{"id":"bbfe75c7-77f6-4614-85c0-b1f039d7953b","arxiv_id":"2501.04023","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"Two theorems give sufficient shallow-network width to reach a prescribed error in a Fréchet metric of semi-norms, applied to exponential spectral Barron, Gelfand-Shilov, and bandlimited (Paley-Wiener type) symbol classes.","lead":"This math paper proves how wide a neural network must be to approximate signal-changing rules (Fourier symbols) accurately across all smoothness scales at once. It matters because such rules appear when machine learning tries to solve physics equations.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 4.8's unit-coefficient network class does not match [57, Theorem 2], so the N^{-1/2} rate and Theorem 4.9 are not established.","rationale":"The reader's weakest assumption was the bounded-growth condition (4.1). That is a valid limitation, but it is an acknowledged scope restriction, not a flaw in the conditional theorems. The more load-bearing problem is in the proof of Proposition 4.8: the approximating class is defined without coefficients, while the cited result [57, Theorem 2] applies to networks with coefficients and a total-variation budget. Since the infimum over the larger coefficient class is ≤ the infimum over the unit-coefficient subclass, the cited theorem cannot provide an upper bound for the subclass. The paper's strongest concrete claim, Theorem 4.9, depends entirely on this rate; if the rate is not established for Σ_N, the dimension-independent Monte Carlo claim falls. I agree with the reader that Proposition 4.2's proof has an unjustified step (not all extensions are Barron) and that the 'first Fréchet extension' statement conflicts with the cited Korolev work; however, these affect novelty and a negative example, whereas the coefficient gap affects the main positive application. The fix is straightforward — add coefficients and a budget to Σ_N and restate Theorem 4.9 accordingly — so a conditional acceptance is appropriate rather than rejection. The verification step would be to rerun the proof with the coefficient class and confirm that the width formula still yields a dimension-independent bound, or to exhibit a counterexample for the unit-coefficient class.","tokens_in":84,"tokens_out":25585,"duration_ms":327091,"concrete_test":"Re-derive Proposition 4.8 with the coefficient class Σ_N := {∑ a_n F^{-1}{χ_{[-Ω,Ω]^d} \\hat{σ}(⟨w_n,·⟩+b_n)} : ∑|a_n| ≤ ∥f∥_{B*_1(Ω)}} and check whether the proof chain and the width estimate in Theorem 4.9 still go through. If the proof requires the unit-coefficient class, test the target g ≡ c on [-Ω,Ω]^d with |c| > ∥\\hat{σ}∥_∞: the unit-coefficient class cannot represent that constant, so the L2 error cannot decay, contradicting the claimed N^{-1/2} rate.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The main bandlimited application rests on Proposition 4.8, which defines the approximation class as Σ_N := {∑_{n=1}^N F^{-1}{χ_{[-Ω,Ω]^d}(·) \\hat{σ}(⟨w_n,·⟩+b_n)}} with no coefficients a_n, and similarly bΣ_N with unit coefficients. The proof then invokes [57, Theorem 2] for a rate N^{-1/2}. However, [57, Theorem 2] concerns networks ∑ a_n σ(w_n·x+b_n) with a coefficient budget ∑|a_n| ≤ ∥g∥_{B1}; the class bΣ_N is a strict subset of that class (all coefficients equal to 1). An upper bound on the infimum over the larger class does not imply an upper bound on the infimum over the smaller class; in fact the infimum over the larger class is smaller or equal, so the cited theorem cannot yield the claimed rate for bΣ_N. Because Theorem 4.9's width formula uses exactly r(N)=N^{-1/2} from Proposition 4.8, the headline 'dimension-independent Monte Carlo rate' for Barron-Bandlimited functions is unsupported as written. This is a correctness gap in the central example, not merely a restrictive assumption. The bounded-growth condition (4.1) is indeed restrictive, but the coefficient issue is more urgent: without free coefficients the dictionary cannot even scale arbitrary targets, so the Monte Carlo rate is not guaranteed.