The Importance of Good Starting Solutions in the Minimum Sum of Squares Clustering Problem
Steven G. Mihaylo College of Business and Economics（史蒂文·G·米哈伊洛商业与经济学院）；California State University-Fullerton（加州州立大学富勒顿分校）；The Royal Military College of Canada（加拿大皇家军事学院）
The clustering problem has many applications in Machine Learning, Operations Research, and Statistics. We propose three algorithms to create starting solutions for improvement algorithms for this problem. We test the algorithms on 72 instances that were investigated in the literature. Forty eight of them are relatively easy to solve and we found the best known solution many times for all of them. Twenty four medium and large size instances are more challenging. We found five new best known solutions and matched the best known solution for 18 of the remaining 19 instances.
The development of algorithms for unsupervised pattern recognition by nonlinear clustering is a notable problem in data science. Markov clustering (MCL) is a renowned algorithm that simulates stochastic flows on a network of sample similarities to detect the structural organization of clusters in the data, but it has never been generalized to deal with data nonlinearity. Minimum Curvilinearity (MC) is a principle that approximates nonlinear sample distances in the high-dimensional feature space by curvilinear distances, which are computed as transversal paths over their minimum spanning tree, …
Universidad de Extremadura（埃斯特雷马杜拉大学）；IES Sierra La Calera（谢拉-拉卡莱拉中学）
The adjusted Rand index (ARI) is commonly used in cluster analysis to measure the degree of agreement between two data partitions. Since its introduction, exploring the situations of extreme agreement and disagreement under different circumstances has been a subject of interest, in order to achieve a better understanding of this index. Here, an explicit formula for the lowest possible value of the ARI for two clusterings of given sizes is shown, and moreover a specific pair of clusterings achieving such a bound is provided.
Clustering is an unsupervised learning task that aims to partition data into a set of clusters. In many applications, these clusters correspond to real-world constructs (e.g. electoral districts) whose benefit can only be attained by groups when they reach a minimum level of representation (e.g. 50\% to elect their desired candidate). This paper considers the problem of performing k-means clustering while ensuring groups (e.g. demographic groups) have that minimum level of representation in a specified number of clusters. We show that the popular $k$-means algorithm, Lloyd's algorithm, can res…
University of California Los Angeles（加利福尼亚大学洛杉矶分校）；University of California Davis（加利福尼亚大学戴维斯分校）；University of California at Davis（加利福尼亚大学戴维斯分校）
The two-step spectral clustering method, which consists of the Laplacian eigenmap and a rounding step, is a widely used method for graph partitioning. It can be seen as a natural relaxation to the NP-hard minimum ratio cut problem. In this paper we study the central question: when is spectral clustering able to find the global solution to the minimum ratio cut problem? First we provide a condition that naturally depends on the intra- and inter-cluster connectivities of a given partition under which we may certify that this partition is the solution to the minimum ratio cut problem. Then we dev…
Sapienza University of Rome（罗马第一大学）；University of Rome Tor Vergata（罗马第二大学）
The minimum sum-of-squares clustering (MSSC), or k-means type clustering, is traditionally considered an unsupervised learning task. In recent years, the use of background knowledge to improve the cluster quality and promote interpretability of the clustering process has become a hot research topic at the intersection of mathematical optimization and machine learning research. The problem of taking advantage of background information in data clustering is called semi-supervised or constrained clustering. In this paper, we present a branch-and-cut algorithm for semi-supervised MSSC, where backg…
University of Electronic Science and Technology of China（电子科技大学）；University of Minnesota（明尼苏达大学）
One of the main challenges for hierarchical clustering is how to appropriately identify the representative points in the lower level of the cluster tree, which are going to be utilized as the roots in the higher level of the cluster tree for further aggregation. However, conventional hierarchical clustering approaches have adopted some simple tricks to select the "representative" points which might not be as representative as enough. Thus, the constructed cluster tree is less attractive in terms of its poor robustness and weak reliability. Aiming at this issue, we propose a novel hierarchical …
TU Dortmund University（多特蒙德工业大学）；Technische Universiteit Eindhoven（埃因霍温理工大学）
