We examine overlapping clustering schemes with functorial constraints, in the spirit of Carlsson--Memoli. This avoids issues arising from the chaining required by partition-based methods. Our principal result shows that any clustering functor is naturally constrained to refine single-linkage clusters and be refined by maximal-linkage clusters. We work in the context of metric spaces with non-expansive maps, which is appropriate for modeling data processing which does not increase information content.
Larodec, ISG University of Tunis（突尼斯ISG大学Larodec）；Ceremade, Université Paris-Dauphine（巴黎第九大学Ceremade）
Producing overlapping schemes is a major issue in clustering. Recent proposed overlapping methods relies on the search of an optimal covering and are based on different metrics, such as Euclidean distance and I-Divergence, used to measure closeness between observations. In this paper, we propose the use of another measure for overlapping clustering based on a kernel similarity metric .We also estimate the number of overlapped clusters using the Gram matrix. Experiments on both Iris and EachMovie datasets show the correctness of the estimation of number of clusters and show that measure based o…
Sungkyunkwan University（成均馆大学）；The University of Texas at Austin（德克萨斯大学奥斯汀分校）
The goal of co-clustering is to simultaneously identify a clustering of rows as well as columns of a two dimensional data matrix. A number of co-clustering techniques have been proposed including information-theoretic co-clustering and the minimum sum-squared residue co-clustering method. However, most existing co-clustering algorithms are designed to find pairwise disjoint and exhaustive co-clusters while many real-world datasets contain not only a large overlap between co-clusters but also outliers which should not belong to any co-cluster. In this paper, we formulate the problem of Non-Exha…
Clustering is one of the most universal approaches for understanding complex data. A pivotal aspect of clustering analysis is quantitatively comparing clusterings; clustering comparison is the basis for many tasks such as clustering evaluation, consensus clustering, and tracking the temporal evolution of clusters. In particular, the extrinsic evaluation of clustering methods requires comparing the uncovered clusterings to planted clusterings or known metadata. Yet, as we demonstrate, existing clustering comparison measures have critical biases which undermine their usefulness, and no measure a…
In a standard cluster analysis, such as k-means, in addition to clusters locations and distances between them, it's important to know if they are connected or well separated from each other. The main focus of this paper is discovering the relations between the resulting clusters. We propose a new method which is based on pairwise overlapping k-means clustering, that in addition to means of clusters provides the graph structure of their relations. The proposed method has a set of parameters that can be tuned in order to control the sensitivity of the model and the desired relative size of the p…
People belong to multiple communities, words belong to multiple topics, and books cover multiple genres; overlapping clusters are commonplace. Many existing overlapping clustering methods model each person (or word, or book) as a non-negative weighted combination of "exemplars" who belong solely to one community, with some small noise. Geometrically, each person is a point on a cone whose corners are these exemplars. This basic form encompasses the widely used Mixed Membership Stochastic Blockmodel of networks (Airoldi et al., 2008) and its degree-corrected variants (Jin et al., 2017), as well…
Air Force Research Laboratory（空军研究实验室）；University of Pennsylvania（宾夕法尼亚大学）；Texas A&M University（得克萨斯农工大学）
This work draws inspiration from three important sources of research on dissimilarity-based clustering and intertwines those three threads into a consistent principled functorial theory of clustering. Those three are the overlapping clustering of Jardine and Sibson, the functorial approach of Carlsson and Mémoli to partition-based clustering, and the Isbell/Dress school's study of injective envelopes. Carlsson and Mémoli introduce the idea of viewing clustering methods as functors from a category of metric spaces to a category of clusters, with functoriality subsuming many desirable properties…
Institut de Robòtica i Informàtica Industrial, CSIC-UPC（西班牙国家研究委员会-加泰罗尼亚理工大学工业机器人与信息技术研究所）；University of Southampton（南安普顿大学）；Institute for Life Sciences, University of Southampton（南安普顿大学生命科学研究所）；The Alan Turing Institute（艾伦·图灵研究所）
Kleinberg introduced three natural clustering properties, or axioms, and showed they cannot be simultaneously satisfied by any clustering algorithm. We present a new clustering property, Monotonic Consistency, which avoids the well-known problematic behaviour of Kleinberg's Consistency axiom, and the impossibility result. Namely, we describe a clustering algorithm, Morse Clustering, inspired by Morse Theory in Differential Topology, which satisfies Kleinberg's original axioms with Consistency replaced by Monotonic Consistency. Morse clustering uncovers the underlying flow structure on a set or…