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies neural-network approximation of symbols/functions measured in a Fréchet metric induced by a separating sequence of semi-norms. The two main results are meta-theorems: Theorem 3.1 shows that if approximation rates are known for every semi-norm and the seminorms satisfy a monotonic-growth condition (3.1), then an explicit width N suffices to make the Fréchet error dV(f-f_N) < ε; Theorem 4.1 shows that under the stronger bounded-growth condition (4.1), knowledge of the rate in only the first semi-norm suffices, again with an explicit width. The paper applies the first result to the exponential spectral Barron space with Sobolev semi-norms and to Gelfand-Shilov spaces, and the second result to a newly introduced class of Barron-Bandlimited functions, for which it claims a dimension-independent Monte Carlo rate N^{-1/2} in the L2 norm and hence in the Fréchet metric.","tokens_in":22917,"tokens_out":28535,"duration_ms":289653,"significance":"The general reduction in Theorems 3.1 and 4.1 is a useful and clean idea: splitting the Fréchet sum into a finite leading part and a geometric tail, then using monotonicity of t/(1+t), converts any per-semi-norm rate into a sufficient width. The paper is also transparent about limitations: it proves that the bounded-growth condition (4.1) fails for the exponential spectral Barron space with Sobolev semi-norms (Propositions 4.2 and 4.3), and it does not rely on fitted parameters—all constants come from cited external rates. However, the headline bandlimited application is currently not established: Proposition 4.8 invokes a coefficient-budget theorem for a unit-coefficient approximation class, and the width formula of Theorem 4.9 depends on that unsupported rate. The manuscript also contains a self-referential appendix proposition. These issues are local and repairable, but they are load-bearing for the main example.","major_comments":[{"comment":"The approximation class bΣ_N defined in the proof of Proposition 4.8 consists of unit-coefficient sums Σ_{n=1}^N σ̂(⟨w_n, ·⟩+b_n), with no coefficients a_n. The proof invokes [57, Theorem 2] to assert an N^{-1/2} rate for this class. As stated in the cited theorem, however, the rate applies to networks with arbitrary coefficients a_n satisfying a bounded ℓ1 budget; the unit-coefficient class is a strict subset of that class for fixed N, and the infimum over a superset does not control the infimum over the subset. Moreover, unit coefficients force the ℓ1 budget to grow with N, so applying the cited theorem with budget N would not yield the claimed rate. Since Theorem 4.9 uses exactly r(N)=N^{-1/2} from Proposition 4.8, the dimension-independent Monte Carlo result is not established as written. The fix is local: add coefficients a_n with a constraint such as Σ|a_n| ≲ ||f||_{B*_1} to the definitions of Σ_N and bΣ_N, and then apply [57, Theorem 2] directly.","section":"Proposition 4.8 and Theorem 4.9"},{"comment":"In the proof of Theorem 3.1, after applying the approximation rate the manuscript writes inf dV ≤ (2-2^{-ℓ}) C_f M_ℓ r_ℓ(N)/(1+C_f M_ℓ r_ℓ(N)) + 2^{-ℓ} ≤ C_f M_ℓ r_ℓ(N) + 2^{-ℓ}. The last inequality is not valid in general: the factor (2-2^{-ℓ})/(1+C_f M_ℓ r_ℓ(N)) can exceed 1, for example when C_f M_ℓ r_ℓ(N) is small. With the threshold C_f M_ℓ r_ℓ(N) ≤ 2^{-ℓ} used in the proof, the correct bound is at most 3·2^{-ℓ}, so the displayed choice ℓ_ε = ⌈−log2 ε⌉+1 does not, as proven, guarantee an error below ε. The same issue appears in the proof of Theorem 4.1. This is repairable by taking a slightly larger ℓ_ε or a smaller threshold (e.g., replacing 2^{-ℓ} by 2^{-ℓ}/3), and the qualitative statements of the theorems survive, but the explicit width formulas need correction.","section":"Theorem 3.1 and Theorem 4.1, proofs"},{"comment":"Proposition B.1 asserts that the weighted Gevrey class is either trivial or intractable to approximate, but its proof is self-referential: it states only that the proof 'could be found in the Appendix,' which is the same appendix containing the proposition. The surrounding discussion also contains an internal inconsistency: the differential-inequality argument |f'| ≤ M_1|f| implies that a C∞ function with a zero at a finite point vanishes on its connected component, which rules out compactly supported bump functions, whereas the preceding paragraph claims that in the non-analytic case