When it comes to clustering nonconvex shapes, two paradigms are used to find the most suitable clustering: minimum cut and maximum density. The most popular algorithms incorporating these paradigms are Spectral Clustering and DBSCAN. Both paradigms have their pros and cons. While minimum cut clusterings are sensitive to noise, density-based clusterings have trouble handling clusters with varying densities. In this paper, we propose \textsc{SpectACl}: a method combining the advantages of both approaches, while solving the two mentioned drawbacks. Our method is easy to implement, such as spectra…
Monash University（莫纳什大学）；Allstate Research and Planning Center（好事达研究与规划中心）
The K-Mean and EM algorithms are popular in clustering and mixture modeling, due to their simplicity and ease of implementation. However, they have several significant limitations. Both coverage to a local optimum of their respective objective functions (ignoring the uncertainty in the model space), require the apriori specification of the number of classes/clsuters, and are inconsistent. In this work we overcome these limitations by using the Minimum Message Length (MML) principle and a variation to the K-Means/EM observation assignment and parameter calculation scheme. We maintain the simpli…
Chongqing University of Telecommunications and Posts（重庆邮电大学）
Most of the existing clustering methods are based on a single granularity of information, such as the distance and density of each data. This most fine-grained based approach is usually inefficient and susceptible to noise. Therefore, we propose a clustering algorithm that combines multi-granularity Granular-Ball and minimum spanning tree (MST). We construct coarsegrained granular-balls, and then use granular-balls and MST to implement the clustering method based on "large-scale priority", which can greatly avoid the influence of outliers and accelerate the construction process of MST. Experim…
Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio)（里约热内卢天主教大学）
Minimum sum-of-squares clustering (MSSC) is a widely used clustering model, of which the popular K-means algorithm constitutes a local minimizer. It is well known that the solutions of K-means can be arbitrarily distant from the true MSSC global optimum, and dozens of alternative heuristics have been proposed for this problem. However, no other algorithm has been predominantly adopted in the literature. This may be related to differences of computational effort, or to the assumption that a near-optimal solution of the MSSC has only a marginal impact on clustering validity. In this article, we …
Fair Clustering with Minimum Representation Constraints
Department of Management Science and Engineering, Stanford University（斯坦福大学管理科学与工程系）；H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology（佐治亚理工学院H. Milton Stewart工业与系统工程学院）
Lancaster University（兰卡斯特大学）；Liverpool John Moores University（利物浦约翰摩尔大学）
Associating distinct groups of objects (clusters) with contiguous regions of high probability density (high-density clusters), is central to many statistical and machine learning approaches to the classification of unlabelled data. We propose a novel hyperplane classifier for clustering and semi-supervised classification which is motivated by this objective. The proposed minimum density hyperplane minimises the integral of the empirical probability density function along it, thereby avoiding intersection with high density clusters. We show that the minimum density and the maximum margin hyperp…
MIT CSAIL（麻省理工学院计算机科学与人工智能实验室）；UC Riverside（加州大学河滨分校）
This paper presents new parallel algorithms for generating Euclidean minimum spanning trees and spatial clustering hierarchies (known as HDBSCAN$^*$). Our approach is based on generating a well-separated pair decomposition followed by using Kruskal's minimum spanning tree algorithm and bichromatic closest pair computations. We introduce a new notion of well-separation to reduce the work and space of our algorithm for HDBSCAN$^*$. We also present a parallel approximate algorithm for OPTICS based on a recent sequential algorithm by Gan and Tao. Finally, we give a new parallel divide-and-conquer …
We present a novel model called One Class Minimum Spanning Tree (OCmst) for novelty detection problem that uses a Convolutional Neural Network (CNN) as deep feature extractor and graph-based model based on Minimum Spanning Tree (MST). In a novelty detection scenario, the training data is no polluted by outliers (abnormal class) and the goal is to recognize if a test instance belongs to the normal class or to the abnormal class. Our approach uses the deep features from CNN to feed a pair of MSTs built starting from each test instance. To cut down the computational time we use a parameter $γ$ to…
Minimum Description Length based Granular-Ball Tree Regularization for Spectral Clustering
Department of Mathematics and Computer Science, Gannan Normal University（赣南师范大学数学与计算机科学学院）；Key Laboratory of Data Science and Artificial Intelligence of Jiangxi Education Institutes, Gannan Normal University（江西省教育研究院数据科学与人工智能重点实验室）
United Arab Emirates University（阿拉伯联合酋长国大学）；Toronto Rehabilitation Institute, University Health Network（多伦多康复研究所，大学健康网络）
