{"id":"0fdbd2ce-9cf9-4b71-8972-425861d6e9f1","arxiv_id":"2507.11944","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"A finite set of data-adaptive basis functions is proven to represent the least-squares, Tikhonov, and conjugate-gradient estimators for learning convolution kernels, removing manual reproducing-kernel selection.","lead":"This paper builds a data-defined 'lens' so that learning a convolution kernel from noisy input-output pairs no longer requires hand-picking a Gaussian or Matérn kernel and tuning its bandwidth. The authors prove that the resulting estimators live in a finite set of automatic basis functions, then provide scalable iterative algorithms and test them on three operator-learning problems.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 2.5 is internally sound for semi-continuum data, but the practical claim for discrete observations depends on an unquantified piecewise-constant approximation; the computed estimator need not minimize (2.6) over H_G.","rationale":"I verified the core algebra of Theorem 2.5 and its proof: the representation φ=ξ+Σ c_kj ξ_kj, the identities xξ_kj,φy_HG=Rφ[u_k](x_j), and the coefficient normal equations (2.9)-(2.10) are correct. I therefore do not see a flaw in the central representer theorem itself. The reader's weakest-assumption identification, exploration identifiability, is a genuine limitation but it is explicitly stated in the paper and is inherent to any inverse problem: components of the kernel in N(L_G) cannot be recovered. That limitation does not undermine the finite-representation claim. The more load-bearing issue is the unquantified discretization in Section 3: the theorem applies to continuum functions g[u_k](x_j,·), while the implemented algorithm uses piecewise-constant reconstructions ξD_kj that are not shown to lie in H_G and are not connected to the theorem by any error estimate. This gap separates the mathematical headline from the practical algorithms and numerical demonstrations, and it supports the reader's CONDITIONAL verdict. I recommend keeping the verdict unchanged pending a convergence check of the discrete approximation against the exact semi-continuum estimator.","tokens_in":23372,"tokens_out":25061,"duration_ms":300751,"concrete_test":"On Example 1.1 with a known input u_k(t)=Σ_{n=2}^{n_u} X_n cos(2πnt), compute a high-accuracy reference solution: form exact ξ_kj(s) and Σ by adaptive quadrature in s, solve (2.10), and record the estimator. Then run Algorithm 1/Tikhonov with piecewise-constant g on uniform s-meshes with ns=J,2J,4J,8J. If the relative L2 error between the discrete estimator and the exact semi-continuum estimator does not decrease at a first-order-in-1/ns rate (or at least by a consistent factor), the Section 3 approximation is not controlled and the practical implementation is not connected to Theorem 2.5.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 2.5 is verified: for the idealized semi-continuum data {g[u_k](x_j,·)}, the minimizers of (2.6) over H_G live in span{ξ_kj} and the coefficient formulas (2.9)-(2.10) are correct. The load-bearing gap is the transition to the actual discrete-data setting of (1.2). The theorem requires the functions s ↦ g[u_k](x_j,s) on the whole interval; the data only provide values at mesh points. Section 3 replaces g by piecewise constants (3.1), producing discrete basis functions ξD_kj that are not, generically, elements of H_G. The computed estimator (3.4) is therefore not proved to be the minimizer of (2.6), and no discretization error bound is supplied. Thus the advertised reduction of the inverse problem to a finite matrix problem, and the numerical conclusions of Section 5, are justified only by the numerical experiments, not by Theorem 2.5. If the quadrature error is not controlled, a finer s-mesh could change the estimator in ways the theory does not predict; the 'mesh-free' and 'automatic' claims are stronger than what is proved.