arXiv 2008.067002020-08-15cs.DScs.CCcs.CGcs.LGmath.MG

高效低失真超测量嵌入技术研究

On Efficient Low Distortion Ultrametric Embedding

Google（谷歌）；Tel Aviv University（特拉维夫大学）；LaBRI Bordeaux（波尔多计算机科学实验室）

Vincent Cohen-Addad, Karthik C. S., Guillaume Lagarde

摘要：

无监督学习和数据分析中的一个经典问题是找到更简单，更易于可视化的数据表示，以保持其基本属性。一个广泛使用的方法，以保持底层的层次结构的数据，同时降低其复杂性是找到一个嵌入到树或超度量的数据。用于此任务的最流行的算法是经典的链接算法（单个，平均或完全）。然而，这些方法在$Ω(\log n)$维度中的$n$点的数据集上表现出$Θ(n^2)$的相当令人望而却步的运行时间。本文提出了一种新的算法，它以$\mathbb{R}^d$中的一组点$P$作为输入，对每个$c\ge 1$，在时间上运行$n^{1+\fracρ{c^2}}$（对于某个普适常数$ρ>1$）输出超度量$Δ$，使得对于$P$中的任何两点$u,v$，我们有$Δ(u,v)$是在$5c$的乘法因子内的$u$和$v$之间的距离在“最佳”$P$的超度量表示。这里，最佳超度量是最小化相对于$\ell_2$距离的最大距离失真，即最小化$\underset{u,v \in P}{\max}\ \frac{\tildeΔ(u,v)}{\|u-v\|_2}$的超度量$\tildeΔ$。我们补充上述结果表明，在流行的复杂性理论的假设下，对于每个常数$\varepsilon>0$，没有算法的运行时间$n^{2-\varepsilon}$可以区分输入的$\ell_\infty$ -度量，允许等距嵌入和那些招致失真的$\frac{3}{2}$。最后，我们对经典机器学习数据集进行了实证评估，并表明我们算法的输出与链接算法的输出相当，同时实现了更快的运行时间。

英文摘要：

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.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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英文摘要：

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