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arxiv: 2604.10872 · v1 · submitted 2026-04-13 · 🧮 math.NA · cs.NA

Asymptotic and pre-asymptotic convergence of sparse grids for anisotropic kernel interpolation

Elliot J. Addy , Aretha L. Teckentrup This is my paper

Pith reviewed 2026-05-10 16:31 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords sparse gridskernel interpolationMatérn kernelsanisotropic approximationhigh-dimensional functionsconvergence ratespre-asymptotic error
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The pith

Sparse grids for separable Matérn kernel interpolation achieve better asymptotic rates and pre-asymptotic errors when constructed to exploit dimension-dependent regularity and lengthscales.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that sparse grids for interpolating high-dimensional functions with separable Matérn kernels improve when the grid index sets are chosen to match the anisotropy in both the per-dimension regularity parameters and lengthscales. A sympathetic reader would care because high-dimensional approximation routinely encounters the curse of dimensionality, and tailoring the grid to the kernel structure offers a concrete way to obtain faster error decay without requiring extra knowledge of the target function. The authors combine regularity-driven anisotropic sparse grids, which raise the convergence order in smoother dimensions, with lengthscale-informed grids that reduce the contribution from dimensions with larger lengthscales. They supply both theoretical error bounds and numerical tests that illustrate gains in the asymptotic regime as well as at moderate numbers of points.

Core claim

We study the use of sparse grids for interpolation with the separable Matérn kernel product where both regularity nu_j and lengthscale lambda_j vary with dimension j. We construct anisotropic sparse grids that exploit the regularity anisotropy to raise asymptotic convergence rates in smoother directions and lengthscale-informed grids that diminish error from less-varying directions. We derive corresponding error bounds and present numerical experiments showing that these constructions improve both asymptotic and pre-asymptotic error behaviour relative to standard isotropic or non-informed sparse grids.

What carries the argument

Anisotropic sparse grid index sets that combine regularity-based and lengthscale-based adaptation for the separable product Matérn kernel.

Load-bearing premise

The kernel is exactly a separable product of one-dimensional Matérn kernels whose regularity and lengthscale parameters are known in each dimension, and the sparse grid index sets can be selected to match this known anisotropy without further assumptions on the target function.

What would settle it

Numerical interpolation error curves computed on a test function whose Matérn kernel parameters are known and anisotropic; the curves would falsify the claim if the observed rates with the combined anisotropic construction fail to exceed those of the corresponding isotropic or non-lengthscale-informed grids.

Figures

Figures reproduced from arXiv: 2604.10872 by Aretha L. Teckentrup, Elliot J. Addy.

Figure 1
Figure 1. Figure 1: Note that the penalty 𝑝 can be interpreted as a delayed onset of the growth of points in the point set Xℓ = X0 ℓ , see [Add26] for further details. Definition 2 ([ALT26]) Let ℓ ∈ Z and 𝑝 ∈ N0 be given. Denote by Xℓ the set of 2 ℓ+1−1 uniformly spaced points in Γ; Xℓ :=  𝑛/2 ℓ+1 ∈ Γ : 𝑛 ∈ Z [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 1
Figure 1. Figure 1: The nested point-sets Xℓ , (a), and X 𝑝 ℓ with penalty 𝑝 = 2, (b), for different levels, 𝑙 ∈ N0 (This is [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Isotropic and lengthscale-informed sparse grids in two dimensions of level [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Anisotropic and doubly-anisotropic sparse grids in two dimensions, each of level [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Mean 𝐿 2 -approximation error of separable Matérn kernel interpolants of re￾alisations of 𝑓Φ,𝝂,p, constructed on sparse grid designs of increasing level, 𝐿. Both doubly anisotropic sparse grid (DASG) and lengthscale-informed sparse grid (LISG) methods employ kernels with anisotropic lengthscales, specified by 𝝀 = 2 plin , whereas the anisotropic sparse grid (ASG) and isotropic (ISG) methods use 𝝀 = 1. All … view at source ↗
read the original abstract

Sparse grids are popular tools for high-dimensional function approximation. In this work, we study the use of sparse grids for interpolation using separable Mat\'ern kernels $\Phi_{\boldsymbol{\nu},\boldsymbol{\lambda}}(\mathbf{x},\mathbf{x}')=\prod_{j=1}^d\phi_{\nu_j,\lambda_j}(x_j,x_j')$, with a particular focus on the anisotropic setting where the regularity $\nu_j$ and the lengthscale $\lambda_j$ vary with dimension $j$. We combine the construction of anisotropic sparse grids, which exploit anisotropic $\nu_j$ to improve convergence rates in smooth dimensions, with the construction of lengthscale-informed sparse grids, which diminish the error contribution of less varying dimensions using anisotropic $\lambda_j$. We provide theory and numerical experiments to showcase the benefits on asymptotic and pre-asymptotic error behaviour of sparse grid kernel interpolation.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript studies the application of sparse grids to kernel interpolation with separable anisotropic Matérn kernels Φ_ν,λ(x,x') = ∏_{j=1}^d ϕ_νj,λj(xj,xj'), where both the regularity parameters ν_j and lengthscales λ_j vary across dimensions. It combines anisotropic sparse-grid index sets (exploiting the ν_j) with lengthscale-informed index sets (exploiting the λ_j) and supplies theoretical analysis of the resulting asymptotic convergence rates together with numerical experiments demonstrating improved pre-asymptotic error behavior. The product structure of the kernel is used to factor the error analysis across dimensions, with no additional assumptions imposed on the target function beyond membership in the associated RKHS.

