{"id":"a3e5d11f-d037-4059-bb53-95a4ee11bd92","arxiv_id":"2509.19830","paper_version":3,"verdict":"REJECT","confidence":"HIGH","novelty_score":5.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper claims spline-parameterized KAN least-squares estimators achieve the minimax univariate regression rate O(n^{-2r/(2r+1)}) for additive and multiplicative KAN structures, independent of dimension.","lead":"An analysis claims that spline-based Kolmogorov-Arnold Networks fit nonparametric regressions at the minimax-optimal rate n^{-2r/(2r+1)} when each univariate piece is r-smooth, with the rate independent of input dimension. If true, this would put KANs on the same theoretical footing as classical spline and additive-model estimators, but the proof has gaps around the nonlinear sieve and component identification.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1's log-free rate rests on an unproved O(p_n/n) estimation-error bound; standard entropy-based sieve bounds give a log factor.","rationale":"The reader's weakest assumption correctly identifies the estimation-error step in Appendix A.1 as the load-bearing premise: the paper moves from a metric-entropy bound to a log-free O(p_n/n) least-squares risk for a nonconvex nonlinear sieve. My independent reading confirms that this step is not justified by the cited standard empirical-process results, and the paper's own metadata abstract contradicts the body's no-log rate. The lower-bound part of Corollary 1 is fine, but the upper-bound proof does not establish the claimed log-free minimax optimality. The concrete test—applying the standard entropy-based bound and balancing—would either validate the concern or show that one can invoke a stronger uniform-entropy/bracketing condition that the paper never states. Since the reader's REJECT verdict is aligned with this analysis, I recommend no change to the verdict.","tokens_in":14495,"tokens_out":3406,"duration_ms":26542,"concrete_test":"Re-derive Step 2 using the standard empirical-process bound for sieve least squares: for log N(eps, F_n, ||.||_inf) <= C p_n log(1/eps), the L2 estimation error is at most C (p_n log n)/n (Gyorfi et al. 2002, Theorem 9.2; Chen 2007, Thm 4.1). Substitute p_n = k_n, balance k_n^{-2r} + k_n log n / n, and compute the resulting exponent. If the resulting rate is O((log n/n)^{2r/(2r+1)}), then Theorem 1's stated O(n^{-2r/(2r+1)}) and Corollary 1's '≍' fail as written. Also verify that the manuscript's own metadata abstract, which already says O((log n/n)^{2r/(2r+1)}) and 'minimax optimal up to the logarithmic factor', is inconsistent with the body's log-free claim—this settles whether the no-log rate is actually proved.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim, Theorem 1 and Corollary 1, is the log-free rate O(n^{-2r/(2r+1)}) for the spline KAN sieve estimator. The proof's Step 2 (Appendix A.1, eqs. 39-41) asserts that from metric entropy log N(eps, F_n, ||.||_inf) <= C p_n log(1/eps) it follows that E||fhat - f*_n||^2 = O(p_n/n). This is not a valid empirical-process inference for a nonconvex, nonlinear sieve. For a class satisfying only this entropy bound, the standard bound (e.g., Gyorfi et al. Theorem 9.2; van der Vaart/Wellner entropy integral) is O(p_n log n / n). The log factor is material: balancing k_n^{-2r} + (k_n log n)/n gives k_n ~ (n/log n)^{1/(2r+1)} and rate (log n/n)^{2r/(2r+1)}, not n^{-2r/(2r+1)}. This is exactly the discrepancy between the paper's metadata abstract (which states the rate with log and says the lower bound matches 'up to a logarithmic factor') and the body abstract/Theorem 1 (which omit the log). The claim that O(p_n/n) holds for this nonconvex class would require bracketing entropy or VC-type conditions with a dimension-independent uniform entropy integral, which is not supplied. A secondary issue is that Step 1 (eq. 35) bounds the composition error only through the Lipschitz constant of g_q while ignoring the spline approximation error of g_q itself; this also needs a sup-norm or L_infty bound on g_q - g_q,k_n, which the L2 spline approximation does not provide in the stated form. The lower bound in Corollary 1 is standard, but the upper bound is not established without the log-free estimation error.