Hierarchical Agglomerative Clustering (HAC) algorithms are extensively utilized in modern data science, and seek to partition the dataset into clusters while generating a hierarchical relationship between the data samples. HAC algorithms are employed in many applications, such as biology, natural language processing, and recommender systems. Thus, it is imperative to ensure that these algorithms are fair -- even if the dataset contains biases against certain protected groups, the cluster outputs generated should not discriminate against samples from any of these groups. However, recent work in clustering fairness has mostly focused on center-based clustering algorithms, such as k-median and k-means clustering. In this paper, we propose fair algorithms for performing HAC that enforce fairness constraints 1) irrespective of the distance linkage criteria used, 2) generalize to any natural measures of clustering fairness for HAC, 3) work for multiple protected groups, and 4) have competitive running times to vanilla HAC. Through extensive experiments on multiple real-world UCI datasets, we show that our proposed algorithm finds fairer clusterings compared to vanilla HAC as well as other state-of-the-art fair clustering approaches.
Science Foundation Ireland（爱尔兰科学基金会）；Université de Montréal（蒙特利尔大学）
The Ward error sum of squares hierarchical clustering method has been very widely used since its first description by Ward in a 1963 publication. It has also been generalized in various ways. However there are different interpretations in the literature and there are different implementations of the Ward agglomerative algorithm in commonly used software systems, including differing expressions of the agglomerative criterion. Our survey work and case studies will be useful for all those involved in developing software for data analysis using Ward's hierarchical clustering method.
This manuscript develops the theory of agglomerative clustering with Bregman divergences. Geometric smoothing techniques are developed to deal with degenerate clusters. To allow for cluster models based on exponential families with overcomplete representations, Bregman divergences are developed for nondifferentiable convex functions.
Conventional multi-view clustering methods seek for a view consensus through minimizing the pairwise discrepancy between the consensus and subviews. However, the pairwise comparison cannot portray the inter-view relationship precisely if some of the subviews can be further agglomerated. To address the above challenge, we propose the agglomerative analysis to approximate the optimal consensus view, thereby describing the subview relationship within a view structure. We present Agglomerative Neural Network (ANN) based on Constrained Laplacian Rank to cluster multi-view data directly while avoidi…
Agglomerative hierarchical clustering (AHC) requires only the similarity between objects to be known. This is attractive when clustering signals of varying length, such as speech, which are not readily represented in fixed-dimensional vector space. However, AHC is characterised by $O(N^2)$ space and time complexity, making it infeasible for partitioning large datasets. This has recently been addressed by an approach based on the iterative re-clustering of independent subsets of the larger dataset. We show that, due to its iterative nature, this procedure can sometimes lead to unchecked growth …
Standard agglomerative clustering suggests establishing a new reliable linkage at every step. However, in order to provide adaptive, density-consistent and flexible solutions, we study extracting all the reliable linkages at each step, instead of the smallest one. Such a strategy can be applied with all common criteria for agglomerative hierarchical clustering. We also study that this strategy with the single linkage criterion yields a minimum spanning tree algorithm. We perform experiments on several real-world datasets to demonstrate the performance of this strategy compared to the standard …
Large scale agglomerative clustering is hindered by computational burdens. We propose a novel scheme where exact inter-instance distance calculation is replaced by the Hamming distance between Kernelized Locality-Sensitive Hashing (KLSH) hashed values. This results in a method that drastically decreases computation time. Additionally, we take advantage of certain labeled data points via distance metric learning to achieve a competitive precision and recall comparing to K-Means but in much less computation time.
