An $N \log N$ Parallel Fast Direct Solver for Kernel Matrices

TL;DR

This paper introduces an efficient parallel algorithm for approximate factorization of large kernel matrices, significantly reducing computational costs in machine learning tasks by leveraging low-rank off-diagonal block properties.

Contribution

It presents an $ ext{O}(dN ext{log}N)$-work parallel algorithm for kernel matrix factorization that does not require global low-rank structure, only off-diagonal low-rank properties.

Findings

Factorizes an 11 million by 11 million kernel matrix in 2 minutes on 3,072 cores.

Factorizes a 4.5 million by 4.5 million matrix in 1 minute on 4,352 cores.

Achieves $ ext{O}(N ext{log}N)$ complexity for solving linear systems with kernel matrices.

Abstract

Kernel matrices appear in machine learning and non-parametric statistics. Given $N$ points in $d$ dimensions and a kernel function that requires $\mathcal{O}(d)$ work to evaluate, we present an $\mathcal{O}(dN\log N)$-work algorithm for the approximate factorization of a regularized kernel matrix, a common computational bottleneck in the training phase of a learning task. With this factorization, solving a linear system with a kernel matrix can be done with $\mathcal{O}(N\log N)$ work. Our algorithm only requires kernel evaluations and does not require that the kernel matrix admits an efficient global low rank approximation. Instead our factorization only assumes low-rank properties for the off-diagonal blocks under an appropriate row and column ordering. We also present a hybrid method that, when the factorization is prohibitively expensive, combines a partial factorization with iterative methods. As a highlight, we are able to approximately factorize a dense $11M\times11M$ kernel matrix in 2 minutes on 3,072 x86 "Haswell" cores and a $4.5M\times4.5M$ matrix in 1 minute using 4,352 "Knights Landing" cores.