Low-rank approximation of Rippa method for RBF interpolation

TL;DR

This paper introduces an efficient low-rank approximation method for RBF interpolation that accelerates leave-one-out cross-validation for shape parameter selection, enabling practical use on larger datasets.

Contribution

It combines Nyström approximation with the Woodbury identity to create a surrogate objective for LOOCV, reducing computational cost and improving scalability.

Findings

Accelerates LOOCV for RBF shape parameter tuning.

Maintains qualitative behavior of full LOOCV in larger datasets.

Demonstrates effectiveness across 1D, 2D, and 3D problems.

Abstract

We study the problem of selecting the shape parameter in Radial Basis function (RBF) interpolation using leave-one-out-cross-validation (LOOCV). Since the classical LOOCV formula requires repeated solves with a dense $N \times N$ kernel matrix, we combine a Nystr\"{o}m approximation with the Woodbury identity to obtain an efficient surrogate objective that avoids large matrix inversions. Based on this reduced form, we compare a grid-based search with a gradient descent strategy and examine their behavior across different dimensions. Numerical experiments are performed in 1D, 2D, and 3D using the Inverse Multiquadratic RBF to illustrate the computational advantages of the approximation as well as the situations in which it may introduce additional sensitivity. These results show that the proposed acceleration makes LOOCV-based parameter tuning practical for larger datasets while preserving the qualitative behavior of the full method.