On Efficient Low Distortion Ultrametric Embedding

Abstract

A classic problem in unsupervised learning and data analysis is to find simpler and easy-to-visualize representations of the data that preserve its essential properties. A widely-used method to preserve the underlying hierarchical structure of the data while reducing its complexity is to find an embedding of the data into a tree or an ultrametric. The most popular algorithms for this task are the classic linkage algorithms (single, average, or complete). However, these methods on a data set of $n$ points in $Ω(\log n)$ dimensions exhibit a quite prohibitive running time of $Θ(n^2)$. In this paper, we provide a new algorithm which takes as input a set of points $P$ in $\mathbb{R}^d$, and for every $c\ge 1$, runs in time $n^{1+\fracρ{c^2}}$ (for some universal constant $ρ>1$) to output an ultrametric $Δ$ such that for any two points $u,v$ in $P$, we have $Δ(u,v)$ is within a multiplicative factor of $5c$ to the distance between $u$ and $v$ in the "best" ultrametric representation of $P$. Here, the best ultrametric is the ultrametric $\tildeΔ$ that minimizes the maximum distance distortion with respect to the $\ell_2$ distance, namely that minimizes $\underset{u,v \in P}{\max}\ \frac{\tildeΔ(u,v)}{\|u-v\|_2}$. We complement the above result by showing that under popular complexity theoretic assumptions, for every constant $\varepsilon>0$, no algorithm with running time $n^{2-\varepsilon}$ can distinguish between inputs in $\ell_\infty$-metric that admit isometric embedding and those that incur a distortion of $\frac{3}{2}$. Finally, we present empirical evaluation on classic machine learning datasets and show that the output of our algorithm is comparable to the output of the linkage algorithms while achieving a much faster running time.