{"id":"e50af870-3d58-4952-a0ee-56b40ba6097f","arxiv_id":"2505.00351","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Sobolev functions of order (d+2k+1)/2 are exactly L2-weighted integrals of ReLU^k ridge functions, and linearized networks with well-distributed fixed weights achieve the optimal n^{-1/2-(2k+1)/(2d)} approximation rate.","lead":"This paper proves that smooth functions can be represented as weighted averages of simple kinked building blocks called ReLU^k ridges, and that adding more evenly or randomly chosen building blocks improves accuracy at the mathematically optimal rate. It explains why linearized neural networks, where only the final weights are trained, can be as powerful as fully trained networks for these functions.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The parity-preserving Sobolev extension on which Theorem 2.2 rests is asserted, not proved; without it the ball-to-sphere transfer and hence the optimal FNS rate do not follow.","rationale":"The reader's weakest_assumption identifies exactly this unproved parity-preserving Sobolev extension as the load-bearing step, and I agree that it is the single most important gap. I verified the algebra of the parity condition on the equator: the natural candidate g_sym does not agree with T_k f_E on G unless T_k f_E vanishes where both eta and -eta lie in G. I also checked that Theorem 4.2's isomorphism is only for the parity subspace and that all main results (Theorems 2.2, 2.3, 2.4, Corollary 2.2, Theorem 6.2) depend on this transfer. The concern is not a disagreement with standard results in the field; it is an internal gap in the proof as written. It is likely repairable: one might extend f_E to be supported away from the boundary of the ball (so the lift vanishes on the equator), or use an extension operator that commutes with the involution eta -> -eta on S^d, or prove the domain result directly without the sphere. Because the repair is plausible and the surrounding technical machinery (Legendre coefficient bounds, quadrature, Cesaro summation) appears sound, a conditional acceptance with a request for this extension proof is the right verdict, matching the reader's CONDITIONAL.","tokens_in":32665,"tokens_out":2731,"duration_ms":23925,"concrete_test":"Check the extension step for the specific lift T_k: for a smooth f on B^d with T_k f_E not vanishing on the equator partial G (e.g., f(x)=1 so T_k f_E(eta)=eta_{d+1}^k), compute the minimal H^{1/2}(S^d) norm of any g with g(eta)=(-1)^{k+1}g(-eta) and g=T_k f_E on G. If the infimum is not bounded by a constant times ||T_k f_E||_{H^{1/2}(G)}, the asserted extension in the proof of Theorem 2.2 is false as stated. A cheaper analytical test: write the odd/even extension explicitly using spherical coordinates (phi, eta) near the equator and verify whether the parity condition forces a jump in the trace that cannot be absorbed by H^{1/2}(S^d) when f is not zero on the boundary of B^d.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The proof of Theorem 2.2 (Section 5.1) transfers the spherical approximation result Theorem 4.1 to the ball by extending T_k f_E from the cap G = {eta : eta_{d+1} >= 1/sqrt(2)} to the whole sphere S^d. The text states: 'By the extension theorems again, there exists a function g on S^d, which is an extension of T_k f_E, such that ||g||_{H^r(S^d)} <= C ||T_k f_E||_{H^r(G)} and g(eta) = (-1)^{k+1} g(-eta).' This is the only bridge between Theorem 4.1, which requires the parity condition (4.1), and the domain statement of Theorem 2.2. The asserted extension combines two nontrivial requirements: a bounded extension operator from H^r(G) to H^r(S^d) and exact odd/even symmetry depending on k. Exact parity is not preserved by standard Sobolev extension operators (e.g., Stein's extension, [89], or reflection-based extensions) because they are constructed locally and depend on the domain, not on the parity class of the function. Symmetrization after extension, g_sym(eta) = (g(eta) + (-1)^{k+1} g(-eta))/2, restores parity but only yields ||g_sym||_{H^r(S^d)} <= C ||g||_{H^r(S^d)}, and critically g_sym equals T_k f_E on G requires T_k f_E to already be parity-compatible on G: for eta in G with -eta also in G (the equator eta_{d+1} = 1/sqrt(2)), we must have T_k f_E(eta) = (-1)^{k+1} T_k f_E(-eta). But T_k f_E(-eta) = (-eta_{d+1})^k f_E(eta/(eta_{d+1})) = (-1)^k eta_{d+1}^k f_E(...) = (-1)^k T_k f_E(eta), while the desired parity is (-1)^{k+1} T_k f_E(eta). These coincide only if T_k f_E(eta)=0 on the equator, which is not guaranteed. So the symmetrized extension does not match T_k f_E on G. The paper's own Theorem 4.2 shows that for the sphere equality H^{(d+2k+1)/2}_*(S^d) = {G(psi): psi in L^2} holds only on the parity subspace (4.1). On a general bounded Lipschitz Omega there is no apparent reason the lift T_k f_E should lie in that subspace.