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper analyzes least-squares regression over Kolmogorov-Arnold networks whose univariate components are B-splines. It claims that for target functions admitting an additive or multiplicative KAN representation with univariate components in W^r([0,1]), the spline-based KAN sieve estimator achieves squared L2 risk O(n^{-2r/(2r+1)}), that this rate is minimax optimal over the KAN class, that k_n ≍ n^{1/(2r+1)} is the optimal knot scaling, and that the univariate components can be estimated at the same rate. The lower bound is obtained by embedding the univariate Sobolev class into the KAN class. The upper bound is decomposed into an approximation error step and an estimation error step. The paper also contains simulation evidence and a discussion of non-identifiability of the KAN representation.","tokens_in":14986,"tokens_out":7191,"duration_ms":60681,"significance":"If the main theorems were established, the result would be significant: it would show that spline-based KANs achieve a dimension-free minimax rate for a structured compositional class, in contrast to the usual n^{-2r/(2r+d)} rate for general Sobolev classes. The lower bound and the non-identifiability discussion are useful, and the paper provides reproducible code and simulations that directly check the predicted knot scaling. However, the central upper-bound proof contains a load-bearing step that is not justified, and the claims in the body contradict the paper's own metadata abstract. The claimed log-free minimax optimality is therefore not established as written.","major_comments":[{"comment":"The proof asserts that the metric entropy bound log N(ε, F_n, ||·||_∞) ≤ C p_n log(1/ε) implies E||f̂_n - f*_n||² = O(p_n/n). This does not follow from the cited 'standard empirical-process argument' for a nonconvex nonlinear sieve. For classes satisfying only this entropy bound, the standard least-squares bound is O(p_n log n / n); the log factor is material because balancing k^{-2r} + k log n / n gives k ≍ (n/log n)^{1/(2r+1)} and rate (log n/n)^{2r/(2r+1)}, not n^{-2r/(2r+1)}. Indeed, the metadata abstract states exactly this log-rate and says the lower bound matches 'up to a logarithmic factor,' while Theorem 1 and Corollary 1 claim the log-free rate. A valid proof of O(p_n/n) for this nonconvex class requires additional structure, such as VC-type or localized Rademacher conditions, which is not supplied.","section":"Appendix A.1, Step 2, Eqs. (39)-(41)"},{"comment":"The approximation step is missing the approximation error of the outer spline g_{q,k_n}. In Theorem 1, with f_{0,q,k_n} = g_{q,k_n}(Σ_j ψ_{qj,k_n}(x_j)), the inequality ||f_{0,q,k_n} - f_{0,q}|| ≤ L_q ||Σ_j(ψ_{qj,k_n}-ψ_{qj})|| omits the term involving g_{q,k_n} - g_q. One cannot conclude the displayed bound from L2 spline approximation of g_q alone; controlling g_{q,k_n}(S_{k_n}) - g_q(S_{k_n}) requires a sup-norm or range/density condition. The same issue appears in the Theorem 2 proof, where the products also require sup-norm control. This affects the approximation error term in both theorems.","section":"Appendix A.1, Eq. (35); Theorem 2 proof, Eqs. (47)-(50)"},{"comment":"The corollary claims E||ψ̂_{qj} - ψ_{qj}||²_{L2} = O(n^{-2r/(2r+1)}) for each univariate spline unit. This is not established by the preceding proofs. Proposition 1 shows that, under the stated centering conditions, the representation is identifiable only up to permutation and constant shifts only after additional normalizations, and Remark 2 explicitly says consistency of the fit does not imply consistency of the components. The proof of Corollary 2 simply asserts that components inherit the overall rate, without addressing the non-identifiability. This claim should either be proved under explicit identifiability constraints or removed/weakened.","section":"Corollary 2, Eq. (70)"},{"comment":"The upper bound in Corollary 1 is justified by saying F_KAN^r is contained in the union of the additive subclass of Theorem 1 and the hybrid subclass of Theorem 2. But F_KAN^r as defined in Eq. (21) allows each node T_q to be either additive or multiplicative, including mixed architectures. Theorem 1 handles all-additive targets and Theorem 2 handles all-multiplicative targets; neither proof covers a target with both additive and multiplicative nodes. Thus the upper bound over the full class is not established as stated. The lower bound via the univariate subclass is fine.","section":"Corollary 1, Eq. (21) and Eq. (60)"}],"minor_comments":[{"comment":"The body abstract and Theorem 1 state the log-free rate O(n^{-2r/(2r+1)}), while the arXiv metadata abstract states O((log n/n)^{2r/(2r+1)}) and describes the lower bound as matching only up to a logarithmic factor. These are materially different claims; the paper should state one consistent rate.","section":"Abstract"},{"comment":"The lower-bound proof cites 'Tsybakov, 2009' but the bibliography lists Tsybakov (2008). Please harmonize.","section":"References"},{"comment":"The proof uses p_n and k_n interchangeably; while p_n ≍ k_n is stated, the notation would be clearer if the dimension of F_n were tracked consistently throughout the appendix.","section":"Notation"},{"comment":"The simulation figure reports empirical slopes but no confidence intervals or repeated-trial variability; adding error bars or multiple seeds would strengthen the empirical claim, though this is not central to the theoretical result.","section":"Figure 1"}],"recommendation":"reject","confidential_remarks":"The gap in Step 2 is not a local omission: the claimed log-free rate is contradicted by the paper's own metadata abstract, and the standard entropy-based bound introduces a log factor that changes the stated theorem. Even if the approximation step were patched, the upper bound would require a substantially different empirical-process argument. I would be willing to reconsider if the authors provide a rigorous proof of O(p_n/n) under explicit conditions, or if the theorems are revised to the log rate."