# Fast Low-Rank Kernel Matrix Factorization through Skeletonized Interpolation

## TL;DR

This paper introduces a nearly optimal, efficient method for low-rank kernel matrix factorization using skeletonized interpolation, improving computational cost and accuracy for integral equation discretizations.

## Contribution

It presents a novel skeletonized interpolation approach that constructs low-rank factorizations with near-optimal cost and stability, outperforming naive algorithms.

## Key findings

- Achieves low-rank factorization at $\\mathcal{O}(nr)$ cost
- Demonstrates asymptotic convergence and stability
- Performs well in numerical experiments, near optimal rank

## Abstract

Integral equations are commonly encountered when solving complex physical problems. Their discretization leads to a dense kernel matrix that is block or hierarchically low-rank. This paper proposes a new way to build a low-rank factorization of those low-rank blocks at a nearly optimal cost of $\mathcal{O}(nr)$ for a $n \times n$ block submatrix of rank r. This is done by first sampling the kernel function at new interpolation points, then selecting a subset of those using a CUR decomposition and finally using this reduced set of points as pivots for a RRLU-type factorization. We also explain how this implicitly builds an optimal interpolation basis for the Kernel under consideration. We show the asymptotic convergence of the algorithm, explain his stability and demonstrate on numerical examples that it performs very well in practice, allowing to obtain rank nearly equal to the optimal rank at a fraction of the cost of the naive algorithm.