Mixed datasets consist of both numeric and categorical attributes. Various k-means-based clustering algorithms have been developed for these datasets. Generally, these algorithms use random partition as a starting point, which tends to produce different clustering results for different runs. In this paper, we propose, initKmix, a novel algorithm for finding an initial partition for k-means-based clustering algorithms for mixed datasets. In the initKmix algorithm, a k-means-based clustering algorithm is run many times, and in each run, one of the attributes is used to create initial clusters fo…
Biclustering is the task of simultaneously clustering the rows and columns of the data matrix into different subgroups such that the rows and columns within a subgroup exhibit similar patterns. In this paper, we consider the case of producing block-diagonal biclusters. We provide a new formulation of the biclustering problem based on the idea of minimizing the empirical clustering risk. We develop and prove a consistency result with respect to the empirical clustering risk. Since the optimization problem is combinatorial in nature, finding the global minimum is computationally intractable. In …
In spectral clustering and spectral image segmentation, the data is partioned starting from a given matrix of pairwise similarities S. the matrix S is constructed by hand, or learned on a separate training set. In this paper we show how to achieve spectral clustering in unsupervised mode. Our algorithm starts with a set of observed pairwise features, which are possible components of an unknown, parametric similarity function. This function is learned iteratively, at the same time as the clustering of the data. The algorithm shows promosing results on synthetic and real data.
Harbin Institute of Technology（哈尔滨工业大学）；Beihang University（北京航空航天大学）；Jilin University（吉林大学）
The purpose of this paper is to improve the traditional K-means algorithm. In the traditional K mean clustering algorithm, the initial clustering centers are generated randomly in the data set. It is easy to fall into the local minimum solution when the initial cluster centers are randomly generated. The initial clustering center selected by K-means clustering algorithm which based on density is more representative. The experimental results show that the improved K clustering algorithm can eliminate the dependence on the initial cluster, and the accuracy of clustering is improved.
IBM Thomas J. Watson Research Center（IBM托马斯·J·沃森研究中心）
Spectral clustering has shown a superior performance in analyzing the cluster structure. However, its computational complexity limits its application in analyzing large-scale data. To address this problem, many low-rank matrix approximating algorithms are proposed, including the Nystrom method - an approach with proven approximate error bounds. There are several algorithms that provide recipes to construct Nystrom approximations with variable accuracies and computing times. This paper proposes a scalable Nystrom-based clustering algorithm with a new sampling procedure, Centroid Minimum Sum of …
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…
Imperial College London（伦敦帝国理工学院）；University of Bristol（布里斯托尔大学）
Spectral clustering is a popular method for community detection in network graphs: starting from a matrix representation of the graph, the nodes are clustered on a low dimensional projection obtained from a truncated spectral decomposition of the matrix. Estimating correctly the number of communities and the dimension of the reduced latent space is critical for good performance of spectral clustering algorithms. Furthermore, many real-world graphs, such as enterprise computer networks studied in cyber-security applications, often display heterogeneous within-community degree distributions. Suc…
In this paper, we study the spectrum and the eigenvectors of radial kernels for mixtures of distributions in $\mathbb{R}^n$. Our approach focuses on high dimensions and relies solely on the concentration properties of the components in the mixture. We give several results describing of the structure of kernel matrices for a sample drawn from such a mixture. Based on these results, we analyze the ability of kernel PCA to cluster high dimensional mixtures. In particular, we exhibit a specific kernel leading to a simple spectral algorithm for clustering mixtures with possibly common means but dif…
CEA, LIST（法国原子能和替代能源委员会清单实验室）；Université Paris-Dauphine, PSL Research University, CNRS, LAMSADE（巴黎第九大学（巴黎多芬纳大学）、巴黎文理研究大学、法国国家科学研究中心、拉撒路德实验室）
In this paper, we present the first differentially private clustering method for arbitrary-shaped node clusters in a graph. This algorithm takes as input only an approximate Minimum Spanning Tree (MST) $\mathcal{T}$ released under weight differential privacy constraints from the graph. Then, the underlying nonconvex clustering partition is successfully recovered from cutting optimal cuts on $\mathcal{T}$. As opposed to existing methods, our algorithm is theoretically well-motivated. Experiments support our theoretical findings.