β<1 the class may contain spatial concatenations of bump functions. The proposition should either be given a rigorous proof or removed, and the contradictory discussion should be reconciled.","section":"Appendix B, Proposition B.1"}],"minor_comments":[{"comment":"Definition 4.7 defines B*_s(Ω) for s>1, but Proposition 4.8 states the result for f ∈ B*_1(Ω); if the intended class is B*_1, the definition and the theorem statements should be made consistent.","section":"Definition 4.7 and Proposition 4.8"},{"comment":"In Theorem 4.9 the constant M_ℓ_ε is taken as ⟨Ω⟩^{ℓ_ε}, but for d-dimensional functions with supp f̂ ⊆ [−Ω,Ω]^d the correct Sobolev-growth constant is ⟨√d Ω⟩^{ℓ_ε} (up to a harmless constant); the dimension-free rate is unaffected, but the formula as written understates the constant.","section":"Theorem 4.9"},{"comment":"The claim that f̂ ∈ L1(e^{c|·|^β}) implies lim_{|ξ|→∞} |f̂(ξ)| e^{c|ξ|^β} = 0 and sup_ξ |f̂(ξ)| e^{c|ξ|^β} < ∞ is not valid for general L1 functions; the later inclusion Lemma 3.5 is proved by a different argument and does not depend on this claim, so the paragraph should be revised.","section":"Section 3.2, paragraph before Lemma 3.5"},{"comment":"The sentence claiming that this work is 'to the best of our knowledge the first such extension' to Fréchet spaces is contradicted by the cited works [12] and [33], which already treat neural networks in Fréchet spaces and Barron-type approximation in Fréchet metrics; the wording should be softened.","section":"Section 1, after Theorem 3.1 discussion"},{"comment":"The notation S^{1,ℓ}(U) used in Lemma 3.4 is not defined; please use the notation S^p_{ω,ℓ}(U) from Definition 2.3 or state explicitly which p and weight are meant.","section":"Lemma 3.4 and Definition 2.3"},{"comment":"The displayed distributional Fourier transform of sin(nx) contains incorrect delta arguments (ξ/n instead of ξ∓n), and the subsequent change of variables is hard to follow; the computation should be rewritten with the stated Fourier convention.","section":"Proposition 4.3 proof"}],"recommendation":"major_revision","confidential_remarks":"The paper is within the journal's scope and the general framework is sound, but the main bandlimited example currently rests on a mismatch between the stated network class and the cited coefficient-budget theorem, and the central proofs contain a repairable but real numerical-factor issue. I do not see circularity or hidden fitted parameters; the authors' self-citations are background. I recommend major revision rather than rejection because the defects are local and the stated results can likely be repaired by adding coefficients to the bandlimited network class and adjusting the thresholds in the two main proofs."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThis paper has one genuinely useful core: Theorem 3.1 (and its variant 4.1) shows how to convert per-semi-norm approximation rates into a sufficient width in a Fréchet metric by splitting the series and controlling the tail. The proof is a clean monotonicity argument, and the exponential-spectral-Barron application in Corollary 3.3 follows directly from Siegel-Xu. That part is solid and worth preserving.\n\nThe soft spots are real, though. The advertised dimension-independent rate for Barron-Bandlimited functions is not established. Proposition 4.8 defines the approximating class with all coefficients set to 1, and then invokes Siegel-Xu, whose theorem requires a coefficient budget. An upper bound on a larger dictionary does not bound the infimum over a smaller one; with unit coefficients you cannot even scale a target of arbitrary Barron norm. So Theorem 4.9's width formula is unsupported. This is the load-bearing example, so the paper's headline example collapses until the dictionary is repaired (e.g., by adding coefficients).\n\nSecond, the 'first Fréchet extension' claim is contradicted by the authors' own citations [12,33]; that needs to be reworded to something like 'first for symbol approximation.' Third, Proposition 4.2's proof has an unjustified step: the claim that all L1 extensions of a Barron function are themselves Barron on the whole space. The conclusion may be true, but getting there needs a real analyticity argument (a finite weighted-L1 Fourier transform forces analyticity for β≥1); the present text doesn't make it. Finally, Proposition B.1 is stated with a proof that the appendix never gives—what's there is discussion, not a proof.