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies approximation properties of linearized shallow ReLU^k networks with fixed, well-distributed inner parameters. Its main results are: (i) Theorem 2.2, an approximation estimate showing that finite neuron spaces L^k_{n,M} achieve the rate O(n^{-1/2-(2k+1)/(2d)}) in L2 for functions in H^{(d+2k+1)/2}(Ω), with a coefficient bound M comparable to the Sobolev norm; (ii) Theorem 2.3, an integral representation of H^{(d+2k+1)/2}(Ω) by L2-weighted integrals of ReLU^k ridge functions, with equivalent norms; and (iii) corollaries including an embedding of this Sobolev space into the Barron space, an RKHS interpretation, and a generalization bound. The proofs transfer harmonic-analytic approximation results from the sphere to the ball through an operator T_k mapping functions on the ball to a cap of the sphere.","tokens_in":33050,"tokens_out":23685,"duration_ms":201101,"significance":"If the proofs are completed, the results are significant: they show that linearized networks (convex optimization over fixed weights) match the optimal approximation rates of nonlinear shallow networks on Sobolev spaces, give an explicit L2 integral representation of Sobolev spaces via ReLU^k ridge functions, and provide coefficient bounds that are useful for generalization analysis. The approximation-rate claim is falsifiable and would explain the empirical success of random feature methods as a consequence of the deterministic well-distributed nature of the parameters. The paper also reproduces the embedding H^{(d+2k+1)/2}(Ω) into B^k(Ω) and clarifies the quotient structure between Barron and Sobolev spaces. The main technical gaps, however, are two unproved load-bearing steps that must be supplied before the central claims are fully established.","major_comments":[{"comment":"The step 'By the extension theorems again, there exists a function g on S^d, which is an extension of T_k f_E, such that ||g||_{H^r(S^d)} ≲ ||T_k f_E||_{H^r(G)} ≲ ||f||_{H^r(Ω)} and g(η) = (-1)^{k+1}g(-η)' is asserted without proof or reference. This parity-preserving extension is load-bearing: Theorem 4.1 requires the parity condition (4.1), and this is the only bridge from the spherical approximation result to the ball. Standard Sobolev extension operators do not preserve exact parity, and symmetrization after extension changes the values on G unless additional construction is used. Because G and -G are disjoint, a correct construction is possible: define g on -G by g(-η)=(-1)^{k+1}T_k f_E(η), apply a bounded extension operator from the disjoint union G∪(-G) to S^d, and then use a smooth cutoff to adjust the equatorial belt while preserving the parity. However, this argument is not provided, and the current manuscript therefore leaves a gap in the proofs of both Theorem 2.2 and Theorem 2.3, which rely on the same asserted extension.","section":"Section 5.1, proof of Theorem 2.2"},{"comment":"The construction of piecewise-constant densities ψ_n is only sketched, and the claim 'One could verify the limit of this subsequence is the desired function ψ' omits the key compactness argument. To complete the proof one must show that the weak limit ψ satisfies f(x) = ∫_{S^d} σ_k(θ·x̃)ψ(θ)dθ. This requires two facts: (i) the integral operator G: L^2(S^d) → L^2(Ω) is compact (it is Hilbert–Schmidt on a bounded domain), so Gψ_n → Gψ strongly whenever ψ_n → ψ weakly; and (ii) Gψ_n differs from the neural-network approximant f_n by an error of size O(n^{-1/d} ∑_j |a_j^{(n)}|) ≤ O(M n^{-1/d}) using the quasi-uniform partition of S^d and Lipschitz continuity of θ ↦ σ_k(θ·x̃). Since this