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this paper has a real question and a clean lower bound, but the main upper-bound theorem is not proved. The claimed log-free rate n^{-2r/(2r+1)} depends on asserting O(p_n/n) estimation error from metric entropy log N(eps,F_n,sup) <= C p_n log(1/eps). For a nonconvex, nonlinear sieve that inference does not follow; standard results give O(p_n log n / n). Balance then gives (log n / n)^{2r/(2r+1)}, not n^{-2r/(2r+1)}. The paper even contains the discrepancy: the metadata abstract states the log-factor rate and says the lower bound matches up to a log factor, while the body abstract and Theorem 1 drop the log.\n\nWhat is good: the minimax lower bound via the one-coordinate subclass is standard and clean. The non-identifiability proposition is honest and correctly notes that centering alone doesn't identify components. The simulation setup chooses target functions with exactly known Sobolev smoothness, which is a nice touch, and code and data are released. The knot-selection heuristic k_n ~ n^{1/(2r+1)} is the right classical balancing if the estimation bound were log-free.\n\nSoft spots beyond the main one: the approximation-error step transfers L2 spline errors through the outer Lipschitz map without a sup-norm or distributional condition; L2 control on psi_qj - psi_qj,kn does not control the sup norm of the sum, so the composition error bound in eq. (35) is not justified. Corollary 2's per-component rate clashes with the paper's own Proposition 1: without identifiability, the estimator's components are not well-defined, so E||psi_hat_qj - psi_qj||^2 has no clear target. This is a real inconsistency, not a quibble.\n\nWho is this for? People who want a KAN regression theory will find the lower bound and the structural setup useful. But the central result as stated is not reliable. A serious referee could sort this out, because the issue is localized to the estimation-error step, and the paper is salvageable as a log-rate result with a correct proof. I'd recommend sending it to peer review with a request to fix the estimation-error argument, reconcile the abstracts, and either prove or drop the per-component claim.","headline":"The claimed log-free minimax rate for spline KAN sieves is not established—the upper bound rests on an invalid estimation-error step—but the lower bound and the non-identifiability discussion are decent.","tokens_in":15391,"tokens_out":2706,"would_cite":false,"duration_ms":22260,"reading_group":"maybe","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["62G08","62G20","41A15"],"pacs":[],"model":"deepseek-v4-flash","headline":"Spline-based Kolmogorov–Arnold Networks are claimed to attain the dimension-free minimax rate n^{-2r/(2r+1)} for regression targets whose univariate components have Sobolev smoothness r.","keywords":["Kolmogorov-Arnold networks","nonparametric regression","B-splines","sieve estimation","minimax optimality","Sobolev smoothness","dimension-free rate","identifiability"],"falsifier":"Fit the spline KAN sieve for the univariate target h(x)=x^r with k ≍ n^{1/(2r+1)} knots and measure E||fhat - h||^2_{L2} over n = 10^4 to 10^6. If the squared error is not consistently bounded by C n^{-2r/(2r+1)} but instead tracks C (log n / n)^{2r/(2r+1)}, Theorem 1's no-log claim is false. A direct proof check: verify whether the inequality (39)–(41) can be derived without a log factor; standard chaining arguments for nonconvex sieves introduce one.","tokens_in":14376,"feed_emoji":"📈","tokens_out":6180,"duration_ms":46857,"temperature":0.7,"pith_summary":"This paper tries to prove that Kolmogorov–Arnold Networks with univariate B-spline components, fit by least squares over a growing sieve of spline spaces, converge to the true regression function at the minimax-optimal rate O(n^{-2r/(2r+1)}) whenever the target admits an additive or hybrid additive–multiplicative KAN representation with components of Sobolev smoothness r. If true, this gives KANs a theoretical footing as structured, interpretable alternatives to black-box neural nets, with the rate free of the ambient dimension. The paper also derives a knot-selection rule k ~ n^{1/(2r+1)}, proves a matching lower bound over the KAN-representable class, and shows the univariate components are identifiable only up to constant shifts and permutations. The preprint's metadata abstract states the upper rate as O((log n / n)^{2r/(2r+1)}), attributing the