School of Microelectronics and Communication Engineering, Chongqing University（重庆大学微电子与通信工程学院）
The challenge of imbalanced learning lies not only in class imbalance problem, but also in the class overlapping problem which is complex. However, most of the existing algorithms mainly focus on the former. The limitation prevents the existing methods from breaking through. To address this limitation, this paper proposes an ensemble learning algorithm based on dual clustering and stage-wise hybrid sampling (DCSHS). The DCSHS has three parts. Firstly, we design a projection clustering combination framework (PCC) guided by Davies-Bouldin clustering effectiveness index (DBI), which is used to ob…
In the past few years co-clustering has emerged as an important data mining tool for two way data analysis. Co-clustering is more advantageous over traditional one dimensional clustering in many ways such as, ability to find highly correlated sub-groups of rows and columns. However, one of the overlooked benefits of co-clustering is that, it can be used to extract meaningful knowledge for various other knowledge extraction purposes. For example, building predictive models with high dimensional data and heterogeneous population is a non-trivial task. Co-clusters extracted from such data, which …
Purdue University（普渡大学）；Sungkyunkwan University（成均馆大学）；The University of Texas at Austin（德克萨斯大学奥斯汀分校）
Clustering is one of the most fundamental and important tasks in data mining. Traditional clustering algorithms, such as K-means, assign every data point to exactly one cluster. However, in real-world datasets, the clusters may overlap with each other. Furthermore, often, there are outliers that should not belong to any cluster. We recently proposed the NEO-K-Means (Non-Exhaustive, Overlapping K-Means) objective as a way to address both issues in an integrated fashion. Optimizing this discrete objective is NP-hard, and even though there is a convex relaxation of the objective, straightforward …
Multi-view data analysis has gained increasing popularity because multi-view data are frequently encountered in machine learning applications. A simple but promising approach for clustering of multi-view data is multi-view clustering (MVC), which has been developed extensively to classify given subjects into some clustered groups by learning latent common features that are shared across multi-view data. Among existing approaches, graph-based multi-view clustering (GMVC) achieves state-of-the-art performance by leveraging a shared graph matrix called the unified matrix. However, existing method…
Spectral clustering is widely used in practice due to its flexibility, computational efficiency, and well-understood theoretical performance guarantees. Recently, spectral clustering has been studied to find balanced clusters under population-level constraints. These constraints are specified by additional information available in the form of auxiliary categorical node attributes. In this paper, we consider a scenario where these attributes may not be observable, but manifest as latent features of an auxiliary graph. Motivated by this, we study constrained spectral clustering with the aim of f…
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…
A New Framework for Convex Clustering in Kernel Spaces: Finite Sample Bounds, Consistency and Performance Insights
Indian Statistical Institute, Kolkata（印度统计研究院，加尔各答）；Electronics and Communication Sciences Unit, Indian Statistical Institute（印度统计研究院电子与通信科学单位）；Department of Statistics, University of Oxford（牛津大学统计系）；Department of Statistics, University of Michigan（密歇根大学统计系）
University of California at Davis（加州大学戴维斯分校）；New York University Shanghai（上海纽约大学）
Spectral clustering has become one of the most popular algorithms in data clustering and community detection. We study the performance of classical two-step spectral clustering via the graph Laplacian to learn the stochastic block model. Our aim is to answer the following question: when is spectral clustering via the graph Laplacian able to achieve strong consistency, i.e., the exact recovery of the underlying hidden communities? Our work provides an entrywise analysis (an $\ell_{\infty}$-norm perturbation bound) of the Fielder eigenvector of both the unnormalized and the normalized Laplacian …
Although consistency is a minimum requirement of any estimator, little is known about consistency of the mean partition approach in consensus clustering. This contribution studies the asymptotic behavior of mean partitions. We show that under normal assumptions, the mean partition approach is consistent and asymptotic normal. To derive both results, we represent partitions as points of some geometric space, called orbit space. Then we draw on results from the theory of Fréchet means and stochastic programming. The asymptotic properties hold for continuous extensions of standard cluster criteri…