direction is part of the claimed characterization in Theorem 2.4, the argument should be included explicitly.","section":"Section 2.2, proof of Theorem 2.4 (converse direction)"}],"minor_comments":[{"comment":"The notation 'T_k f' is used for f ∈ H^{(d+2k+1)/2}(Ω) without first extending f to B^d; the representation formula defines an extension implicitly, but this should be stated explicitly to avoid confusion.","section":"Section 5.2, proof of Theorem 2.3"},{"comment":"References [83] and [84] are identical (both cite the same paper by Siegel, Hong, Jin, Hao, and Xu in Journal of Computational Physics, 484:112084, 2023) and should be de-duplicated.","section":"References"},{"comment":"In the definition of the normalization map P, the domain is written as S^{d-1} × [1,1] but should be S^{d-1} × [-1,1]; the same display also contains a minor typo in the final bracket.","section":"Corollary 2.1"},{"comment":"The word 'follwoing' in the sentence 'Standard concentration inequalities applied to the integral representation (1.16) yield the follwoing theorem' should be 'following'.","section":"Section 6"},{"comment":"In the proof, the set I_{-1,j} is defined as {i : ρ(θ*_i, -θ*_j) < h̄}, but later the notation I_{-1,j,-} appears without definition; this is a typographical slip.","section":"Lemma 3.4(c)"},{"comment":"The phrase 'Some calculus estimation yields' in the proof of the sign of the polygamma sums is vague; adding a short justification or a reference would improve reproducibility.","section":"Lemma 3.1"}],"recommendation":"major_revision","confidential_remarks":"The paper is likely correct in its main claims, and the parity-preserving extension issue raised by the stress-test is more benign than suggested: G and -G are disjoint caps, so a parity-preserving extension can be constructed by first extending T_k f_E to -G via the parity relation and then using a standard bounded extension operator with a smooth cutoff. Nevertheless, the manuscript must supply this proof, since the current assertion is load-bearing. The compactness step in Theorem 2.4 is also standard but needs to be written out. Both gaps are fixable within the manuscript's scope, so I recommend major revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper is worth taking seriously. The central claim — that linearized ReLU^k networks with fixed, well-distributed parameters match the optimal Sobolev approximation rate n^{-1/2-(2k+1)/(2d)} — is plausible and, as far as I can see from the derivations, likely correct. What is genuinely new is the finite-neuron-space theorem with general well-distributed weights and the explicit coefficient bound sqrt(n)||a||_2 <= C||f||; the integral representation is a repackaging of known harmonic analysis but is cleanly tied to the RKHS viewpoint, and Corollary 2.1 correctly places Petrushev's theorem as a special case. The heavy estimates in Theorem 4.1 — the Legendre coefficient decay and the quadrature-based construction — are careful and consistent with the published lemmas they cite. That is real work and I would not desk-reject it.\n\nThe soft spots are real but repairable. The biggest gap is the parity-preserving Sobolev extension from the cap G to the sphere in the proofs of Theorems 2.2 and 2.3. The text simply says “by the extension theorems again” without giving the construction. The stress-test note claims this is impossible because symmetrization would fail on the equator; that specific objection is wrong, because for eta in G the antipodal point -eta is not in G, so one can define the extension on -G by parity and fill the equatorial band symmetrically. The extension exists, but it is not trivial for fractional r on Lipschitz domains, and the paper owes the reader an explicit argument. The converse direction of Theorem 2.4 also relies on a sketched compactness limit of piecewise-constant densities; it is fixable but needs detail. The extraction of a quasi-uniform subset in Theorem 4.1 is a standard packing argument, though it should be stated. None of these looks like a load-bearing contradiction; they are places where a referee should demand more writing, not reasons to doubt the result.