log factor to the nonlinear sieve, while the main theorems state the log-free version.","feed_headline":"Spline KANs converge at the minimax rate, free of dimension","feed_subtitle":"If the proof step holds, spline KANs match optimal rates without the curse of dimensionality.","key_machinery":"The engine is the spline-based KAN sieve estimator: each univariate component is a B-spline with k interior knots, so the whole network is a finite-dimensional function class F_n of dimension p_n ≍ k. The proof combines univariate spline approximation theory (which gives O(k^{-2r}) bias through Lipschitz composition) with a sieve least-squares empirical-process bound that gives O(p_n/n) estimation error from the metric entropy bound log N(ε,F_n,‖·‖∞) ≲ p_n log(1/ε). Balancing the two terms at k ≍ n^{1/(2r+1)} yields the rate.","core_discovery":"The central claim is that the spline-based KAN sieve least-squares estimator attains E||fhat - f0||^2_{L2([0,1]^d)} = O(n^{-2r/(2r+1)}) uniformly over a ball of KAN-representable functions with univariate Sobolev smoothness r, for both additive and hybrid multiplicative architectures. The proof splits the risk into a bias term from spline approximation, O(k^{-2r}), and a variance term from the p_n ~ k-dimensional empirical risk minimizer, O(p_n/n), then balances k ~ n^{1/(2r+1)}. Corollary 1 packages the upper bound with a univariate lower bound to assert minimax optimality over the KAN class; Corollary 2 turns the balance into a concrete knot-count rule.","pith_inferences":["If the log-free estimation-error step fails, the true upper rate is likely (log n / n)^{2r/(2r+1)}; then the minimax claim holds only up to log, which matters for theory but is often invisible in practice.","The dimension-free exponent is not a free lunch: it rests on the KAN structure assumption; on a full Sobolev class over [0,1]^d the usual n^{-2r/(2r+d)} curse still applies.","The simulations' 'steeper than theoretical' slopes are consistent with finite-sample constants hiding a log factor, so they do not discriminate between the two rate statements.","One testable byproduct: for univariate targets in the class, the spline sieve should achieve exactly n^{-2r/(2r+1)} squared error; measuring this directly would settle whether the no-log bound holds."],"forward_implications":["KAN sieve estimators are minimax optimal over the additive and hybrid KAN classes at the univariate Sobolev rate, so dimension does not enter the exponent.","Practitioners get a concrete knot budget: k_n ≍ n^{1/(2r+1)}.","Hybrid multiplicative nodes retain the same rate up to a constant overhead that grows with dimension, so expressiveness need not degrade convergence order.","Fitted KANs can be consistent even when their internal univariate components are not; interpretation requires centering constraints or permutation fixes.","The guarantees apply to the empirical risk minimizer, independent of optimizer noise; stochastic or gradient-based training dynamics are outside the paper's scope."],"fun_headline_variants":["Spline KAN estimators: dimension-free minimax rate","KAN spline sieve matches lower bound up to log","Adaptive knot selection for spline KANs at optimal rate","Proof: spline KANs achieve near-optimal convergence","Dimension-free exponent for KAN regression rates"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The proof assumes that the p_n-dimensional KAN sieve is well-behaved enough that its least-squares risk is O(p_n/n) with no log n factor; if the correct empirical-process bound is O(p_n log n / n), the log-free rate collapses to the (log n / n)^{2r/(2r+1)} stated in the abstract.","fun_headline_variants_meta":{"raw":{"variants":["Spline KAN estimators: dimension-free minimax rate","KAN spline sieve matches lower bound up to log","Adaptive knot selection for spline KANs at optimal rate","Proof: spline KANs achieve near-optimal convergence","Dimension-free exponent for KAN regression rates"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000329,"raw_usage":{"total_tokens":1722,"prompt_tokens":846,"completion_tokens":876,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":590,"completion_tokens_details":{"reasoning_tokens":795}},"tokens_in":590,"tokens_out":876,"duration_ms":18254,"temperature":1.0,"reasoning_tokens":795,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T15:20:01.072854+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Fit the spline KAN sieve for the univariate target h(x)=x^r with k ≍ n^{1/(2r+1)} knots and measure E||fhat - h||^2_{L2} over n = 10^4 to 10^6. If the squared error is not consistently bounded by C n^{-2r/(2r+1)} but instead tracks C (log n / n)^{2r/(2r+1)}, Theorem 1's no-log claim is false. A direct proof check: verify whether the inequality (39)–(41) can be derived without a log factor; standard chaining arguments for nonconvex sieves introduce one.","supporting_citations":[],"review_version":1}