Hierarchical Clustering trees have been widely accepted as a useful form of clustering data, resulting in a prevalence of adopting fields including phylogenetics, image analysis, bioinformatics and more. Recently, Dasgupta (STOC 16') initiated the analysis of these types of algorithms through the lenses of approximation. Later, the dual problem was considered by Moseley and Wang (NIPS 17') dubbing it the Revenue goal function. In this problem, given a nonnegative weight $w_{ij}$ for each pair $i,j \in [n]=\{1,2, \ldots ,n\}$, the objective is to find a tree $T$ whose set of leaves is $[n]$ that maximizes the function $\sum_{i<j \in [n]} w_{ij} (n -|T_{ij}|)$, where $|T_{ij}|$ is the number of leaves in the subtree rooted at the least common ancestor of $i$ and $j$.
In our work we consider the revenue goal function and prove the following results. First, we prove the existence of a bisection (i.e., a tree of depth 2 in which the root has two children, each being a parent of $n/2$ leaves) which approximates the general optimal tree solution up to a factor of $\frac{1}{2}$ (which is tight). Second, we apply this result in order to prove a $\frac{2}{3}p$ approximation for the general revenue problem, where $p$ is defined as the approximation ratio of the Max-Uncut Bisection problem. Since $p$ is known to be at least 0.8776 (Wu et al., 2015, Austrin et al., 2016), we get a 0.585 approximation algorithm for the revenue problem. This improves a sequence of earlier results which culminated in an 0.4246-approximation guarantee (Ahmadian et al., 2019).
Improved Hierarchical Clustering on Massive Datasets with Broad Guarantees
层次聚类是当今最有影响力的无监督学习方法之一：聚类的更强扩展。该方法的目标是创建一个层次的集群，从而构建集群的进化历史，同时发现聚类在所有分辨率。我们提出了层次聚类算法的四个特性：（1）经验性能，（2）理论保证，（3）集群平衡，（4）可扩展性。虽然许多算法被设计成一次实现其中一到两个特征，但没有一个算法实现了所有四个特征。灵感来自Bateni et al.的可扩展和经验上成功的亲和聚类[NeurIPs 2017]，我们介绍亲和聚类的继任者，匹配亲和聚类。与其前身一样，匹配亲和聚类保持了强大的经验性能，并使用大规模并行通信作为其分布式模型。旨在保持可证明平衡的集群，我们表明，我们的算法实现了良好的，恒定的因素近似Moseley和王的收入和科恩-Addad等人。的价值。我们表明亲和聚类不能近似任何一个函数。在此过程中，我们还引入了一个有效的$k$大小的最大匹配算法的MPC模型。
Hierarchical Clustering: Objective Functions and Algorithms
University of Copenhagen（哥本哈根大学）；University of Oxford（牛津大学）；The Alan Turing Institute（阿兰·图灵研究所）；École normale supérieure, CNRS, PSL Research University（巴黎高等师范学院，法国国家科学研究中心，巴黎文理研究大学）；Simon Fraser University（西蒙菲莎大学）
Exact and Approximate Hierarchical Clustering Using A*
National Institute of Standards and Technology（美国国家标准与技术研究院）；New York University（纽约大学）；University of Massachusetts Amherst（马萨诸塞大学阿默斯特分校）；Google Research（谷歌研究院）