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a data-adaptive reproducing kernel Hilbert space (DA-RKHS) for learning convolution kernels in linear-in-kernel operators from input-output data. An exploration measure and an automatic kernel are constructed from the input features and the forward operator, and a finite-dimensional representer theorem (Theorem 2.5) is proved for semi-continuum observations, giving coefficient formulas for the minimal-norm least-squares and Tikhonov estimators. Conjugate-gradient and Golub-Kahan iterative/hybrid algorithms are developed, followed by a discrete approximation in Section 3 and numerical experiments on integral, nonlocal, and aggregation operators showing that the proposed HG-norm regularization often outperforms Gaussian-kernel and L2-rho baselines.","tokens_in":23549,"tokens_out":8830,"duration_ms":103718,"significance":"If the main claims hold, the paper offers a useful and practically motivated idea: replacing manual kernel selection by a kernel constructed from the data and forward operator, and reducing an infinite-dimensional variational problem to a finite matrix problem for several estimators. The core identities in Theorems 2.5, 2.8, and 2.9 are internally consistent, the pseudoinverse-based Tikhonov formula correctly blocks null-space noise, and the numerical study is extensive with public code. The significance is reduced, however, by the gap between the idealized semi-continuum setting of the representer theorem and the genuinely discrete data setting of the algorithms and experiments, which is not accompanied by error bounds.","major_comments":[{"comment":"Theorem 2.5 is proved under the assumption that the functions s -> g[u_k](x_j,s) are known on the whole interval S (semi-continuum data). In the actual discrete setting of (1.2), only point values are available, and Section 3 replaces these functions by piecewise constants (3.1), producing discrete basis functions that are not, in general, elements of the infinite-dimensional space H_G. The computed estimator (3.4) is therefore not shown to be the minimizer of the variational problem (2.6) over H_G, and no discretization or quadrature error bound is supplied. Consequently, the advertised reduction to a finite matrix problem, and the 'mesh-free' and 'automatic' claims in the abstract and conclusion, are stronger than what is rigorously proved. The authors should either provide a discretization error analysis (e.g., convergence of the discrete estimator to the H_G-minimizer as the s-mesh and x-mesh sizes tend to zero under suitable regularity conditions) or explicitly restate the practical claims as heuristic for the discrete-data setting.","section":"Section 3, Eqs. (3.1)-(3.4)"},{"comment":"The manuscript states that the true kernels are deliberately chosen 'close to the identifiable spaces H' for each example, making accurate estimation possible. This is an explicit limitation: the experiments live in the favorable regime where the exploration measure covers the support of the kernel and the kernel has no significant component in N(L_G). The abstract and conclusion do not carry this caveat, so a reader may overestimate the method's guarantees. Please state this identifiability restriction in the abstract or conclusion, and ideally add at least one experiment in which the kernel's support extends into a region where the exploration measure vanishes, to demonstrate the behavior of the estimator in that regime.","section":"Section 5, 'Numerical settings'"}],"minor_comments":[{"comment":"The notation 'Σ:' for the pseudoinverse is used without an explicit definition at first occurrence; please define it clearly (e.g., 'Σ: denotes the Moore-Penrose pseudoinverse').","section":"Section 2.3, Eq. (2.9)"},{"comment":"The proof heading reads 'Proof of Theorem 2.4', but the statement in the main text is Lemma 2.4; please correct the label.","section":"Appendix A, first line"},{"comment":"Several displayed formulas contain typographical artifacts such as 'p9ρDpsq' and '9ρ_Discrete'; these should be cleaned up.","section":"Section 3, Table 1 and Eq. (3.2)"},{"comment":"The caption contains the typo 'Nolocal operator'; it should read 'Nonlocal operator'.","section":"Section 5, Figure 1 caption"},{"comment":"The initialization divides by β1 = ||P_{N(Σ_D)^⊥} f||_2; if this quantity is zero (i.e., f is orthogonal to the range of