\n\nThis paper is for approximation theorists and people working on random features and linearized neural networks. It deserves a serious referee: the main theorem is important if true, the proofs are mostly solid, and the gaps are fillable. I would send it to peer review with a request for major revision that forces the authors to write out the parity extension and the compactness argument in full.\n\nMy own view is more optimistic than the skeptic's: the core result stands, and the stress-test note's main attack misfires. Give it a real referee.","headline":"The main approximation-rate theorem is believable and likely correct; the alleged parity-extension contradiction in the stress-test note is misfired, but the proof still needs to spell out the extension and compactness arguments.","tokens_in":33754,"tokens_out":6307,"would_cite":true,"duration_ms":68726,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["41A25","41A30","46E35","68T07"],"pacs":[],"model":"deepseek-v4-flash","headline":"A fixed-parameter linearized ReLU^k network attains the same optimal approximation rate in Sobolev spaces as a fully trained nonlinear shallow network.","keywords":["ReLU^k activation","shallow neural networks","Sobolev spaces","integral representation","finite neuron space","approximation rates","random features","spherical harmonics"],"falsifier":"Test the extension step numerically in the simplest case $d=1$, $k=0$, $r=1$: for a family of smooth functions on $\\Omega=(-1,1)$, compute $g$ as the odd extension of $T_0f_E$ from the cap and measure $\\|g\\|_{H^1(S^1)}/\\|f\\|_{H^1(\\Omega)}$; if this ratio is unbounded as $f$ varies, Theorem 2.2 fails at $k=0$. A second check for the converse direction of Theorem 2.4 is to run the least-squares fit on a fixed quasi-uniform family and test whether the normalized coefficient vectors produce piecewise-constant densities with a convergent subsequence in $L^2(S^d)$; failure of compactness would break the characterization.","tokens_in":32399,"feed_emoji":"🧠","tokens_out":11952,"duration_ms":109972,"temperature":0.7,"pith_summary":"The paper establishes that every function in $H^{(d+2k+1)/2}(\\Omega)$ on a bounded Lipschitz domain $\\Omega\\subset\\mathbb{R}^d$ can be written as an $L^2$-weighted integral of ReLU$^k$ ridge functions $\\sigma_k(\\theta\\cdot\\tilde x)$ over the parameter sphere $S^d$, with the Sobolev norm equivalent to the least $L^2$ norm of the weight. It then proves that a shallow network with inner parameters fixed in advance---only the output coefficients are trained, turning the problem into convex least squares---approximates every such function at rate $O(n^{-1/2-(2k+1)/(2d)})$ in $L^2$. This matches, up to logarithms, the best known rate for fully nonlinear ReLU$^k$ networks that also optimize the inner parameters. A coefficient bound $\\sqrt{n}\\|a\\|_2\\lesssim\\|f\\|_{H^{(d+2k+1)/2}(\\Omega)}$ accompanies the rate and connects the approximation result to generalization. If correct, the paper shows that for Sobolev regularity the nonlinearity of shallow networks is not needed for optimal approximation: well-distributed fixed features suffice.","feed_headline":"Linearized ReLU^k networks hit optimal Sobolev rates","feed_subtitle":"One integral representation turns fixed-parameter least squares into best-possible rates.","key_machinery":"The machine is the spherical harmonic expansion of the truncated power function $\\sigma_k(t)=\\max(t,0)^k$. Its Legendre coefficients $\\hat\\sigma_k(m)$ vanish except when $m\\in\\{0,\\ldots,k\\}$ or $m\\ge k+1$ with $m-k$ odd, and satisfy $m^{d+2k+1}\\hat\\sigma_k(m)^2\\simeq1$; this makes the integral operator $\\psi\\mapsto\\int_{S^d}\\sigma_k(\\theta\\cdot\\eta)\\psi(\\theta)\\,d\\theta$ an isomorphism between the even/odd $L^2$ space and the even/odd Sobolev space $H^{(d+2k+1)/2}(S^d)$. To move from the sphere to the ball, the proof uses the pair of maps $S_k g(x)=|\\tilde x|^k g(\\tilde x/|\\tilde x|)$ on the cap $G=\\{\\eta\\in S^d:\\eta_{d+1}\\ge1/\\sqrt2\\}$ and its inverse $T_k$, which send ReLU$^k$ ridge functions to ReLU$^k$ ridge functions and are norm equivalences for $H^r$. Finite approximation on the sphere uses a scattered-point quadrature rule with weights $\\tau_j\\lesssim h^d$ to match all spherical-harmonic coefficients of degree up to $J\\simeq