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 have $[a] < [b]$ in the produced clustering. The clustering is similarity based and uses standard linkage functions, such as single- and complete linkage, and is an extension of classical hierarchical clustering.
To achieve this, we define the output from running classical hierarchical clustering on strictly ordered data to be partial dendrograms; sub-trees of classical dendrograms with several connected components. We then construct an embedding of partial dendrograms over a set into the family of ultrametrics over the same set. An optimal hierarchical clustering is defined as the partial dendrogram corresponding to the ultrametric closest to the original dissimilarity measure, measured in the p-norm. Thus, the method is a combination of classical hierarchical clustering and ultrametric fitting.
A reference implementation is employed for experiments on both synthetic random data and real world data from a database of machine parts. When compared to existing methods, the experiments show that our method excels both in cluster quality and order preservation.
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 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.
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…
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…
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…
Inria（法国国家信息与自动化研究所）；CEA（法国原子能和替代能源委员会）；University Paris-Saclay（巴黎萨克雷大学）；Institut de Mathématiques de Toulouse（图卢兹数学研究所）；Université de Toulouse（图卢兹大学）；CNRS（法国国家科学研究中心）
In this work, we revisit fast dimension reduction approaches, as with random projections and random sampling. Our goal is to summarize the data to decrease computational costs and memory footprint of subsequent analysis. Such dimension reduction can be very efficient when the signals of interest have a strong structure, such as with images. We focus on this setting and investigate feature clustering schemes for data reductions that capture this structure. An impediment to fast dimension reduction is that good clustering comes with large algorithmic costs. We address it by contributing a linear…
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…
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…
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 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,…
University of California, San Diego（加利福尼亚大学圣地亚哥分校）
Deep embedding methods have influenced many areas of unsupervised learning. However, the best methods for learning hierarchical structure use non-Euclidean representations, whereas Euclidean geometry underlies the theory behind many hierarchical clustering algorithms. To bridge the gap between these two areas, we consider learning a non-linear embedding of data into Euclidean space as a way to improve the hierarchical clustering produced by agglomerative algorithms. To learn the embedding, we revisit using a variational autoencoder with a Gaussian mixture prior, and we show that rescaling the …
There has been a surge in the number of large and flat data sets - data sets containing a large number of features and a relatively small number of observations - due to the growing ability to collect and store information in medical research and other fields. Hierarchical clustering is a widely used clustering tool. In hierarchical clustering, large and flat data sets may allow for a better coverage of clustering features (features that help explain the true underlying clusters) but, such data sets usually include a large fraction of noise features (non-clustering features) that may hide the …
Kansas State University（堪萨斯州立大学）；University of California, Berkeley（加州大学伯克利分校）
As the size $n$ of datasets become massive, many commonly-used clustering algorithms (for example, $k$-means or hierarchical agglomerative clustering (HAC) require prohibitive computational cost and memory. In this paper, we propose a solution to these clustering problems by extending threshold clustering (TC) to problems of instance selection. TC is a recently developed clustering algorithm designed to partition data into many small clusters in linearithmic time (on average). Our proposed clustering method is as follows. First, TC is performed and clusters are reduced into single "prototype" …
Meila (2018) introduces an optimization based method called the Sublevel Set method, to guarantee that a clustering is nearly optimal and "approximately correct" without relying on any assumptions about the distribution that generated the data. This paper extends the Sublevel Set method to the cost-based hierarchical clustering paradigm proposed by Dasgupta (2016).
Université Paul Verlaine-Metz（保罗·魏尔伦-梅斯大学）；Université Paris 1 Panthéon-Sorbonne（巴黎第一大学 先贤祠-索邦）
We introduce in this paper a new way of optimizing the natural extension of the quantization error using in k-means clustering to dissimilarity data. The proposed method is based on hierarchical clustering analysis combined with multi-level heuristic refinement. The method is computationally efficient and achieves better quantization errors than the
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…
Recent work on dissimilarity-based hierarchical clustering has led to the introduction of global objective functions for this classical problem. Several standard approaches, such as average linkage, as well as some new heuristics have been shown to provide approximation guarantees. Here we introduce a broad new class of objective functions which satisfy desirable properties studied in prior work. Many common agglomerative and divisive clustering methods are shown to be greedy algorithms for these objectives, which are inspired by related concepts in phylogenetics.
Bagging and boosting are proved to be the best methods of building multiple classifiers in classification combination problems. In the area of "flat clustering" problems, it is also recognized that multi-clustering methods based on boosting provide clusterings of an improved quality. In this paper, we introduce a novel multi-clustering method for "hierarchical clusterings" based on boosting theory, which creates a more stable hierarchical clustering of a dataset. The proposed algorithm includes a boosting iteration in which a bootstrap of samples is created by weighted random sampling of eleme…
Data clustering is an approach to seek for structure in sets of complex data, i.e., sets of "objects". The main objective is to identify groups of objects which are similar to each other, e.g., for classification. Here, an introduction to clustering is given and three basic approaches are introduced: the k-means algorithm, neighbour-based clustering, and an agglomerative clustering method. For all cases, C source code examples are given, allowing for an easy implementation.
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…
The development of algorithms for hierarchical clustering has been hampered by a shortage of precise objective functions. To help address this situation, we introduce a simple cost function on hierarchies over a set of points, given pairwise similarities between those points. We show that this criterion behaves sensibly in canonical instances and that it admits a top-down construction procedure with a provably good approximation ratio.
In the election of a hierarchical clustering method, theoretic properties may give some insight to determine which method is the most suitable to treat a clustering problem. Herein, we study some basic properties of two hierarchical clustering methods: $α$-unchaining single linkage or $SL(α)$ and a modified version of this one, $SL^*(α)$. We compare the results with the properties satisfied by the classical linkage-based hierarchical clustering methods.
The problem of hierarchical clustering items from pairwise similarities is found across various scientific disciplines, from biology to networking. Often, applications of clustering techniques are limited by the cost of obtaining similarities between pairs of items. While prior work has been developed to reconstruct clustering using a significantly reduced set of pairwise similarities via adaptive measurements, these techniques are only applicable when choice of similarities are available to the user. In this paper, we examine reconstructing hierarchical clustering under similarity observation…