This paper establishes the consistency of spectral approaches to data clustering. We consider clustering of point clouds obtained as samples of a ground-truth measure. A graph representing the point cloud is obtained by assigning weights to edges based on the distance between the points they connect. We investigate the spectral convergence of both unnormalized and normalized graph Laplacians towards the appropriate operators in the continuum domain. We obtain sharp conditions on how the connectivity radius can be scaled with respect to the number of sample points for the spectral convergence t…
This paper examines the applicability in realistic scenarios of two deep learning based solutions to the overlapping speaker separation problem. Firstly, we present experiments that show that these methods are applicable for a broad range of languages. Further experimentation indicates limited performance loss for untrained languages, when these have common features with the trained language(s). Secondly, it investigates how the methods deal with realistic background noise and proposes some modifications to better cope with these disturbances. The deep learning methods that will be examined ar…
LARODEC（拉罗德克实验室）；Institut Supérieur de Gestion de Tunis（突尼斯高等管理学院）；Université de Tunis（突尼斯大学）
Overlapping clustering problem is an important learning issue in which clusters are not mutually exclusive and each object may belongs simultaneously to several clusters. This paper presents a kernel based method that produces overlapping clusters on a high feature space using mercer kernel techniques to improve separability of input patterns. The proposed method, called OKM-K(Overlapping $k$-means based kernel method), extends OKM (Overlapping $k$-means) method to produce overlapping schemes. Experiments are performed on overlapping dataset and empirical results obtained with OKM-K outperform…
Multi-view clustering can make use of multi-source information for unsupervised clustering. Most existing methods focus on learning a fused representation matrix, while ignoring the influence of private information and noise. To address this limitation, we introduce a novel Multi-view Semantic Consistency based Information Bottleneck for clustering (MSCIB). Specifically, MSCIB pursues semantic consistency to improve the learning process of information bottleneck for different views. It conducts the alignment operation of multiple views in the semantic space and jointly achieves the valuable co…
University of Wisconsin - Madison（威斯康星大学麦迪逊分校）；Indian Institute of Science（印度科学学院）
Spectral clustering is one of the most popular methods for community detection in graphs. A key step in spectral clustering algorithms is the eigen decomposition of the $n{\times}n$ graph Laplacian matrix to extract its $k$ leading eigenvectors, where $k$ is the desired number of clusters among $n$ objects. This is prohibitively complex to implement for very large datasets. However, it has recently been shown that it is possible to bypass the eigen decomposition by computing an approximate spectral embedding through graph filtering of random signals. In this paper, we analyze the working of sp…
IBM Research（IBM研究院）；Indian Institute of Science（印度科学学院）
Spectral clustering is popular among practitioners and theoreticians alike. While performance guarantees for spectral clustering are well understood, recent studies have focused on enforcing ``fairness'' in clusters, requiring them to be ``balanced'' with respect to a categorical sensitive node attribute (e.g. the race distribution in clusters must match the race distribution in the population). In this paper, we consider a setting where sensitive attributes indirectly manifest in an auxiliary \textit{representation graph} rather than being directly observed. This graph specifies node pairs th…
China University of Mining and Technology（中国矿业大学）；Nankai University（南开大学）
Community detection in network analysis is an attractive research area recently. Here, under the degree-corrected mixed membership (DCMM) model, we propose an efficient approach called mixed regularized spectral clustering (Mixed-RSC for short) based on the regularized Laplacian matrix. Mixed-RSC is designed based on an ideal cone structure of the variant for the eigen-decomposition of the population regularized Laplacian matrix. We show that the algorithm is asymptotically consistent under mild conditions by providing error bounds for the inferred membership vector of each node. As a byproduc…
DCSI -- An improved measure of cluster separability based on separation and connectedness
Ludwig-Maximilians-Universität München（慕尼黑大学）；Munich Center for Machine Learning（慕尼黑机器学习中心）；Institute for Medical Information Processing, Biometry, and Epidemiology（医学信息处理、生物统计学与流行病学研究所）
Quick Shift is a popular mode-seeking and clustering algorithm. We present finite sample statistical consistency guarantees for Quick Shift on mode and cluster recovery under mild distributional assumptions. We then apply our results to construct a consistent modal regression algorithm.