h^{-1}$, and a summation-by-parts estimate on Cesàro sums of Legendre matrices to control the high-frequency tail. The tail control, combined with the coefficient estimate $\\|a\\|_2^2\\lesssim h^{2r-2k-1}\\|f\\|_{H^r}^2$, produces the final rates.","core_discovery":"At the center is a characterization theorem: $H^{(d+2k+1)/2}(\\Omega)$ coincides with the set of functions $\\int_{S^d}\\sigma_k(\\theta\\cdot\\tilde x)\\psi(\\theta)\\,d\\theta$ with $\\psi\\in L^2(S^d)$, and the two natural norms are equivalent. The same machinery yields approximation by the finite neuron space $L^k_{n,M}$ spanned by $\\sigma_k(\\theta_j^*\\cdot\\tilde x)$ for a well-distributed point set: for $f\\in H^{(d+2k+1)/2}(\\Omega)$ there are coefficients $a_j$ with $\\sqrt{n}\\|a\\|_2\\lesssim\\|f\\|$ such that $\\|f-\\sum_j a_j\\sigma_k(\\theta_j^*\\cdot\\tilde x)\\|_{L^2(\\Omega)} \\lesssim n^{-1/2-(2k+1)/(2d)}\\|f\\|$. More generally, $H^r$-to-$H^s$ approximation holds at rate $h^{r-s}$ for mesh size $h$, hence $n^{-(r-s)/d}$ for well-distributed points. The paper further proves the converse: a function is approximable by such fixed-parameter linear spaces, with uniformly bounded coefficients and quasi-uniform parameters, if and only if it lies in $H^{(d+2k+1)/2}(\\Omega)$. Consequently the Sobolev space is the RKHS of the kernel $\\int_{S^d}\\sigma_k(\\theta\\cdot x)\\sigma_k(\\theta\\cdot y)\\,d\\theta$, random feature sampling reaches the same rate up to a log factor, and an $O(m^{-1/2})$ generalization bound in $H^1$ follows for an elliptic PDE loss.","pith_inferences":["Because the proof relies only on the Legendre-coefficient decay of $\\sigma_k$, the same linearized construction should extend to other ridge activations with comparable coefficient decay, with the exponent adjusted accordingly.","Since $H^{(d+2k+1)/2}(\\Omega)$ is shown to be the RKHS of the ReLU$^k$ kernel, standard kernel-based generalization bounds apply without further work; the paper does not exploit this connection.","The comparison with finite elements suggests a direct numerical benchmark: for a smooth high-dimensional target, plotting least-squares error against $n$ for fixed-parameter ReLU$^k$ spaces versus degree-$k$ FEM should show the predicted polynomial gap.","The contrast with Barron spaces implies that functions in $B_k(\\Omega)\\setminus H^{(d+2k+1)/2}(\\Omega)$ are essentially not approximable at the same rate by fixed-parameter linear spaces; constructing such a function explicitly would mark the boundary between linear and nonlinear approximation."],"forward_implications":["For every $f\\in H^{(d+2k+1)/2}(\\Omega)$ and any well-distributed parameter mesh, the finite neuron space reaches $O(n^{-1/2-(2k+1)/(2d)})$ in $L^2$, the same rate known for fully nonlinear shallow networks up to logarithmic factors.","Because only output coefficients are trained, the optimal rate is achieved by solving a convex least-squares problem rather than a nonconvex parameter optimization.","Random i.i.d. parameters on $S^d$ recover the same rate up to a log factor with probability at least $1-\\delta$, which is why random feature methods work; deterministic well-distributed points remove the log factor and the failure probability.","The coefficient bound $\\sqrt{n}\\|a\\|_2\\lesssim\\|f\\|$ yields an expected $H^1$ generalization error of $O(m^{-1/2})$ for the empirical risk minimizer of an elliptic PDE loss.","The characterization also implies $H^{(d+2k+1)/2}(\\Omega)\\hookrightarrow B_k(\\Omega)$ and that the unit balls of the two spaces have metric entropies of the same order."],"supporting_citations":[{"why":"Supplies the sharp Barron-space rate $n^{-1/2-(2k+1)/(2d)}$ and metric entropy that the linearized result is compared against.","marker":"[87]"},{"why":"Provides the measure-based integral representation of Barron spaces and the variation-space characterization used as contrast.","marker":"[88]"},{"why":"Introduces the $S_k/T_k$ lifting operator between the sphere cap and the ball that carries spherical approximation to the domain.","marker":"[2]"},{"why":"Establishes the embedding $H^{(d+2k+1)/2}(\\Omega)\\subset B_k(\\Omega)$ that Theorem 2.2 recovers as a corollary.","marker":"[58]"},{"why":"Gives the earlier tensor-product