# Fast Cross-validation in Harmonic Approximation

## TL;DR

This paper introduces a fast, FFT-based method for computing cross-validation scores in harmonic approximation problems, significantly reducing computational costs across various geometries and node distributions.

## Contribution

The authors develop a novel FFT-based approach to efficiently compute cross-validation scores for Tikhonov regularization, applicable to scattered and structured nodes on multiple manifolds.

## Key findings

- Algorithm reduces computational cost to that of the regularization problem
- Applicable to nodes on the torus, interval, and sphere
- Numerical experiments confirm efficiency and accuracy

## Abstract

Finding a good regularization parameter for Tikhonov regularization problems is a though yet often asked question. One approach is to use leave-one-out cross-validation scores to indicate the goodness of fit. This utilizes only the noisy function values but, on the downside, comes with a high computational cost. In this paper we present a general approach to shift the main computations from the function in question to the node distribution and, making use of FFT and FFT-like algorithms, even reduce this cost tremendously to the cost of the Tikhonov regularization problem itself. We apply this technique in different settings on the torus, the unit interval, and the two-dimensional sphere. Given that the sampling points satisfy a quadrature rule our algorithm computes the cross-validations scores in floating-point precision. In the cases of arbitrarily scattered nodes we propose an approximating algorithm with the same complexity. Numerical experiments indicate the applicability of our algorithms.