University of Massachusetts Amherst（马萨诸塞大学阿默斯特分校）；Google LLC（谷歌有限责任公司）；Facebook（脸书）；University of Southern California（南加州大学）
The applicability of agglomerative clustering, for inferring both hierarchical and flat clustering, is limited by its scalability. Existing scalable hierarchical clustering methods sacrifice quality for speed and often lead to over-merging of clusters. In this paper, we present a scalable, agglomerative method for hierarchical clustering that does not sacrifice quality and scales to billions of data points. We perform a detailed theoretical analysis, showing that under mild separability conditions our algorithm can not only recover the optimal flat partition, but also provide a two-approximati…
Chinese University of Hong Kong（香港中文大学）；Shenzhen Research Institute of the Chinese University of Hong Kong（香港中文大学深圳研究院）
An agglomerative clustering of random variables is proposed, where clusters of random variables sharing the maximum amount of multivariate mutual information are merged successively to form larger clusters. Compared to the previous info-clustering algorithms, the agglomerative approach allows the computation to stop earlier when clusters of desired size and accuracy are obtained. An efficient algorithm is also derived based on the submodularity of entropy and the duality between the principal sequence of partitions and the principal sequence for submodular functions.
Energy companies often implement various demand response (DR) programs to better match electricity demand and supply by offering the consumers incentives to reduce their demand during critical periods. Classifying clients according to their consumption patterns enables targeting specific groups of consumers for DR. Traditional clustering algorithms use standard distance measurement to find the distance between two points. The results produced by clustering algorithms such as K-means, K-medoids, and Gaussian Mixture Models depend on the clustering parameters or initial clusters. In contrast, ou…
The HCV package implements the hierarchical clustering for spatial data. It requires clustering results not only homogeneous in non-geographical features among samples but also geographically close to each other within a cluster. We modified typically used hierarchical agglomerative clustering algorithms to introduce the spatial homogeneity, by considering geographical locations as vertices and converting spatial adjacency into whether a shared edge exists between a pair of vertices. The main function HCV obeying constraints of the vertex links automatically enforces the spatial contiguity pro…
Partial orders and directed acyclic graphs are commonly recurring data structures that arise naturally in numerous domains and applications and are used to represent ordered relations between entities in the domains. Examples are task dependencies in a project plan, transaction order in distributed ledgers and execution sequences of tasks in computer programs, just to mention a few. We study the problem of order preserving hierarchical clustering of this kind of ordered data. That is, if we have $a < b$ in the original data and denote their respective clusters by $[a]$ and $[b]$, then we shall…
Hierarchical Agglomerative Clustering (HAC) is one of the oldest but still most widely used clustering methods. However, HAC is notoriously hard to scale to large data sets as the underlying complexity is at least quadratic in the number of data points and many algorithms to solve HAC are inherently sequential. In this paper, we propose {Reciprocal Agglomerative Clustering (RAC)}, a distributed algorithm for HAC, that uses a novel strategy to efficiently merge clusters in parallel. We prove theoretically that RAC recovers the exact solution of HAC. Furthermore, under clusterability and balance…
University of São Paulo（圣保罗大学）；Federal University of São Carlos（圣卡洛斯联邦大学）
An important issue in clustering concerns the avoidance of false positives while searching for clusters. This work addressed this problem considering agglomerative methods, namely single, average, median, complete, centroid and Ward's approaches applied to unimodal and bimodal datasets obeying uniform, gaussian, exponential and power-law distributions. A model of clusters was also adopted, involving a higher density nucleus surrounded by a transition, followed by outliers. This paved the way to defining an objective means for identifying the clusters from dendrograms. The adopted model also al…
Pohang University of Science and Technology（浦项科技大学）
Bayesian hierarchical clustering (BHC) is an agglomerative clustering method, where a probabilistic model is defined and its marginal likelihoods are evaluated to decide which clusters to merge. While BHC provides a few advantages over traditional distance-based agglomerative clustering algorithms, successive evaluation of marginal likelihoods and careful hyperparameter tuning are cumbersome and limit the scalability. In this paper we relax BHC into a non-probabilistic formulation, exploring small-variance asymptotics in conjugate-exponential models. We develop a novel clustering algorithm, re…
An agglomerative hierarchical clustering (AHC) framework and algorithm named HOSil based on a new linkage metric optimized by the average silhouette width (ASW) index is proposed. A conscientious investigation of various clustering methods and estimation indices is conducted across a diverse verities of data structures for three aims: a) clustering quality, b) clustering recovery, and c) estimation of number of clusters. HOSil has shown better clustering quality for a range of artificial and real world data structures as compared to k-means, PAM, single, complete, average, Ward, McQuitty, spec…
University of Victoria（维多利亚大学）；Pacific Forestry Centre, National Resources Canada（加拿大自然资源部太平洋林业中心）
A clustering termination procedure which is locally adaptive (with respect to the hierarchical tree of sets representative of the agglomerative merging) is proposed, for agglomerative hierarchical clustering on a set equipped with a distance function. It represents a multi-scale alternative to conventional scale dependent threshold based termination criteria.
