Simultaneously Guaranteed Kernel Interpolation

• SGKI is a kernel-based interpolation framework that guarantees point estimates with non-asymptotic uncertainty quantification using simultaneous confidence bands.
• It unifies deterministic and probabilistic methods to robustly handle noisy, incomplete, and high-dimensional data with explicit error bounds.
• Its scalable algorithms and proven convergence theory make SGKI a practical tool for PDE solvers, image inpainting, and high-dimensional surrogate modeling.

Simultaneously Guaranteed Kernel Interpolation (SGKI) is a class of kernel-based approximation and interpolation methodologies focused on providing point estimates along with rigorous, non-asymptotic uncertainty quantification—typically in the form of simultaneous confidence bands—across potentially large, unstructured, or incomplete datasets. SGKI unifies deterministic and stochastic perspectives in interpolation by rigorously quantifying both interpolation accuracy and prediction uncertainty, even in the presence of noise or when only indirect data (derivatives, integrals) are available. The methodology extends classical deterministic kernel interpolation, kriging, Bayesian estimation, and meshfree PDE solvers, offering a robust, theoretically founded alternative for high-dimensional approximation and learning tasks.

1. Theoretical Foundations and Definitions

SGKI builds on the framework of reproducing kernel Hilbert spaces (RKHS) and kernel-based probability measures. A kernel-based probability measure is a Gaussian measure on a Banach space $B$, specified by mean $\mu$ and covariance kernel $K: B^* \times B^* \to \mathbb{R}$, where $B^*$ is the dual of $B$. This setting enables interpolation problems to be interpreted probabilistically: given functionals $(L_j, f_j)$, one seeks the conditional expectation of the field $S_L(\omega) := \langle \omega, L \rangle_B$ over all $\omega$ matching the (possibly noisy) data constraints.

The SGKI estimator for a functional $L$ given data $(L_j, f_j)$ is

$$s_{A_n}(L) = L\mu + b_{K, n}(L)^T A_{K, n}^\dagger (f_n - n\mu)$$

where $b_{K, n}(L) = (K(L, L_1), \ldots, K(L, L_n))^T$, and $A_{K, n}$ is the kernel matrix.

SGKI extends to generalized functionals, which can include evaluations, derivatives, integrals, or Radon transforms, thus facilitating broad application (e.g., inverse problems, computerized tomography) (1710.05192, 2407.03840).

2. Simultaneity: Error and Uncertainty Guarantees

The core principle is to provide simultaneous guarantees—both on pointwise error and on uncertainty—for all quantities of interest. This is realized via the conditional variance (power function) associated with the kernel-based estimator: $$\sigma_{L|n}^2 = K(L, L) - b_{K, n}(L)^T A_{K, n}^\dagger b_{K, n}(L)$$ which delivers explicit bounds: $|L u - s_{A_n}(L)| \leq C \sigma_{L|n}$ for any function $u$ in the appropriate native space.

SGKI robustly handles noisy data by conditioning over a high-dimensional data ball—leading to confidence bands that remain simultaneously valid over all interpolation points or functionals (non-asymptotic, high-probability guarantees). For instance, in image inpainting and super-resolution, SGKI computes nonparametric confidence intervals simultaneously for all missing pixels with finite-sample reliability (2506.23221).

3. Algorithms and Computational Strategies

SGKI encompasses several algorithmic classes, including:

• Kernel-based Probability Measures (KPM): Estimators are constructed as conditional expectations within a Gaussian measure framework, applicable to various interpolation and PDE settings (1710.05192).
• Greedy Regularized Kernel Interpolation: Sparse, stable representations are obtained by incrementally selecting data sites via power function or residual-based greedy rules, with proven quasi-optimal convergence. These algorithms adapt to noisy or expensive-to-evaluate target functions (1807.09575, 2307.09811).
• Generalized Interpolation via Greedy Data Selection: SGKI algorithms support arbitrary functionals, with minimal assumptions. Nested greedy selection (e.g., power greedy, f-greedy, psr-greedy) provably achieves convergence for totally bounded sets of sampling functionals—even when only Radon or Birkhoff-type data are available (2407.03840).
• Sparse Grid Combination Techniques and Samplet Compression: High-dimensional interpolation is made scalable via sparse grids, hierarchical decomposition, and efficient matrix compression, enabling SGKI on billions of degrees of freedom with rigorous error quantification (2505.12282).
• Volume Sampling and Fekete Points: Near-optimal node sets for interpolation can be efficiently constructed by maximizing determinants of kernel Gram matrices (approximate Fekete points) or by randomized continuous volume sampling, both achieving uniform (simultaneous) error guarantees (1912.07316, 2002.09677).

