: approximation theorists working on mixed smoothness, entropy numbers, or widths. The logarithmic gaps and restriction to d=1,2 are honestly stated, and the open problems are sensible. I would send this to a serious referee—the core claim looks sound, but the referee should verify the endpoint citation and ask for the typos to be fixed. It is not a desk reject.","headline":"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.","tokens_in":15828,"tokens_out":1934,"would_cite":true,"duration_ms":19893,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["41A46","41A63","42B35"],"pacs":[],"model":"deepseek-v4-flash","headline":"For $r>1$, worst-case entropy of integral-operator classes with mixed-smoothness kernels is $k^{-2r}$ up to logarithmic factors.","keywords":["entropy numbers","integral operators","mixed smoothness","hyperbolic cross","Littlewood-Paley blocks","Kolmogorov widths","Bernoulli kernel","multivariate approximation"],"falsifier":"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.","tokens_in":14761,"feed_emoji":"📉","tokens_out":15214,"duration_ms":129784,"temperature":0.7,"pith_summary":"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.","feed_headline":"Worst-case entropy of kernel classes falls like k^{-2r}","feed_subtitle":"Knowing only that a kernel is r-smooth guarantees k^{-2r} entropy up to logs.","key_machinery":"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).","core_discovery":"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 $H^{r,2}_{1,1}$.","pith_inferences":["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."],"forward_implications":["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 $r>3/2$."],"supporting_citations":[{"why":"Supplies the vector-parameter Littlewood-Paley characterization used to convert $H^{r,2d}_{1,\\infty}$ membership into dyadic-block decay, and the Kolmogorov-width theorem used in Section 5.","marker":"[35]"},{"why":"Provides the scalar Littlewood-Paley characterization and the Bernoulli-kernel norm facts behind the definition of the classes.","marker":"[36]"},{"why":"Source for the entropy estimates of hyperbolic-cross polynomial spaces used to verify condition (2.6), and for the lower bound in Theorem 4.1.","marker":"[42]"},{"why":"Supplies the transfer inequality from Kolmogorov widths to entropy numbers that yields the sharp $d=1$ result in Theorem 5.3.","marker":"[5]"},{"why":"Known entropy order for $W^r_q$ in $L_\\infty$ with $1<q<\\infty$; combined with Lemma 4.1 it produces the stronger $d=2$ lower bound.","marker":"[38]"},{"why":"Known endpoint bounds for $W^r_1$ in $L_p$ and $L_\\infty$ that give the lower bounds (5.8) and (5.9).","marker":"[14]"},{"why":"Original formulation of the collection-of-classes problem for Kolmogorov widths, which this paper extends to entropy numbers.","marker":"[32]"}],"fun_headline_variants":["Kernel smoothness r forces k^{-2r} entropy","Worst-case kernel entropy scales as k^{-2r}","Smooth kernels yield k^{-2r} entropy, logs aside","Entropy of kernel classes: k^{-2r} up to logs"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Kernel smoothness r forces k^{-2r} entropy","Worst-case kernel entropy scales as k^{-2r}","Smooth kernels yield k^{-2r} entropy, logs aside","Entropy of kernel classes: k^{-2r} up to logs"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000165,"raw_usage":{"total_tokens":1188,"prompt_tokens":819,"completion_tokens":369,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":435,"completion_tokens_details":{"reasoning_tokens":295}},"tokens_in":435,"tokens_out":369,"duration_ms":3963,"temperature":1.0,"reasoning_tokens":295,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T21:52:35.203614+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Temlyakov, Estimates of the best bilinear approximations of functions of two variables and some of their applications, Mat","cited_arxiv_id":null,"evidence_quote":"Supplies the vector-parameter Littlewood-Paley characterization used to convert $H^{r,2d}_{1,\\infty}$ membership into dyadic-block decay, and the Kolmogorov-width theorem used in Section 5."},{"cited_title":"Carl, Entropy numbers, s-numbers, and eigenvalue problems, J","cited_arxiv_id":null,"evidence_quote":"Supplies the transfer inequality from Kolmogorov widths to entropy numbers that yields the sharp $d=1$ result in Theorem 5.3."},{"cited_title":"Temlyakov, An inequality for trigonometric polynomials and its application for estimating the entropy numbers, J","cited_arxiv_id":null,"evidence_quote":"Known entropy order for $W^r_q$ in $L_\\infty$ with $1<q<\\infty$; combined with Lemma 4.1 it produces the stronger $d=2$ lower bound."},{"cited_title":"Kashin and V.N","cited_arxiv_id":null,"evidence_quote":"Known endpoint bounds for $W^r_1$ in $L_p$ and $L_\\infty$ that give the lower bounds (5.8) and (5.9)."},{"cited_title":"Temlyakov, On widths of function classes, Dokl","cited_arxiv_id":null,"evidence_quote":"Original formulation of the collection-of-classes problem for Kolmogorov widths, which this paper extends to entropy numbers."}],"review_version":1}