Previous works have shown that spatial location information can be complementary to speaker embeddings for a speaker diarisation task. However, the models used often assume that speakers are fairly stationary throughout a meeting. This paper proposes to relax this assumption, by explicitly modelling the movements of speakers within an Agglomerative Hierarchical Clustering (AHC) diarisation framework. Kalman filters, which track the locations of speakers, are used to compute log-likelihood ratios that contribute to the cluster affinity computations for the AHC merging and stopping decisions. Ex…
Indian Institute of Science Education and Research（印度科学教育与研究院）
Author names often suffer from ambiguity owing to the same author appearing under different names and multiple authors possessing similar names. It creates difficulty in associating a scholarly work with the person who wrote it, thereby introducing inaccuracy in credit attribution, bibliometric analysis, search-by-author in a digital library, and expert discovery. A plethora of techniques for disambiguation of author names has been proposed in the literature. I try to focus on the research efforts targeted to disambiguate author names. I first go through the conventional methods, then I discus…
Schloss Dagstuhl – Leibniz Center for Informatics（达格斯托尔城堡–莱布尼茨信息学中心）；University of Paderborn（帕德博恩大学）；TU Dortmund（多特蒙德工业大学）
The diameter $k$-clustering problem is the problem of partitioning a finite subset of $\mathbb{R}^d$ into $k$ subsets called clusters such that the maximum diameter of the clusters is minimized. One early clustering algorithm that computes a hierarchy of approximate solutions to this problem (for all values of $k$) is the agglomerative clustering algorithm with the complete linkage strategy. For decades, this algorithm has been widely used by practitioners. However, it is not well studied theoretically. In this paper, we analyze the agglomerative complete linkage clustering algorithm. Assuming…
We study the widely used hierarchical agglomerative clustering (HAC) algorithm on edge-weighted graphs. We define an algorithmic framework for hierarchical agglomerative graph clustering that provides the first efficient $\tilde{O}(m)$ time exact algorithms for classic linkage measures, such as complete- and WPGMA-linkage, as well as other measures. Furthermore, for average-linkage, arguably the most popular variant of HAC, we provide an algorithm that runs in $\tilde{O}(n\sqrt{m})$ time. For this variant, this is the first exact algorithm that runs in subquadratic time, as long as $m=n^{2-ε}$…
Carnegie Mellon University（卡内基梅隆大学）；Georgia Institute of Technology（佐治亚理工学院）；Google, Inc.（谷歌公司）
One of the most widely used techniques for data clustering is agglomerative clustering. Such algorithms have been long used across many different fields ranging from computational biology to social sciences to computer vision in part because their output is easy to interpret. Unfortunately, it is well known, however, that many of the classic agglomerative clustering algorithms are not robust to noise. In this paper we propose and analyze a new robust algorithm for bottom-up agglomerative clustering. We show that our algorithm can be used to cluster accurately in cases where the data satisfies …
Microsoft Research（微软研究院）；Massachusetts Institute of Technology（麻省理工学院）
We examine methods for clustering in high dimensions. In the first part of the paper, we perform an experimental comparison between three batch clustering algorithms: the Expectation-Maximization (EM) algorithm, a winner take all version of the EM algorithm reminiscent of the K-means algorithm, and model-based hierarchical agglomerative clustering. We learn naive-Bayes models with a hidden root node, using high-dimensional discrete-variable data sets (both real and synthetic). We find that the EM algorithm significantly outperforms the other methods, and proceed to investigate the effect of va…
Heidelberg Collaboratory for Image Processing（海德堡图像处理合作实验室）；EMBL（欧洲分子生物学实验室）；University Göttingen（哥廷根大学）；MRC Laboratory of Molecular Biology（英国医学研究委员会分子生物学实验室）
We propose a theoretical framework that generalizes simple and fast algorithms for hierarchical agglomerative clustering to weighted graphs with both attractive and repulsive interactions between the nodes. This framework defines GASP, a Generalized Algorithm for Signed graph Partitioning, and allows us to explore many combinations of different linkage criteria and cannot-link constraints. We prove the equivalence of existing clustering methods to some of those combinations and introduce new algorithms for combinations that have not been studied before. We study both theoretical and empirical …
Khulna University of Engineering & Technology（库尔纳工程技术大学）