\n\nThe bounded-growth condition (4.1) is restrictive, and the paper is honest about that. The problem is that, once the bandlimited example fails, the second theorem has no verified concrete application in the current version. That is fixable, but it is a genuine gap, not a stylistic quibble.\n\nWho is this for: researchers working on neural network approximation rates in metric spaces and operator learning. The general theorem is a useful tool; I'd bring it to a reading group. It deserves serious peer review, but the version I read should not be accepted without major revision.\n\nRecommendation: send it out, with the expectation that the authors fix the coefficient class, correct the novelty claim, and either repair or remove the unproven parts.","headline":"A clean Fréchet-width theorem that is currently oversold by a broken bandlimited example and an overclaimed novelty statement.","tokens_in":23498,"tokens_out":11085,"would_cite":true,"duration_ms":112963,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["41A25","41A46","41A65","46E10","68T05","68T07"],"pacs":[],"model":"deepseek-v4-flash","headline":"Shallow networks can approximate Fourier symbols of linear operators in Fréchet metrics to any prescribed accuracy, with explicit sufficient widths, and the Barron-bandlimited class attains the dimension-independent $N^{-1/2}$ Monte Carlo…","keywords":["Neural networks","Approximation rates","Operator learning","Symbol approximation","Barron spaces","Fréchet spaces","Paley-Wiener spaces","Fourier multipliers"],"falsifier":"Compute the best $N$-term approximation error in the $L^2$ semi-norm for a concrete Barron-bandlimited function, for instance a function whose Fourier transform is a truncated Gaussian on $[-\\Omega,\\Omega]^d$, and check whether the error decays as $N^{-1/2}$ with the implied constant independent of dimension; if the decay is slower, Proposition 4.8 would be contradicted. For Theorem 4.9, evaluate the full Fréchet metric $d_V(f-f_N)$ for the width given by the closed-form formula and test whether it indeed falls below $\\varepsilon$; a single violation would falsify the suffiency claim.","tokens_in":22350,"feed_emoji":"🧠","tokens_out":10749,"duration_ms":84274,"temperature":0.7,"pith_summary":"This paper studies how well shallow neural networks can approximate the Fourier symbols of linear differential operators, where the error is measured not by a single norm but by a whole Fréchet metric built from a sequence of semi-norms. The main theorems give sufficient conditions, in closed form, on the network width $N$ needed to drive the Fréchet error below any prescribed $\\varepsilon$: one theorem requires approximation rates in every semi-norm under a mild monotonic-growth condition, and a second theorem requires only the rate for the first semi-norm under a stronger bounded-growth condition. Applied to exponentially weighted spectral Barron spaces and Gelfand-Shilov spaces, the first theorem yields explicit width formulas; applied to a new class of Barron-bandlimited functions, the second yields a dimension-independent width formula with the Monte Carlo rate $N^{-1/2}$. The upshot is that infinite-dimensional operator approximation can be reduced to explicit finite-network-size guarantees, with the caveat that the bounded-growth condition is narrow and fails for the natural exponential-spectral-Barron example.","feed_headline":"Width formula guarantees Fréchet-accurate symbol approximation","feed_subtitle":"Bandlimited symbols reach the N^{-1/2} Monte Carlo rate with no curse of dimensionality.","key_machinery":"The central object is the Fréchet metric $d_V(f)=\\sum_{\\ell\\in\\mathbb{Z}_+} 2^{-\\ell} p_\\ell(f)/(1+p_\\ell(f))$ induced by a separating sequence of semi-norms $\\{p_\\ell\\}$, which lets the authors measure symbol approximation in a topology strictly finer-grained than any single Banach norm. The proof machinery splits the infinite sum in the metric at some level $\\ell$: the tail is bounded by $2^{-\\ell}$ by direct geometric