For large linear systems or high sample sizes, techniques such as Schur complement-based inversion and samplet-based matrix sparsification are crucial for scalability.

4. Extensions and Practical Applications

SGKI has been deployed across numerous contexts:

• Uncertainty Quantification in Inverse Problems and UQ: Quasi-optimal point selection (e.g., Cholesky/greedy Fekete point analogs) stabilizes kernel interpolation in high-dimensional parametric PDE models, outperforming sparse grid collocation in accuracy and robustness (2104.06291).
• Image Inpainting and Super-Resolution: SGKI constructs minimum-norm kernel interpolants for missing pixels and simultaneously delivers finite-sample confidence bands, using Paley-Wiener kernels for band-limited image models (2506.23221).
• Elliptic and Parabolic PDEs: Meshfree solution methods via SGKI accommodate broad classes of data functionals, including derivative and weak-form constraints (1710.05192).
• Gaussian Process Regression (online/streaming): SGKI enables scalable, constant-time posterior updates while preserving exact inference and full posterior quantification (2103.01454).
• Interpolation on Manifolds/Spheres: Distributed SGKI approaches (e.g., block-wise local interpolation + aggregation) overcome the uncertainty barrier observed in classical RBF interpolation, providing noise-robust, scalable approximations (2310.16384).

5. Convergence Theory and Optimality

SGKI methods provide provable convergence under minimal assumptions. Notably, convergence in the RKHS norm (and, under suitable kernel boundedness, in the uniform norm) is guaranteed for arbitrary sequences of data functionals whose fill distance decays. For Sobolev-equivalent kernels, greedy SGKI schemes match best possible rates $n^{-\tau/d}$ for $f$-adaptive rules (target adaptive) and nearly-optimal rates for uniform worst-case ($P$-greedy) rules (2307.09811). Recent advances remove prior logarithmic gaps in the theory, ensuring both uniform and nonlinear approximation optimality.

A summary table of convergence rates:

Selection Strategy	Convergence Rate	Guarantee Type
P-greedy	$n^{-\tau/d+1/2}$	Worst-case, all $f$
f-greedy	$n^{-\tau/d}$	Adaptive, target-specific
Nonlinear best	$n^{-\tau/d}$	Theoretical optimum

6. Robustness, Stability, and Distributed Approaches

SGKI enables robust interpolation in the presence of noise and numerical instability. Results such as Schaback’s uncertainty relation—that small interpolation error and a well-conditioned kernel matrix cannot coexist as sample size increases—are addressed via distributed or blockwise methods. These approaches combine local interpolants computed on disjoint quasi-uniform data subsets, aggregating them to achieve variance reduction and robust prediction, especially on manifolds with noisy data (2310.16384).

Additionally, regularization (e.g., Tikhonov) and samplet-based matrix compression further stabilize the interpolation process.

7. Significance and Future Directions

SGKI unifies deterministic and probabilistic kernel interpolation, providing a comprehensive methodology for simultaneous approximation and uncertainty quantification under general types of data. Its theoretical flexibility, computational scalability, and general applicability position it as a cornerstone technique for modern scientific computing, data-driven PDEs, high-dimensional surrogate modeling, and real-world uncertainty quantification. Future research aims to extend convergence theory, adaptivity, and practical utility to broader classes of kernels and data functionals, as well as to deepen connections with Bayesian learning and operator-theoretic frameworks.