Handwritten numerals of different languages have various characteristics. Similarities and dissimilarities of the languages can be measured by analyzing the extracted features of the numerals. Handwritten numeral datasets are available and accessible for many renowned languages of different regions. In this paper, several handwritten numeral datasets of different languages are collected. Then they are used to find the similarity among those written languages through determining and comparing the similitude of each handwritten numerals. This will help to find which languages have the same or ad…
Georgia Institute of Technology（佐治亚理工学院）；Magic Leap, Inc.（Magic Leap公司）
Mean Shift today, is widely used for mode detection and clustering. The technique though, is challenged in practice due to assumptions of isotropicity and homoscedasticity. We present an adaptive Mean Shift methodology that allows for full anisotropic clustering, through unsupervised local bandwidth selection. The bandwidth matrices evolve naturally, adapting locally through agglomeration, and in turn guiding further agglomeration. The online methodology is practical and effecive for low-dimensional feature spaces, preserving better detail and clustering salience. Additionally, conventional Me…
MIT CSAIL（麻省理工学院计算机科学与人工智能实验室）；UC Riverside（加利福尼亚大学河滨分校）
This paper studies the hierarchical clustering problem, where the goal is to produce a dendrogram that represents clusters at varying scales of a data set. We propose the ParChain framework for designing parallel hierarchical agglomerative clustering (HAC) algorithms, and using the framework we obtain novel parallel algorithms for the complete linkage, average linkage, and Ward's linkage criteria. Compared to most previous parallel HAC algorithms, which require quadratic memory, our new algorithms require only linear memory, and are scalable to large data sets. ParChain is based on our paralle…
University of Heilbronn（海尔布隆大学）；Refinitiv Labs（路孚特实验室）；University of Sheffield（谢菲尔德大学）
We introduce Topic Grouper as a complementary approach in the field of probabilistic topic modeling. Topic Grouper creates a disjunctive partitioning of the training vocabulary in a stepwise manner such that resulting partitions represent topics. It is governed by a simple generative model, where the likelihood to generate the training documents via topics is optimized. The algorithm starts with one-word topics and joins two topics at every step. It therefore generates a solution for every desired number of topics ranging between the size of the training vocabulary and one. The process represe…
Columbia University（哥伦比亚大学）；Recognition Technologies, Inc.（识别技术公司）
Majority of speech signals across different scenarios are never available with well-defined audio segments containing only a single speaker. A typical conversation between two speakers consists of segments where their voices overlap, interrupt each other or halt their speech in between multiple sentences. Recent advancements in diarization technology leverage neural network-based approaches to improvise multiple subsystems of speaker diarization system comprising of extracting segment-wise embedding features and detecting changes in the speaker during conversation. However, to identify speaker…
Convex clustering is an attractive clustering algorithm with favorable properties such as efficiency and optimality owing to its convex formulation. It is thought to generalize both k-means clustering and agglomerative clustering. However, it is not known whether convex clustering preserves desirable properties of these algorithms. A common expectation is that convex clustering may learn difficult cluster types such as non-convex ones. Current understanding of convex clustering is limited to only consistency results on well-separated clusters. We show new understanding of its solutions. We pro…
Howard Hughes Medical Institute（霍华德·休斯医学研究所）；University of Pennsylvania（宾夕法尼亚大学）
We aim to improve segmentation through the use of machine learning tools during region agglomeration. We propose an active learning approach for performing hierarchical agglomerative segmentation from superpixels. Our method combines multiple features at all scales of the agglomerative process, works for data with an arbitrary number of dimensions, and scales to very large datasets. We advocate the use of variation of information to measure segmentation accuracy, particularly in 3D electron microscopy (EM) images of neural tissue, and using this metric demonstrate an improvement over competing…
Hierarchical graph clustering is a common technique to reveal the multi-scale structure of complex networks. We propose a novel metric for assessing the quality of a hierarchical clustering. This metric reflects the ability to reconstruct the graph from the dendrogram, which encodes the hierarchy. The optimal representation of the graph defines a class of reducible linkages leading to regular dendrograms by greedy agglomerative clustering.
