HODLR$d$D: A new Black-box fast algorithm for $N$-body problems in $d$-dimensions with guaranteed error bounds

TL;DR

This paper introduces HODLR$d$D, a new kernel-independent algorithm for $N$-body problems in $d$-dimensions with guaranteed error bounds, achieving nearly linear complexity and extending hierarchical matrix techniques to higher dimensions.

Contribution

The paper proves new rank bounds for kernel matrices, extends weak-admissibility to higher dimensions, and develops a scalable, kernel-independent HODLR$d$D algorithm for $N$-body problems.

Findings

Theoretical rank bounds are validated numerically across dimensions.

HODLR$d$D achieves $ ext{O}(pN ext{log}(N))$ complexity with $p$ nearly logarithmic in $N$.

Algorithm successfully applied to 4D integral equations and SVM training with high efficiency.

Abstract

In this article, we prove new theorems bounding the rank of different sub-matrices arising from these kernel functions. Bounds like these are often useful for analyzing the complexity of various hierarchical matrix algorithms. We also plot the numerical rank growth of different sub-matrices arising out of various kernel functions in $1$D, $2$D, $3$D and $4$D, which, not surprisingly, agrees with the proposed theorems. Another significant contribution of this article is that, using the obtained rank bounds, we also propose a way to extend the notion of \textbf{\emph{weak-admissibility}} for hierarchical matrices in higher dimensions. Based on this proposed \textbf{\emph{weak-admissibility}} condition, we develop a black-box (kernel-independent) fast algorithm for $N$-body problems, hierarchically off-diagonal low-rank matrix in $d$ dimensions (HODLR$d$D), which can perform matrix-vector products with $\mathcal{O}(pN \log (N))$ complexity in any dimension $d$, where $p$ doesn't grow with any power of $N$. More precisely, our theorems guarantee that $p \in \mathcal{O} (\log (N) \log^d (\log (N)))$, which implies our HODLR$d$D algorithm scales almost linearly. The $\texttt{C++}$ implementation with \texttt{OpenMP} parallelization of the HODLR$d$D is available at \url{https://github.com/SAFRAN-LAB/HODLRdD}. We also discuss the scalability of the HODLR$d$D algorithm and showcase the applicability by solving an integral equation in $4$ dimensions and accelerating the training phase of the support vector machines (SVM) for the data sets with four and five features.