Scripps Institution of Oceanography, UCSD（斯克里普斯海洋研究所，加州大学圣迭戈分校）；Naval Undersea Warfare Center Newport（纽波特海军水下作战中心）
Deep clustering was applied to unlabeled, automatically detected signals in a coral reef soundscape to distinguish fish pulse calls from segments of whale song. Deep embedded clustering (DEC) learned latent features and formed classification clusters using fixed-length power spectrograms of the signals. Handpicked spectral and temporal features were also extracted and clustered with Gaussian mixture models (GMM) and conventional clustering. DEC, GMM, and conventional clustering were tested on simulated datasets of fish pulse calls (fish) and whale song units (whale) with randomized bandwidth, …
Anchor-based techniques reduce the computational complexity of spectral clustering algorithms. Although empirical tests have shown promising results, there is currently a lack of theoretical support for the anchoring approach. We define a specific anchor-based algorithm and show that it is amenable to rigorous analysis, as well as being effective in practice. We establish the theoretical consistency of the method in an asymptotic setting where data is sampled from an underlying continuous probability distribution. In particular, we provide sharp asymptotic conditions for the algorithm paramete…
The Hong Kong Polytechnic University（香港理工大学）；Dalian University of Technology（大连理工大学）
Clustering uncertain data is an essential task in data mining for the internet of things. Possible world based algorithms seem promising for clustering uncertain data. However, there are two issues in existing possible world based algorithms: (1) They rely on all the possible worlds and treat them equally, but some marginal possible worlds may cause negative effects. (2) They do not well utilize the consistency among possible worlds, since they conduct clustering or construct the affinity matrix on each possible world independently. In this paper, we propose a representative possible world bas…
Many complex systems in the real world can be characterized by attributed networks. To mine the potential information in these networks, deep embedded clustering, which obtains node representations and clusters simultaneously, has been paid much attention in recent years. Under the assumption of consistency for data in different views, the cluster structure of network topology and that of node attributes should be consistent for an attributed network. However, many existing methods ignore this property, even though they separately encode node representations from network topology and node attr…
The problem of clustering is considered, for the case when each data point is a sample generated by a stationary ergodic process. We propose a very natural asymptotic notion of consistency, and show that simple consistent algorithms exist, under most general non-parametric assumptions. The notion of consistency is as follows: two samples should be put into the same cluster if and only if they were generated by the same distribution. With this notion of consistency, clustering generalizes such classical statistical problems as homogeneity testing and process classification. We show that, for th…
Multi-view Clustering via Bi-level Decoupling and Consistency Learning
School of Computer Science（计算机科学学院）；Nanjing University of Information Science（信息科学与技术大学）；The State Key Laboratory of Tibetan Intelligence（藏族智能国家重点实验室）；Qinghai Normal University（青海师范大学）；Technology（技术）；Zhejiang Normal University（浙江师范大学）
Deep Incomplete Multi-view Clustering with Distribution Dual-Consistency Recovery Guidance
School of Computer, National University of Defense Technology（国防科技大学计算机学院）；Academy of Military Sciences（军事科学院）；Chinese PLA General Hospital（中国人民解放军总医院）
Central China Normal University（华中师范大学）；Kharkiv National University of Radio Electronics（哈尔科夫国立无线电电子大学）
A task of clustering data given in the ordinal scale under conditions of overlapping clusters has been considered. It's proposed to use an approach based on memberhsip and likelihood functions sharing. A number of performed experiments proved effectiveness of the proposed method. The proposed method is characterized by robustness to outliers due to a way of ordering values while constructing membership functions.