Missouri University of Science and Technology（密苏里科技大学）
This paper presents a novel adaptive resonance theory (ART)-based modular architecture for unsupervised learning, namely the distributed dual vigilance fuzzy ART (DDVFA). DDVFA consists of a global ART system whose nodes are local fuzzy ART modules. It is equipped with the distinctive features of distributed higher-order activation and match functions, using dual vigilance parameters responsible for cluster similarity and data quantization. Together, these allow DDVFA to perform unsupervised modularization, create multi-prototype clustering representations, retrieve arbitrarily-shaped clusters…
We study the question of fair clustering under the {\em disparate impact} doctrine, where each protected class must have approximately equal representation in every cluster. We formulate the fair clustering problem under both the $k$-center and the $k$-median objectives, and show that even with two protected classes the problem is challenging, as the optimum solution can violate common conventions---for instance a point may no longer be assigned to its nearest cluster center! En route we introduce the concept of fairlets, which are minimal sets that satisfy fair representation while approximat…
University of Washington（华盛顿大学）；Rutgers University（罗格斯大学）；Google（谷歌）
A common distinction in fair machine learning, in particular in fair classification, is between group fairness and individual fairness. In the context of clustering, group fairness has been studied extensively in recent years; however, individual fairness for clustering has hardly been explored. In this paper, we propose a natural notion of individual fairness for clustering. Our notion asks that every data point, on average, is closer to the points in its own cluster than to the points in any other cluster. We study several questions related to our proposed notion of individual fairness. On t…
National Tsing Hua University（国立清华大学）；National Taiwan University（国立台湾大学）
Clustering is one of the most fundamental problems in data analysis and it has been studied extensively in the literature. Though many clustering algorithms have been proposed, clustering theories that justify the use of these clustering algorithms are still unsatisfactory. In particular, one of the fundamental challenges is to address the following question: What is a cluster in a set of data points? In this paper, we make an attempt to address such a question by considering a set of data points associated with a distance measure (metric). We first propose a new cohesion measure in terms of t…
We propose a general variational framework of fair clustering, which integrates an original Kullback-Leibler (KL) fairness term with a large class of clustering objectives, including prototype or graph based. Fundamentally different from the existing combinatorial and spectral solutions, our variational multi-term approach enables to control the trade-off levels between the fairness and clustering objectives. We derive a general tight upper bound based on a concave-convex decomposition of our fairness term, its Lipschitz-gradient property and the Pinsker's inequality. Our tight upper bound can…
Department of Computer Science, UC Davis（加州大学戴维斯分校计算机科学系）
Fair clustering under the disparate impact doctrine requires that population of each protected group should be approximately equal in every cluster. Previous work investigated a difficult-to-scale pre-processing step for $k$-center and $k$-median style algorithms for the special case of this problem when the number of protected groups is two. In this work, we consider a more general and practical setting where there can be many protected groups. To this end, we propose Deep Fair Clustering, which learns a discriminative but fair cluster assignment function. The experimental results on three pu…
We propose a nearest neighbor based clustering algorithm that results in a naturally defined hierarchy of clusters. In contrast to the agglomerative and divisive hierarchical clustering algorithms, our approach is not dependent on the iterative working of the algorithm, in the sense that the partitions of the hierarchical clusters are purely defined in accordance with the input dataset. Our method is a universal hierarchical clustering approach since it can be implemented as bottom up or top down versions, both of which result in the same clustering. We show that for certain types of datasets,…