Σ_D), the algorithm divides by zero. This degenerate case should be addressed or excluded explicitly.","section":"Algorithm 2, line 4"},{"comment":"The abstract refers to 'Gaussian process methods' as baselines, but the experiments compare against Tikhonov and iterative regularization using a Gaussian kernel norm, which is not the same as Gaussian process regression; please adjust the wording for accuracy.","section":"Abstract and Section 5"}],"recommendation":"major_revision","confidential_remarks":"The paper's central representer theorem is sound for the semi-continuum idealization, and the numerical studies are careful and reproducible. The main gap is the transition from that idealization to the genuinely discrete data setting of the title and algorithms: without discretization error estimates, the practical claims outrun the theorems. I would encourage the editor to request a revision that either supplies such estimates or substantially tempers the claims, and to ensure the identifiability limitation noted in Section 5 is carried into the abstract and conclusion."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this paper is worth a serious referee, and the central theorem survives contact. The genuinely new content is Theorem 2.5, the automatic-basis representer result, plus the coefficient-space CG/GKB equivalences (Theorems 2.8-2.9) and the practical discrete-data formulation. I checked the core algebra; (2.9) and (2.10) are right, and the Tikhonov form (Sigma^2 + n0 J lambda Sigma)^+ Sigma f correctly avoids the null-space contamination that plain ridge would suffer. The GKB derivations also hang together. That is real progress: learning a convolution kernel in a linear-in-kernel operator becomes a finite matrix problem with no manual choice of reproducing kernel.\n\nThe soft spots are mostly about claims versus proof. The abstract says \"obviating manual kernel selection,\" but lambda, the stopping iteration, lmax, the rank threshold, and the discrepancy constant tau are still selected; only the kernel itself is automatic. The GP baseline uses one fixed bandwidth sigma0 = 0.1, so \"consistently outperforms GP\" is a statement about one untuned baseline, not about the method class. Section 5 also says the true kernels are chosen close to the identifiable space H; that is the favorable regime, and the paper owns it.\n\nThe more serious gap is the discrete transition. Theorem 2.5 is proved for semi-continuum data {g[u_k](x_j, .)}. In Section 3, the functions are replaced by piecewise constants, and the discrete basis functions are not shown to lie in H_G. So the computed estimator (3.4) is not proved to minimize (2.6) over H_G, and there is no discretization error bound. The \"mesh-free\" claim is stronger than what is proved. This does not sink the paper—the numerical experiments are reproducible and the algebra on the discrete side is sensible—but it means the practical claim rests on numerics for now.\n\nWho is this for: anyone doing kernel learning for integral, nonlocal, or aggregation operators, or data-driven nonlocal PDE identification. The paper deserves a serious referee and probably a round of revision. I would send it out, asking for a discretization error estimate or at least an honest statement that the discrete estimator is a heuristic approximation of the Theorem 2.5 minimizer, plus a toned-down abstract.","headline":"The central representer theorem and the CG/GKB machinery are sound and genuinely useful, but the discrete-data transition and the \"no hyperparameter tuning\" claim need honest qualification before the practical conclusions can be taken at face value.","tokens_in":24261,"tokens_out":1958,"would_cite":true,"duration_ms":22467,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["65J22","45Q05","62G08"],"pacs":[],"model":"deepseek-v4-flash","headline":"Convolution kernels can be learned from discrete noisy data without manual reproducing-kernel selection, because the data and forward operator themselves induce an automatic RKHS in which all estimators reduce to finite matrix problems.","keywords":["automatic reproducing kernel","convolution kernel learning","data-adaptive