We present a parallel algorithm for computing the approximate factorization of an $N$-by-$N$ kernel matrix. Once this factorization has been constructed (with $N log^2 N $ work), we can solve linear systems with this matrix with $N log N $ work. Kern
el matrices represent pairwise interactions of points in metric spaces. They appear in machine learning, approximation theory, and computational physics. Kernel matrices are typically dense (matrix multiplication scales quadratically with $N$) and ill-conditioned (solves can require 100s of Krylov iterations). Thus, fast algorithms for matrix multiplication and factorization are critical for scalability. Recently we introduced ASKIT, a new method for approximating a kernel matrix that resembles N-body methods. Here we introduce INV-ASKIT, a factorization scheme based on ASKIT. We describe the new method, derive complexity estimates, and conduct an empirical study of its accuracy and scalability. We report results on real-world datasets including COVTYPE ($0.5$M points in 54 dimensions), SUSY ($4.5$M points in 8 dimensions) and MNIST (2M points in 784 dimensions) using shared and distributed memory parallelism. In our largest run we approximately factorize a dense matrix of size 32M $times$ 32M (generated from points in 64 dimensions) on 4,096 Sandy-Bridge cores. To our knowledge these results improve the state of the art by several orders of magnitude.
Dual-tree algorithms are a widely used class of branch-and-bound algorithms. Unfortunately, developing dual-tree algorithms for use with different trees and problems is often complex and burdensome. We introduce a four-part logical split: the tree, t
he traversal, the point-to-point base case, and the pruning rule. We provide a meta-algorithm which allows development of dual-tree algorithms in a tree-independent manner and easy extension to entirely new types of trees. Representations are provided for five common algorithms; for k-nearest neighbor search, this leads to a novel, tighter pruning bound. The meta-algorithm also allows straightforward extensions to massively parallel settings.