Many clustering problems enjoy solutions by semidefinite programming. Theoretical results in this vein frequently consider data with a planted clustering and a notion of signal strength such that the semidefinite program exactly recovers the planted clustering when the signal strength is sufficiently large. In practice, semidefinite programs are notoriously slow, and so speedups are welcome. In this paper, we show how to sketch a popular semidefinite relaxation of a graph clustering problem known as minimum bisection, and our analysis supports a meta-claim that the clustering task is less comp…
In recent years, the problem of fuzzy clustering has been widely concerned. The membership iteration of existing methods is mostly considered globally, which has considerable problems in noisy environments, and iterative calculations for clusters with a large number of different sample sizes are not accurate and efficient. In this paper, starting from the strategy of large-scale priority, the data is fuzzy iterated using granular-balls, and the membership degree of data only considers the two granular-balls where it is located, thus improving the efficiency of iteration. The formed fuzzy granu…
Spectral clustering is a fast and popular algorithm for finding clusters in networks. Recently, Chaudhuri et al. (2012) and Amini et al.(2012) proposed inspired variations on the algorithm that artificially inflate the node degrees for improved statistical performance. The current paper extends the previous statistical estimation results to the more canonical spectral clustering algorithm in a way that removes any assumption on the minimum degree and provides guidance on the choice of the tuning parameter. Moreover, our results show how the "star shape" in the eigenvectors--a common feature of…
This letter presents a new spectral-clustering-based approach to the subspace clustering problem. Underpinning the proposed method is a convex program for optimal direction search, which for each data point d finds an optimal direction in the span of the data that has minimum projection on the other data points and non-vanishing projection on d. The obtained directions are subsequently leveraged to identify a neighborhood set for each data point. An alternating direction method of multipliers framework is provided to efficiently solve for the optimal directions. The proposed method is shown to…
Many clustering algorithms are guided by certain cost functions such as the widely-used $k$-means cost. These algorithms divide data points into clusters with often complicated boundaries, creating difficulties in explaining the clustering decision. In a recent work, Dasgupta, Frost, Moshkovitz, and Rashtchian (ICML 2020) introduced explainable clustering, where the cluster boundaries are axis-parallel hyperplanes and the clustering is obtained by applying a decision tree to the data. The central question here is: how much does the explainability constraint increase the value of the cost funct…
Massachusetts Institute of Technology（麻省理工学院）；Institute of Advanced Study（高等研究院）
Random dimensionality reduction is a versatile tool for speeding up algorithms for high-dimensional problems. We study its application to two clustering problems: the facility location problem, and the single-linkage hierarchical clustering problem, which is equivalent to computing the minimum spanning tree. We show that if we project the input pointset $X$ onto a random $d = O(d_X)$-dimensional subspace (where $d_X$ is the doubling dimension of $X$), then the optimum facility location cost in the projected space approximates the original cost up to a constant factor. We show an analogous stat…
In recent years, crowdsourcing, aka human aided computation has emerged as an effective platform for solving problems that are considered complex for machines alone. Using human is time-consuming and costly due to monetary compensations. Therefore, a crowd based algorithm must judiciously use any information computed through an automated process, and ask minimum number of questions to the crowd adaptively. One such problem which has received significant attention is {\em entity resolution}. Formally, we are given a graph $G=(V,E)$ with unknown edge set $E$ where $G$ is a union of $k$ (again un…
Categorical data clustering: 25 years beyond K-modes
The Kyoto College of Graduate Studies for Informatics（京都信息学研究生院）；College of Computer Science and Software Engineering, Shenzhen University（深圳大学计算机与软件学院）；University of Gothenburg（哥德堡大学）；Japan Advanced Institute of Science and Technology（日本先进科学技术学院）
There has been much recent research on human activity re\-cog\-ni\-tion (HAR), due to the proliferation of wearable sensors in watches and phones, and the advances of deep learning methods, which avoid the need to manually extract features from raw sensor signals. A significant disadvantage of deep learning applied to HAR is the need for manually labelled training data, which is especially difficult to obtain for HAR datasets. Progress is starting to be made in the unsupervised setting, in the form of deep HAR clustering models, which can assign labels to data without having been given any lab…
Approximation Algorithms for Fair Range Clustering
African Institute for Mathematical Sciences–Rwanda（卢旺达非洲数学科学研究所）；Microsoft Research–Redmond（微软研究院（雷德蒙德））；Toyota Technological Institute at Chicago（芝加哥丰田技术研究所）