construction of optimal ridge-function approximation whose ReLU$^k$ case follows from the present theorem.","marker":"[73]"},{"why":"Provides the positive quadrature rule on scattered spherical points used to match low-degree spherical-harmonic coefficients.","marker":"[64]"},{"why":"Supplies the Cesàro summation estimates for Jacobi and Legendre kernels that localize the tail error in the spherical approximation proof.","marker":"[14]"},{"why":"Provides the spherical-harmonic and Legendre polynomial framework used throughout the proof.","marker":"[20]"},{"why":"The classical Sobolev extension theorem used to move from the domain to the ball and then to the sphere.","marker":"[89]"}],"fun_headline_variants":["ReLU^k networks match optimal Sobolev approximation rates","Integral representation pins down Sobolev spaces for ReLU^k nets","Fixed-parameter ReLU^k linear nets achieve optimal rates","Sobolev spaces equal ReLU^k ridge integrals, yielding optimal rates","Optimal linearized ReLU^k approximation in Sobolev spaces"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof of Theorem 2.2 assumes an unproved extension step: after lifting the Sobolev function to the cap $G$ of the sphere, a function $g$ must exist on the whole sphere with the parity $g(\\eta)=(-1)^{k+1}g(-\\eta)$ and with $\\|g\\|_{H^r(S^d)}\\le C\\|f\\|_{H^r(\\Omega)}$; if such an extension fails for some Lipschitz domain or fractional $r$, the main approximation theorem collapses.","fun_headline_variants_meta":{"raw":{"variants":["ReLU^k networks match optimal Sobolev approximation rates","Integral representation pins down Sobolev spaces for ReLU^k nets","Fixed-parameter ReLU^k linear nets achieve optimal rates","Sobolev spaces equal ReLU^k ridge integrals, yielding optimal rates","Optimal linearized ReLU^k approximation in Sobolev spaces"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00053,"raw_usage":{"total_tokens":2605,"prompt_tokens":1046,"completion_tokens":1559,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":662,"completion_tokens_details":{"reasoning_tokens":1469}},"tokens_in":662,"tokens_out":1559,"duration_ms":10875,"temperature":1.0,"reasoning_tokens":1469,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T04:47:41.030061+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Test the extension step numerically in the simplest case $d=1$, $k=0$, $r=1$: for a family of smooth functions on $\\Omega=(-1,1)$, compute $g$ as the odd extension of $T_0f_E$ from the cap and measure $\\|g\\|_{H^1(S^1)}/\\|f\\|_{H^1(\\Omega)}$; if this ratio is unbounded as $f$ varies, Theorem 2.2 fails at $k=0$. A second check for the converse direction of Theorem 2.4 is to run the least-squares fit on a fixed quasi-uniform family and test whether the normalized coefficient vectors produce piecewise-constant densities with a convergent subsequence in $L^2(S^d)$; failure of compactness would break the characterization.","supporting_citations":[{"cited_title":"Sharp bounds on the approx imation rates, metric entropy, and n-widths of shallow neural networks","cited_arxiv_id":null,"evidence_quote":"Supplies the sharp Barron-space rate $n^{-1/2-(2k+1)/(2d)}$ and metric entropy that the linearized result is compared against."},{"cited_title":"Characterization of the var iation spaces corresponding to shallow neural networks","cited_arxiv_id":null,"evidence_quote":"Provides the measure-based integral representation of Barron spaces and the variation-space characterization used as contrast."},{"cited_title":"Approximation by ridge functions and neu ral networks","cited_arxiv_id":null,"evidence_quote":"Gives the earlier tensor-product construction of optimal ridge-function approximation whose ReLU$^k$ case follows from the present theorem."},{"cited_title":"Spherical marcinkiewicz-z ygmund inequalities and positive quadrature","cited_arxiv_id":null,"evidence_quote":"Provides the positive quadrature rule on scattered spherical points used to match low-degree spherical-harmonic coefficients."},{"cited_title":"Approximation theory and harmonic analysis on spheres and b alls","cited_arxiv_id":null,"evidence_quote":"Provides the spherical-harmonic and Legendre polynomial framework used throughout the proof."}],"review_version":1}