RKHS","deconvolution inverse problem","finite-dimensional representer theorem","Tikhonov regularization","Golub-Kahan bidiagonalization","operator learning"],"falsifier":"For a fixed operator and data, compute the Gram matrix $\\Sigma$ by high-accuracy quadrature and compare the coefficient-form estimator $\\hat{c}_\\lambda^T\\xi$ against a direct discretized solution of the infinite-dimensional Tikhonov problem (2.6) on a very fine grid of $S$; if the two do not converge to the same function as the grid refines, the finite representer theorem would be false. Alternatively, test with a true kernel supported where $\\dot\\rho=0$ and noise-free data: the paper predicts the estimator is identically zero there, and any data-driven method that recovers the kernel in that region would require a hypothesis space larger than $H_G$.","tokens_in":22968,"feed_emoji":"📐","tokens_out":11603,"duration_ms":109967,"temperature":0.7,"pith_summary":"This paper claims that for learning a convolution kernel from discrete, noisy input–output data, the data and the forward operator themselves determine a data-adaptive reproducing kernel Hilbert space (DA-RKHS), removing the usual need to pick a Gaussian or Matérn kernel by hand. The central result is a finite-dimensional representer theorem: even though the minimization runs over an infinite-dimensional space, the minimal-norm least-squares, Tikhonov, and conjugate-gradient estimators all lie in the span of finitely many automatic basis functions built from the data. Learning the kernel therefore collapses to a finite matrix problem whose solution is given by the pseudoinverse formulas $\\hat{c}=\\Sigma^{+}f$ or $\\hat{c}_\\lambda=(\\Sigma^2+n_0J\\lambda\\Sigma)^{+}\\Sigma f$. If this is right, the main practical bottleneck of kernel methods for deconvolution-type inverse problems—kernel and hyperparameter selection—disappears. Experiments on integral, nonlocal, and aggregation operators show the automatic regularization consistently beating ridge regression and Gaussian-process methods with preselected kernels.","feed_headline":"Automatic kernel from data ends manual kernel selection","feed_subtitle":"Data-built basis functions represent least-squares and Tikhonov estimators, replacing hand-picked kernels.","key_machinery":"The central object is the automatic reproducing kernel $G(s,s')=\\bar G(s,s')/(\\dot\\rho(s)\\dot\\rho(s'))\\mathbf{1}_{\\dot\\rho(s)\\dot\\rho(s')>0}$, built from the empirical normal operator $\\bar G(s,s')$ and the exploration measure $\\dot\\rho(s)$; its RKHS is $H_G=L_G^{1/2}(L^2_\\rho)$ with closure $H=N(L_G)^\\perp$, the identifiable subspace. The finite representer theorem is the load-bearing mechanism: the functions $\\xi_{kj}(s)=\\int_S G(s,s')g[u_k](x_j,s')\\,ds'$ are representers of the bounded linear functionals $\\varphi\\mapsto R_\\varphi[u_k](x_j)$ on $H_G$, so the variational loss becomes $\\frac{1}{n_0J}\\|\\Sigma c-f\\|_2^2$ with penalty $\\lambda c^T\\Sigma c$ in coefficient space. The pseudoinverse formulas $\\hat{c}=\\Sigma^{+}f$ and $\\hat{c}_\\lambda=(\\Sigma^2+n_0J\\lambda\\Sigma)^{+}\\Sigma f$ are the resulting finite-dimensional implementations, and the Golub–Kahan bidiagonalization of $\\tilde{T}:x\\mapsto\\Sigma x$ on $N(\\Sigma)^\\perp$ yields the iterative and hybrid solvers.","core_discovery":"The paper establishes that for estimating the kernel $\\varphi$ in $R_\\varphi[u](x)=\\int_S \\varphi(s)g[u](x,s)\\,ds$ from finite noisy pairs, the variational estimator over the infinite-dimensional RKHS $H_G$ of the automatic kernel $G$ has an explicit finite-dimensional representation. The automatic kernel is $G(s,s')=\\bar G(s,s')/(\\dot\\rho(s)\\dot\\rho(s'))\\mathbf{1}_{\\{\\dot\\rho(s)\\dot\\rho(s')>0\\}}$ with $\\bar G(s,s')=\\frac{1}{n_0}\\sum_k\\int_X g[u_k](x,s)g[u_k](x,s')\\,\\nu(dx)$, weighted by the exploration measure $\\dot\\rho(s)=\\frac{1}{n_0Z}\\sum_{k,j}|g[u_k](x_j,s)|\\Delta x$. Theorem 2.5 constructs automatic basis functions $\\xi_{kj}(s)=\\int_S G(s,s')g[u_k](x_j,s')\\,ds'$ that represent the linear functionals $R_\\varphi[u_k](x_j)$ on $H_G$; the Gram matrix $\\Sigma=(\\langle\\xi_{kj},\\xi_{k'j'}\\rangle_{H_G})$ then