Generation of point sets by convex optimization for interpolation in reproducing kernel Hilbert spaces

Kernel methods have become fundamental in approximation theory, machine learning, and scientific computing, especially for high-dimensional data. Early approaches like Fekete points and P-贪婪 algorithms focus on selecting well-distributed points to optimize interpolation accuracy. However, these methods often involve non-convex optimization, which is computationally expensive and difficult to scale. Recent developments have explored convex relaxations and spectral techniques, but a unified, efficient framework for generating high-quality point sets remains elusive. Mercer expansion offers a spectral decomposition of kernels, enabling new analytical and computational strategies. This paper builds on these ideas, proposing a convex optimization-based approach that leverages Mercer expansion to approximate the determinant of the kernel matrix, a key measure of point distribution quality.

The core challenge is to efficiently generate point sets that ensure stable and accurate kernel interpolation, particularly in high-dimensional spaces. Existing methods struggle with non-convex optimization landscapes, leading to local minima and high computational costs. Moreover, ensuring the resulting point distribution is both well-spread and numerically stable (low condition number) remains difficult. The problem becomes more acute when dealing with complex domains or kernels lacking explicit spectral decompositions. Therefore, a scalable, theoretically sound method that guarantees near-optimal point distribution is highly desirable but remains an open problem.

The main innovation lies in transforming the determinant maximization problem into a convex form via Mercer expansion, enabling the use of SOCP solvers. This approach approximates the kernel matrix determinant, a proxy for point distribution quality, within a convex optimization framework. It introduces both a one-shot algorithm for generating all points simultaneously and a sequential variant that iteratively adds points, enhancing flexibility. The method bridges classical polynomial point designs with modern kernel techniques, providing theoretical guarantees and computational efficiency. Additionally, the paper develops a rigorous approximation of the determinant, ensuring the convex problem remains faithful to the original non-convex goal.

• �� Expand the kernel function into Mercer series, truncating to obtain a finite spectral approximation. • Approximate the determinant of the kernel matrix using the spectral components and the feature matrix. • Formulate a convex optimization problem (SOCP) to maximize this approximate determinant, subject to domain constraints. • Solve the SOCP using modern solvers like MOSEK, obtaining optimal weights for candidate points. • Select points corresponding to the largest weights, ensuring well-distributed configurations. • Extend to a sequential algorithm by fixing weights for previously selected points and re-optimizing for new points, iteratively refining the distribution. • Analyze the relationship between the determinant approximation and point distribution quality, providing theoretical bounds.

Experiments utilized Gaussian, Brownian motion, and spherical inverse multiquadric kernels across 1D, 2D, and 3D domains. Candidate points were sampled uniformly or on spheres, with neighborhood structures defined for local maxima detection. The algorithms were evaluated based on condition numbers of kernel matrices, maximum power function values, and interpolation errors on test functions. Comparisons with P-贪婪 and random sampling demonstrated the proposed methods' efficiency and distribution quality. Hyperparameters such as the number of candidate points and spectral truncation levels were tuned to balance computational cost and accuracy. The experiments confirmed the algorithms' robustness across diverse settings, highlighting their scalability and effectiveness.

Results showed that the convex optimization algorithms consistently produced point sets with lower condition numbers—up to 20% reduction—and smaller maximum power function values, indicating better distribution and stability. In high-dimensional tests, the point sets achieved more uniform coverage and improved interpolation accuracy, with errors reduced by 10-12% relative to baseline methods. The sequential algorithm effectively handled larger point counts by incremental refinement, maintaining distribution quality. Computational times remained practical, benefiting from the convex reformulation, with the total optimization time significantly lower than traditional non-convex approaches. These findings validate the approach's potential for scalable, high-quality point set design in kernel interpolation.

The algorithms are directly applicable to high-precision numerical solutions of PDEs, kernel regression, and machine learning tasks requiring stable and accurate interpolation. They are suitable for complex geometries and high-dimensional data, provided the kernel admits Mercer expansion. The method can be integrated into existing scientific computing workflows to improve the stability and accuracy of kernel-based approximations, especially when large datasets or complex domains are involved. Long-term, this approach could enable automated, scalable design of point distributions in various scientific and engineering applications, including geostatistics, sensor placement, and meshless methods.

The reliance on Mercer expansion restricts applicability to kernels with known spectral decompositions. High-dimensional spectral computations can be costly, limiting scalability. The method assumes domain regularity; irregular geometries may degrade performance. Additionally, the approximation quality depends on spectral truncation, which may introduce errors in some cases. Future work should address these limitations by developing adaptive spectral methods, extending to non-Mercer kernels, and improving robustness in complex geometries.