Low-rank approximation of Rippa method for RBF interpolation

TL;DR

The paper tackles efficient tuning of the RBF shape parameter ε for interpolation by reframing LOOCV evaluation with a Nyström low-rank approximation of the interpolation matrix and the Woodbury identity, reducing the single-evaluation cost from O($N^3$) to O($N m^2$ + $m^3$) and achieving linear scaling in $N$ for fixed $m$. It supplements this with a gradient-based optimization in log-space to avoid predefined ε-search ranges, plus robust landmark stability analysis using k-means++ to justify the Nyström construction. Across 1D–3D experiments with Inverse Multiquadratic RBFs, the Nyström-accelerated LOOCV preserves the qualitative behavior of the full method while offering substantial speedups, particularly when combined with cosine-type node mappings in higher dimensions to mitigate ill-conditioning. The work demonstrates practical, scalable parameter-tuning for large datasets, and suggests directions for improving optimization landscapes and landmark sampling to further enhance robustness and efficiency.

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 Nyström 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.

Figures (9)

• Figure 1: NMI values for different numbers of landmarks
• Figure 2: Performance plot for function $f_{1}$
• Figure 3: Performance plot for function $f_{2}$
• Figure 4: Performance plot for function $f_{3}$
• Figure 5: Performance plot for function $f_{4}$