carries the whole optimization, with coefficients $\\hat{c}=\\Sigma^{+}f$ and $\\hat{c}_\\lambda=(\\Sigma^2+n_0J\\lambda\\Sigma)^{+}\\Sigma f$. These formulas project out the null-space component of the noise, avoid the small-noise blow-up of plain ridge regression when $\\Sigma$ is singular, and reduce to the ridge estimator when $\\Sigma$ is invertible. The paper adds conjugate-gradient/Golub–Kahan iterative and hybrid algorithms and piecewise-constant discrete-data approximations, and demonstrates convergence and speed advantages in three examples.","pith_inferences":["The finite representer theorem suggests the effective dimension of the deconvolution problem is at most $n_0J$ (and no larger than the numerical rank of $\\Sigma$), so one would expect minimax rates governed by the eigenvalue decay of $L_G$; the paper does not state such rates.","The exploration measure $\\dot\\rho$ acts as an automatic adaptive quadrature weight: a practical consequence, left implicit in the paper, is that one can design input data $u_k$ to push $\\dot\\rho$ toward regions of $S$ where kernel resolution is needed.","Because the learned kernels are exactly the convolution kernels used in nonlocal and neural-operator layers, the automatic basis could serve as a feature map for operator learning, though the paper does not explore this connection.","A sharp test of the method's limits would be to apply it to a real peridynamics or particle-system dataset where the true kernel is not known to lie close to the identifiable space $H$; the numerical demonstrations use test kernels close to $H$, leaving that regime untested."],"forward_implications":["No manual kernel or bandwidth selection is needed: the data and forward operator supply both the hypothesis space and the regularizer, so Tikhonov and iterative estimators are computed directly from the Gram matrix $\\Sigma$.","When $\\Sigma$ is singular, the Tikhonov estimator $(\\Sigma^2+n_0J\\lambda\\Sigma)^{+}\\Sigma f$ stays in $N(\\Sigma)^\\perp$ and avoids contamination by $P_{N(\\Sigma)}f$, whereas ridge regression $(\\Sigma+n_0J\\lambda I)^{-1}f$ can blow up as noise vanishes.","The conjugate-gradient iterates on $H_G$ correspond to coefficient-space Krylov subspaces $\\mathcal{K}_l=\\mathrm{span}\\{\\Sigma^i\\Sigma^{+}f\\}_{i=1}^l$, so the seemingly infinite-dimensional regularization is exactly a finite-dimensional matrix problem at every iteration.","The methods are mesh-free in the sense that the automatic basis functions $\\xi_{kj}$ are used directly when the functions $g[u_k](x_j,\\cdot)$ are known analytically, and the piecewise-constant approximation in Section 3 makes them computable from raw discrete data.","Experiments on integral, nonlocal, and aggregation operators indicate the $H_G$-norm estimator converges as noise decreases and outperforms Gaussian-kernel and $L^2_\\rho$ regularizers, with the hybrid method the most stable."],"supporting_citations":[{"why":"Supplies the data-adaptive RKHS prior and the decomposition of $\\varphi_D$ into signal and noise components that motivates the automatic kernel and its null-space filtering.","marker":"[8]"},{"why":"Establishes data-adaptive RKHS Tikhonov regularization for learning kernels in operators, the immediate predecessor that this paper generalizes to finite data and iterative methods.","marker":"[36]"},{"why":"Provides the adaptive RKHS regularization result for Fredholm integral equations that the numerical convergence comparisons invoke.","marker":"[37]"},{"why":"The classical finite-dimensional representer theorem for smoothing splines that Theorem 2.5 extends to the automatic basis.","marker":"[44]"},{"why":"Standard RKHS characterization $H_G=L_G^{1/2}(L^2_\\rho)$ used in Lemma 2.4(c).","marker":"[13]"},{"why":"Supplies the regularization theory—Tikhonov regularization, discrepancy principle, and Krylov subspace methods—underpinning the iterative schemes.","marker":"[20]"},{"why":"The LSQR algorithm whose Golub–Kahan bidiagonalization structure the paper's iterative implementation follows.","marker":"[40]"},{"why":"The GCV criterion used for Tikhonov regularization parameter selection in the numerical