Indian Institute of Technology Madras（印度理工学院马德拉斯分校）；Queen’s University Belfast（贝尔法斯特女王大学）
A clustering may be considered as fair on pre-specified sensitive attributes if the proportions of sensitive attribute groups in each cluster reflect that in the dataset. In this paper, we consider the task of fair clustering for scenarios involving multiple multi-valued or numeric sensitive attributes. We propose a fair clustering method, \textit{FairKM} (Fair K-Means), that is inspired by the popular K-Means clustering formulation. We outline a computational notion of fairness which is used along with a cluster coherence objective, to yield the FairKM clustering method. We empirically evalua…
Deep clustering has the potential to learn a strong representation and hence better clustering performance compared to traditional clustering methods such as $k$-means and spectral clustering. However, this strong representation learning ability may make the clustering unfair by discovering surrogates for protected information which we empirically show in our experiments. In this work, we study a general notion of group-level fairness for both binary and multi-state protected status variables (PSVs). We begin by formulating the group-level fairness problem as an integer linear programming form…
Department of Statistics, Seoul National University, Republic of Korea（统计系，首尔国立大学，大韩民国）；School of Law, Seoul National University, Republic of Korea（法学院，首尔国立大学，大韩民国）
Within the relatively busy area of fair machine learning that has been dominated by classification fairness research, fairness in clustering has started to see some recent attention. In this position paper, we assess the existing work in fair clustering and observe that there are several directions that are yet to be explored, and postulate that the state-of-the-art in fair clustering has been quite parochial in outlook. We posit that widening the normative principles to target for, characterizing shortfalls where the target cannot be achieved fully, and making use of knowledge of downstream p…
Given the widespread popularity of spectral clustering (SC) for partitioning graph data, we study a version of constrained SC in which we try to incorporate the fairness notion proposed by Chierichetti et al. (2017). According to this notion, a clustering is fair if every demographic group is approximately proportionally represented in each cluster. To this end, we develop variants of both normalized and unnormalized constrained SC and show that they help find fairer clusterings on both synthetic and real data. We also provide a rigorous theoretical analysis of our algorithms on a natural vari…
Google（谷歌）；University of Maryland（马里兰大学）；Carnegie Mellon University（卡内基梅隆大学）
As machine learning has become more prevalent, researchers have begun to recognize the necessity of ensuring machine learning systems are fair. Recently, there has been an interest in defining a notion of fairness that mitigates over-representation in traditional clustering. In this paper we extend this notion to hierarchical clustering, where the goal is to recursively partition the data to optimize a specific objective. For various natural objectives, we obtain simple, efficient algorithms to find a provably good fair hierarchical clustering. Empirically, we show that our algorithms can find…
University of Massachusetts Amherst（马萨诸塞大学阿默斯特分校）；Microsoft Research（微软研究院）；IBM（国际商业机器公司）
We introduce Grinch, a new algorithm for large-scale, non-greedy hierarchical clustering with general linkage functions that compute arbitrary similarity between two point sets. The key components of Grinch are its rotate and graft subroutines that efficiently reconfigure the hierarchy as new points arrive, supporting discovery of clusters with complex structure. Grinch is motivated by a new notion of separability for clustering with linkage functions: we prove that when the model is consistent with a ground-truth clustering, Grinch is guaranteed to produce a cluster tree containing the ground…
