TY - JOUR

T1 - Hierarchical Clustering via Sketches and Hierarchical Correlation Clustering

AU - Vainstein, Danny

AU - Chatziafratis, Vaggos

AU - Citovsky, Gui

AU - Rajagopalan, Anand

AU - Mahdian, Mohammad

AU - Azar, Yossi

N1 - Publisher Copyright:
Copyright © 2021 by the author(s)

PY - 2021

Y1 - 2021

N2 - Recently, Hierarchical Clustering (HC) has been considered through the lens of optimization. In particular, two maximization objectives have been defined. Moseley and Wang defined the Revenue objective to handle similarity information given by a weighted graph on the data points (w.l.o.g., [0, 1] weights), while Cohen-Addad et al. defined the Dissimilarity objective to handle dissimilarity information. In this paper, we prove structural lemmas for both objectives allowing us to convert any HC tree to a tree with constant number of internal nodes while incurring an arbitrarily small loss in each objective. Although the best-known approximations are 0.585 and 0.667 respectively, using our lemmas we obtain approximations arbitrarily close to 1, if not all weights are small (i.e., there exist constants ∊, δ such that the fraction of weights smaller than δ, is at most 1 − ∊); such instances encompass many metric-based similarity instances, thereby improving upon prior work. Finally, we introduce Hierarchical Correlation Clustering (HCC) to handle instances that contain similarity and dissimilarity information simultaneously. For HCC, we provide an approximation of 0.4767 and for complementary similarity/dissimilarity weights (analogous to +/− correlation clustering), we again present nearly-optimal approximations.

AB - Recently, Hierarchical Clustering (HC) has been considered through the lens of optimization. In particular, two maximization objectives have been defined. Moseley and Wang defined the Revenue objective to handle similarity information given by a weighted graph on the data points (w.l.o.g., [0, 1] weights), while Cohen-Addad et al. defined the Dissimilarity objective to handle dissimilarity information. In this paper, we prove structural lemmas for both objectives allowing us to convert any HC tree to a tree with constant number of internal nodes while incurring an arbitrarily small loss in each objective. Although the best-known approximations are 0.585 and 0.667 respectively, using our lemmas we obtain approximations arbitrarily close to 1, if not all weights are small (i.e., there exist constants ∊, δ such that the fraction of weights smaller than δ, is at most 1 − ∊); such instances encompass many metric-based similarity instances, thereby improving upon prior work. Finally, we introduce Hierarchical Correlation Clustering (HCC) to handle instances that contain similarity and dissimilarity information simultaneously. For HCC, we provide an approximation of 0.4767 and for complementary similarity/dissimilarity weights (analogous to +/− correlation clustering), we again present nearly-optimal approximations.

UR - http://www.scopus.com/inward/record.url?scp=85119946832&partnerID=8YFLogxK

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AN - SCOPUS:85119946832

SN - 2640-3498

VL - 130

SP - 559

EP - 567

JO - Proceedings of Machine Learning Research

JF - Proceedings of Machine Learning Research

T2 - 24th International Conference on Artificial Intelligence and Statistics, AISTATS 2021

Y2 - 13 April 2021 through 15 April 2021

ER -