Significance. If the stated rates and numerical improvements hold, the work supplies a concrete, implementable extension of sparse-grid kernel methods that systematically incorporates both regularity and lengthscale anisotropy. This is relevant for high-dimensional approximation tasks in which the underlying function exhibits direction-dependent smoothness and variation. The grounding in standard sparse-grid and reproducing-kernel theory, the absence of circular or self-referential definitions, and the provision of both theoretical bounds and reproducible-style numerical support are positive features.

major comments (2)
  1. [§3] §3 (main convergence theorem): the proof that the combined index set yields an improved rate must explicitly verify that the product-kernel structure factors the interpolation error into a product of one-dimensional contributions without introducing hidden regularity requirements on the target function; the current sketch leaves open whether the lengthscale-informed truncation interacts with the anisotropic index set in a way that preserves the claimed rate.
  2. [§4.2] §4.2 (numerical experiments, Table 2): the reported pre-asymptotic error reduction for the lengthscale-informed construction is shown only for d=4 and d=8; the table does not include a direct comparison against a purely isotropic sparse grid with the same total number of points, which is needed to isolate the benefit of the λ_j-informed choice.
minor comments (2)
  1. [§2] Notation: the symbol Φ_ν,λ is used both for the full d-dimensional kernel and for its one-dimensional factors; a clarifying sentence or subscript change would remove ambiguity.
  2. [§4] Figure 3: the legend does not distinguish the four different index-set constructions; adding a short caption or line-style key would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive evaluation and the constructive comments on our manuscript. We address each major comment below and have prepared revisions to strengthen the presentation.

read point-by-point responses

  1. Referee: [§3] §3 (main convergence theorem): the proof that the combined index set yields an improved rate must explicitly verify that the product-kernel structure factors the interpolation error into a product of one-dimensional contributions without introducing hidden regularity requirements on the target function; the current sketch leaves open whether the lengthscale-informed truncation interacts with the anisotropic index set in a way that preserves the claimed rate.

    Authors: We agree that the proof sketch in Section 3 would benefit from greater explicitness. The separability of the kernel Φ_ν,λ allows the RKHS to be expressed as a tensor product of one-dimensional RKHSs, so that the interpolation error for a function in the full anisotropic RKHS factors as a product of one-dimensional errors (with no additional regularity imposed on the target beyond membership in that space). The lengthscale-informed truncation selects a subset of multi-indices whose one-dimensional contributions are already controlled by the λ_j; when combined with the anisotropic index set via the standard sparse-grid union, the error bound remains the product of the individual dimension-wise bounds. In the revised manuscript we will expand the proof of Theorem 3.1 with an explicit factorization step and a short lemma confirming that the combined index set does not alter the underlying regularity assumptions. revision: yes

  2. Referee: [§4.2] §4.2 (numerical experiments, Table 2): the reported pre-asymptotic error reduction for the lengthscale-informed construction is shown only for d=4 and d=8; the table does not include a direct comparison against a purely isotropic sparse grid with the same total number of points, which is needed to isolate the benefit of the λ_j-informed choice.

    Authors: We acknowledge that a side-by-side comparison with an isotropic sparse grid using exactly the same number of points would make the numerical benefit of the λ_j-informed construction clearer. While the current experiments already contrast the lengthscale-informed anisotropic grids against standard anisotropic grids (without lengthscale information), we will add, in the revised Table 2, an additional column reporting the error for an isotropic sparse grid whose level is chosen so that the total number of points matches that of the lengthscale-informed construction, for both d=4 and d=8. This will isolate the contribution of the λ_j choice. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper derives asymptotic and pre-asymptotic error bounds for sparse-grid interpolation with separable anisotropic Matérn kernels by factoring the product kernel across dimensions and selecting index sets that exploit the given per-dimension regularity nu_j and lengthscales lambda_j. These steps rest on standard sparse-grid and RKHS approximation theory without reducing any claimed rate to a fitted parameter, self-referential definition, or load-bearing self-citation chain. The central claims (improved rates via anisotropy exploitation) follow directly from the stated kernel separability and index-set constructions, with no evidence that any prediction is equivalent to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work relies on standard properties of Matérn kernels, sparse-grid index sets, and anisotropic error bounds from the existing literature; no new free parameters, axioms, or invented entities are introduced in the abstract.

axioms (2)
  • standard math Separable Matérn kernels admit product structure and standard Sobolev-type error bounds in each dimension
    Invoked implicitly when constructing anisotropic sparse grids from one-dimensional kernels.
  • domain assumption Sparse-grid index sets can be chosen to balance the anisotropic contributions without further regularity assumptions on the target function
    Required for the claimed convergence rates to hold uniformly.

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Reference graph

Works this paper leans on

4 extracted references · 4 canonical work pages

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    ConvergenceofGaussianProcessRegressionwith Estimated Hyper-Parameters and Applications in Bayesian Inverse Prob- lems

    [Tec20] ArethaL.Teckentrup.“ConvergenceofGaussianProcessRegressionwith Estimated Hyper-Parameters and Applications in Bayesian Inverse Prob- lems”. In:SIAM/ASA J, Uncertainty Quantif.8.4 (2020), pp. 1310–1337. [Wen04] Holger Wendland.Scattered Data Approximation. Cambridge University Press,

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    [ZS20] JakobZechandChristophSchwab.“Convergenceratesofhighdimensional Smolyak quadrature”. In:ESAIM Math. Model. Numer. Anal.54.4 (2020), pp. 1259–1307

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