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）
Vincent Cohen-Addad, Varun Kanade, Frederik Mallmann-Trenn, Claire Mathieu
Hierarchical clustering is a recursive partitioning of a dataset into clusters at an increasingly finer granularity. Motivated by the fact that most work on hierarchical clustering was based on providing algorithms, rather than optimizing a specific objective, Dasgupta framed similarity-based hierarchical clustering as a combinatorial optimization problem, where a `good' hierarchical clustering is one that minimizes some cost function. He showed that this cost function has certain desirable properties.
We take an axiomatic approach to defining `good' objective functions for both similarity and dissimilarity-based hierarchical clustering. We characterize a set of "admissible" objective functions (that includes Dasgupta's one) that have the property that when the input admits a `natural' hierarchical clustering, it has an optimal value.
Equipped with a suitable objective function, we analyze the performance of practical algorithms, as well as develop better algorithms. For similarity-based hierarchical clustering, Dasgupta showed that the divisive sparsest-cut approach achieves an $O(\log^{3/2} n)$-approximation. We give a refined analysis of the algorithm and show that it in fact achieves an $O(\sqrt{\log n})$-approx. (Charikar and Chatziafratis independently proved that it is a $O(\sqrt{\log n})$-approx.). This improves upon the LP-based $O(\log n)$-approx. of Roy and Pokutta. For dissimilarity-based hierarchical clustering, we show that the classic average-linkage algorithm gives a factor 2 approx., and provide a simple and better algorithm that gives a factor 3/2 approx..
Finally, we consider `beyond-worst-case' scenario through a generalisation of the stochastic block model for hierarchical clustering. We show that Dasgupta's cost function has desirable properties for these inputs and we provide a simple 1 + o(1)-approximation in this setting.
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。
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）