summation, while the head is bounded through the monotonic-growth or bounded-growth condition together with the monotonicity of $t\\mapsto t/(1+t)$ and the assumed rate $r_\\ell$. Inverting the rate gives the closed-form width $N_{\\ell_\\varepsilon}$ for any target accuracy $\\varepsilon$. The bounded-growth condition $p_\\ell(f)\\le M_\\ell p_0(f)$ is the specific bridge that the second theorem uses to transfer an approximation rate known only for $p_0$ to the whole Fréchet metric.","core_discovery":"At the center of the paper are two general approximation theorems. Theorem 3.1 assumes the monotonic-growth condition $p_k(f) \\le M_\\ell p_\\ell(f)$ for $k \\le \\ell$ and, for every semi-norm, a decreasing bijective rate $r_\\ell$ with $p_\\ell(f-f_N) \\le C_f r_\\ell(N)$; it then proves that choosing $N \\ge r_{\\ell_\\varepsilon}^{-1}\\bigl(\\min\\{r_{\\ell_\\varepsilon}(1), 2^{-\\ell_\\varepsilon}/(C_f M_{\\ell_\\varepsilon})\\bigr)\\bigr)$, where $\\ell_\\varepsilon = \\lceil -\\log_2\\varepsilon\\rceil + 1$, guarantees $d_V(f-f_N)<\\varepsilon$. Theorem 4.1 replaces the monotonic condition by the bounded-growth condition $p_\\ell(f)\\le M_\\ell p_0(f)$ and uses only the rate for $p_0$, with the sufficient width $N \\ge r^{-1}\\bigl(\\min\\{r(1), 2^{-\\ell_\\varepsilon}/(C_f M_{\\ell_\\varepsilon})\\bigr)\\bigr)$. The authors then show that the exponential spectral Barron space satisfies the monotonic condition with Sobolev semi-norms, producing a width of order $\\bigl(\\tfrac{1}{c_{\\ell_\\varepsilon}}\\ln(2^{\\ell_\\varepsilon}C_{\\ell_\\varepsilon}\\|f\\|_{B_{\\beta,c}})\\bigr)^{d/\\beta}$, but that it violates the bounded-growth condition: cosine units are not in the space for $\\beta\\ge 1$, and derivatives are unbounded for $\\beta<1$. For Barron-bandlimited functions $B_1^*(\\Omega)$, the bounded-growth condition holds with $M_\\ell = \\langle\\Omega\\rangle^\\ell$, and a dimension-independent width $N \\ge \\bigl(2^{-\\ell_\\varepsilon}/(\\|f\\|_{B_1^*(\\Omega)}\\langle\\Omega\\rangle^{\\ell_\\varepsilon})\\bigr)^{-2}$ guarantees the Fréchet error bound.","pith_inferences":["Editorial inference: the width formula in Theorem 4.9 still depends on the bandwidth through $M_{\\ell_\\varepsilon}=\\langle\\Omega\\rangle^{\\ell_\\varepsilon}$; for families of symbols whose bandwidth grows with dimension or target smoothness, the sufficient width inherits that growth, so the \"no curse of dimensionality\" claim applies to fixed bandwidth at fixed accuracy.","Editorial inference: the paper approximates symbols, not solution operators end-to-end; in an operator-learning pipeline, the Fréchet symbol error would need to be combined with a stability estimate for the underlying PDE to guarantee control of the operator's output error, which the paper leaves implicit.","Editorial inference: Definition 4.7 writes the class condition with $B^1$ but the norm with $B^s$; a natural clarification is whether the rate in Proposition 4.8 and the width in Theorem 4.9 require $s=1$ or hold uniformly over $s>1$.","Editorial inference: a natural next step would be to characterize which weighted symbol classes satisfy the bounded-growth condition while remaining rich enough to contain non-trivial approximation classes, since the self-weighted Gevrey class considered in the appendix turns out to be trivial or intractable."],"forward_implications":["If the two main theorems are correct, then every symbol class with known rate functions in each semi-norm, such as the exponential spectral Barron space and the Gelfand-Shilov spaces considered here, has an explicit shallow-network width for a preassigned Fréchet accuracy.","If the bounded-growth condition holds for a class, a single $L^2$ (or first semi-norm) approximation rate suffices to control the entire Fréchet metric, so existing Monte Carlo rates transfer to the infinite-dimensional symbol-approximation setting.","For the Barron-bandlimited class, the sufficient width formula gives approximation in all Sobolev semi-norms simultaneously with the dimension-independent rate $N^{-1/2}$.","The counterexamples for the exponential spectral Barron space show that the bounded-growth condition is not automatically satisfied by natural classes, so applications