experiments.","marker":"[23]"}],"fun_headline_variants":["Data-driven kernel ends manual kernel picking","Automatic kernel from convolution data beats tuned rivals","Finite basis from data: no kernel selection needed","Data-built kernel proves better than preselected ones","Convolution data yields automatic kernel, no tuning"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The data must actually explore every direction of the kernel's domain that one wants to recover: wherever the exploration measure $\\dot\\rho(s)$ is zero, the automatic kernel is set to zero and the estimator cannot detect kernel components living there.","fun_headline_variants_meta":{"raw":{"variants":["Data-driven kernel ends manual kernel picking","Automatic kernel from convolution data beats tuned rivals","Finite basis from data: no kernel selection needed","Data-built kernel proves better than preselected ones","Convolution data yields automatic kernel, no tuning"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000793,"raw_usage":{"total_tokens":3560,"prompt_tokens":1077,"completion_tokens":2483,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":693,"completion_tokens_details":{"reasoning_tokens":2413}},"tokens_in":693,"tokens_out":2483,"duration_ms":21932,"temperature":1.0,"reasoning_tokens":2413,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T17:01:46.451298+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For a fixed operator and data, compute the Gram matrix $\\Sigma$ by high-accuracy quadrature and compare the coefficient-form estimator $\\hat{c}_\\lambda^T\\xi$ against a direct discretized solution of the infinite-dimensional Tikhonov problem (2.6) on a very fine grid of $S$; if the two do not converge to the same function as the grid refines, the finite representer theorem would be false. Alternatively, test with a true kernel supported where $\\dot\\rho=0$ and noise-free data: the paper predicts the estimator is identically zero there, and any data-driven method that recovers the kernel in that region would require a hypothesis space larger than $H_G$.","supporting_citations":[{"cited_title":"A data-adaptive RKHS prior for Bayesian learning of kernels in operators.Journal of Machine Learning Research, 25(317):1–37, 2024","cited_arxiv_id":null,"evidence_quote":"Supplies the data-adaptive RKHS prior and the decomposition of $\\varphi_D$ into signal and noise components that motivates the automatic kernel and its null-space filtering."},{"cited_title":"Data adaptive RKHS Tikhonov regularization for learning kernels in operators.Proceedings of Mathematical and Scientific Machine Learning, PMLR 190:158- 172, 2022","cited_arxiv_id":null,"evidence_quote":"Establishes data-adaptive RKHS Tikhonov regularization for learning kernels in operators, the immediate predecessor that this paper generalizes to finite data and iterative methods."},{"cited_title":"An adaptive RKHS regularization for the Fredholm integral equations","cited_arxiv_id":null,"evidence_quote":"Provides the adaptive RKHS regularization result for Fredholm integral equations that the numerical convergence comparisons invoke."},{"cited_title":"Learning theory: an approximation theory viewpoint, volume24","cited_arxiv_id":null,"evidence_quote":"Standard RKHS characterization $H_G=L_G^{1/2}(L^2_\\rho)$ used in Lemma 2.4(c)."},{"cited_title":"Regularization of inverse problems, volume 375","cited_arxiv_id":null,"evidence_quote":"Supplies the regularization theory—Tikhonov regularization, discrepancy principle, and Krylov subspace methods—underpinning the iterative schemes."},{"cited_title":"LSQR: An algorithm for sparse linear equations and sparse least squares.ACM Transactions on Mathematical Software (TOMS), 8(1):43–71, 1982","cited_arxiv_id":null,"evidence_quote":"The LSQR algorithm whose Golub–Kahan bidiagonalization structure the paper's iterative implementation follows."},{"cited_title":"Generalized cross-validation as a method for choosing a good ridge parameter.Technometrics, 21(2):215–223, 1979","cited_arxiv_id":null,"evidence_quote":"The GCV criterion used for Tikhonov regularization parameter selection in the numerical experiments."}],"review_version":1}