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In this article, we prove new theorems bounding the rank of differentsub-matrices arising from these kernel functions. Bounds like these are oftenuseful for analyzing the complexity of various hierarchical matrix algorithms.We also plot the numerical rank growth of different sub-matrices arising out ofvarious kernel functions in $1$D, $2$D, $3$D and $4$D, which, not surprisingly,agrees with the proposed theorems. Another significant contribution of thisarticle is that, using the obtained rank bounds, we also propose a way toextend the notion of \textbf{\emph{weak-admissibility}} for hierarchicalmatrices 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, hierarchicallyoff-diagonal low-rank matrix in $d$ dimensions (HODLR$d$D), which can performmatrix-vector products with $\mathcal{O}(pN \log (N))$ complexity in anydimension $d$, 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 theHODLR$d$D is available at \url{https://github.com/SAFRAN-LAB/HODLRdD}. We alsodiscuss the scalability of the HODLR$d$D algorithm and showcase theapplicability by solving an integral equation in $4$ dimensions andaccelerating the training phase of the support vector machines (SVM) for thedata sets with four and five features.

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