# A Multigrid Preconditioner for Tensor Product Spline Smoothing

## TL;DR

This paper introduces a matrix-free multigrid preconditioner for tensor product spline smoothing that reduces memory usage and achieves scalable convergence in high-dimensional nonparametric regression.

## Contribution

It presents a novel multigrid preconditioned conjugate gradient method for tensor product B-spline smoothing that is memory-efficient and scalable across fixed dimensions.

## Key findings

- Achieves grid independent convergence in fixed dimensions.
- Requires moderate memory, suitable for high-dimensional data.
- Demonstrates scalability and efficiency in multivariate smoothing.

## Abstract

Uni- and bivariate data smoothing with spline functions is a well established method in nonparametric regression analysis. The extension to multivariate data is straightforward, but suffers from exponentially increasing memory and computational complexity. Therefore, we consider a matrix-free implementation of a geometric multigrid preconditioned conjugate gradient method for the regularized least squares problem resulting from tensor product B-spline smoothing with multivariate and scattered data. The algorithm requires a moderate amount of memory and is therefore applicable also for high-dimensional data. Moreover, for arbitrary but fixed dimension, we achieve grid independent convergence which is fundamental to achieve algorithmic scalability.