Fast Low-Rank Kernel Matrix Factorization through Skeletonized Interpolation
For researchers solving integral equations via discretization, this provides a faster way to compute low-rank approximations of dense kernel matrices.
This paper proposes a new method for building low-rank factorizations of kernel matrix blocks at nearly optimal cost O(nr), achieving ranks close to optimal with significantly reduced computational cost compared to naive algorithms.
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.