of the second theorem must verify condition (4.1) explicitly.","The embedding results connect Barron spaces to the multiplier class $S^{1,\\ell}(U)$ and to Gelfand-Shilov spaces, so the approximation guarantees apply to standard classes of pseudodifferential symbols."],"supporting_citations":[{"why":"Establishes the form of the Fréchet metric and its equivalence with the semi-norm topology, which the main theorems use throughout.","marker":"[53]"},{"why":"Supplies the exponential approximation rate for the exponential spectral Barron space that Corollary 3.3 inverts into a width bound.","marker":"[58]"},{"why":"Supplies the dimension-independent $L^2$ approximation rate for Barron functions with general activation functions, which Proposition 4.8 transfers to the bandlimited class.","marker":"[57]"},{"why":"Provides the original Barron space and the Monte Carlo $L^2$ rate that the bandlimited example builds on.","marker":"[10]"},{"why":"Defines the semi-norms that make the Hörmander symbol class a Fréchet space, motivating the use of the Fréchet metric.","marker":"[27]"},{"why":"Defines the Hörmander symbol class that the paper takes as its target for symbol and operator approximation.","marker":"[26]"}],"fun_headline_variants":["Fréchet metric yields explicit operator learning widths","Barron-bandlimited symbols avoid curse of dimensionality","Two theorems for Fréchet-accurate symbol approximation","Exponential Barron space gives dimension-free width bound","Fourier symbols hit Monte Carlo rate in Fréchet metric"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise of the second main theorem is that each derivative-level semi-norm of the target function is bounded by a fixed multiple of the semi-norm at level zero; the paper shows that this fails for the natural exponential spectral Barron space, so the theorem only applies to classes specially made to satisfy it, such as the Barron-bandlimited class.","fun_headline_variants_meta":{"raw":{"variants":["Fréchet metric yields explicit operator learning widths","Barron-bandlimited symbols avoid curse of dimensionality","Two theorems for Fréchet-accurate symbol approximation","Exponential Barron space gives dimension-free width bound","Fourier symbols hit Monte Carlo rate in Fréchet metric"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001049,"raw_usage":{"total_tokens":4527,"prompt_tokens":1186,"completion_tokens":3341,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":802,"completion_tokens_details":{"reasoning_tokens":3263}},"tokens_in":802,"tokens_out":3341,"duration_ms":21537,"temperature":1.0,"reasoning_tokens":3263,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T23:48:58.098703+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the best $N$-term approximation error in the $L^2$ semi-norm for a concrete Barron-bandlimited function, for instance a function whose Fourier transform is a truncated Gaussian on $[-\\Omega,\\Omega]^d$, and check whether the error decays as $N^{-1/2}$ with the implied constant independent of dimension; if the decay is slower, Proposition 4.8 would be contradicted. For Theorem 4.9, evaluate the full Fréchet metric $d_V(f-f_N)$ for the width given by the closed-form formula and test whether it indeed falls below $\\varepsilon$; a single violation would falsify the suffiency claim.","supporting_citations":[{"cited_title":"Rudin, Functional analysis(International series in pure and applied math- ematics), 2nd ed","cited_arxiv_id":null,"evidence_quote":"Establishes the form of the Fréchet metric and its equivalence with the semi-norm topology, which the main theorems use throughout."},{"cited_title":"High-order approximation rates for shallow neural net- works with cosine and ReLU activation functions,","cited_arxiv_id":null,"evidence_quote":"Supplies the exponential approximation rate for the exponential spectral Barron space that Corollary 3.3 inverts into a width bound."},{"cited_title":"Howdo infinitewidthbounded norm networks look in function space?","cited_arxiv_id":null,"evidence_quote":"Supplies the dimension-independent $L^2$ approximation rate for Barron functions with general activation functions, which Proposition 4.8 transfers to the bandlimited class."}],"review_version":1}