Well-Separated Pair Decompositions for High-Dimensional Datasets

Well-separated pair decomposition (WSPD) is a well known geometric decomposition used for encoding distances, introduced in a seminal paper by Paul B. Callahan and S. Rao Kosaraju in 1995. WSPD compresses $O\left(n^2\right)$ pairwise distances of n given points from ${\mathbb{R}}^d$ in $O\left(n\right)$ space for a fixed dimension d. However, the main problem with this remarkable decomposition is the “hidden” dependence on the dimension d, which in practice does not allow for the computation of a WSPD for any dimension $d>2$ or $d>3$ at best. In this work, I will show how to compute a WSPD for points in ${\mathbb{R}}^d$ and for any dimension d. Instead of computing a WSPD directly in ${\mathbb{R}}^d$, I propose to learn nonlinear mapping and transform the data to a lower-dimensional space ${\mathbb{R}}^{d^{\prime }}$, $d^{\prime }=2$ or $d^{\prime }=3$, since only in such low-dimensional spaces can a WSPD be efficiently computed. Furthermore, I estimate the quality of the computed WSPD in the original $R^d$ space. My experiments show that for different synthetic and real-world datasets my approach allows that a WSPD of size $O\left(n\right)$ can still be computed for points in ${\mathbb{R}}^d$ for dimensions d much larger than two or three in practice.