Distributed controllers are oftentimes used in large-scale SDN deployments where they run a myriad of network applications simultaneously. Such applications could have different consistency and availability preferences. These controllers need to communicate via east/west interfaces in order to synchronize their state information. The consistency and the availability of the distributed state information are governed by an underlying consistency model. Earlier, we suggested the use of adaptively-consistent controllers that can autonomously tune their consistency parameters in order to meet the p…
Consistency between ordering and clustering methods for graphs
Artificial Intelligence Research Center, National Institute of Advanced Industrial Science and Technology（国立先进工业科学技术研究所人工智能研究中心）；The University of Tokyo（东京大学）；Kobe University（神户大学）
The problem of clustering is considered, for the case when each data point is a sample generated by a stationary ergodic process. We propose a very natural asymptotic notion of consistency, and show that simple consistent algorithms exist, under most general non-parametric assumptions. The notion of consistency is as follows: two samples should be put into the same cluster if and only if they were generated by the same distribution. With this notion of consistency, clustering generalizes such classical statistical problems as homogeneity testing and process classification. We show that, for th…
Microsoft（微软公司）；The Ohio State University（俄亥俄州立大学）
In this paper, we focus on unsupervised representation learning for clustering of images. Recent advances in deep clustering and unsupervised representation learning are based on the idea that different views of an input image (generated through data augmentation techniques) must be close in the representation space (exemplar consistency), and/or similar images must have similar cluster assignments (population consistency). We define an additional notion of consistency, consensus consistency, which ensures that representations are learned to induce similar partitions for variations in the repr…
The present paper is devoted to clustering geometric graphs. While the standard spectral clustering is often not effective for geometric graphs, we present an effective generalization, which we call higher-order spectral clustering. It resembles in concept the classical spectral clustering method but uses for partitioning the eigenvector associated with a higher-order eigenvalue. We establish the weak consistency of this algorithm for a wide class of geometric graphs which we call Soft Geometric Block Model. A small adjustment of the algorithm provides strong consistency. We also show that our…
Manifold-based Incomplete Multi-view Clustering via Bi-Consistency Guidance
College of Information Science and Technology, Dalian Maritime University（大连海事大学信息科学技术学院）；University of Technology, Sydney（悉尼科技大学）；Hefei University of Technology（合肥工业大学）
Discovering Operational Patterns Using Image-Based Convolutional Clustering and Composite Evaluation: A Case Study in Foundry Melting Processes
SDU Center for Energy Informatics, The Maersk Mc-Kinney Moller Institute, Faculty of Engineering, University of Southern Denmark（南丹麦大学工程学院马士基麦克-金尼·穆勒研究所SDU能源信息学中心）
School of Electronic Engineering, Xidian University（西安电子科技大学电子工程学院）
Bi-clustering refers to the task of finding sub-matrices (indexed by a group of columns and a group of rows) within a matrix of data such that the elements of each sub-matrix (data and features) are related in a particular way, for instance, that they are similar with respect to some metric. In this paper, after analyzing the well-known Cheng and Church (CC) bi-clustering algorithm which has been proved to be an effective tool for mining co-expressed genes. However, Cheng and Church bi-clustering algorithm and summarizing its limitations (such as interference of random numbers in the greedy st…
Compactness and Consistency: A Conjoint Framework for Deep Graph Clustering
College of Computer Science, Sichuan University（四川大学计算机学院）；Wellcome Sanger Institute（惠康桑格研究所）；University of International Business and Economics（对外经济贸易大学）；School of Computing and Information Technology, Great Bay University（湾区大学计算与信息技术学院）；School of Cyber Science and Technology, Shenzhen Campus of Sun Yat-sen University（中山大学深圳校区网络空间安全学院）；NITFID, School of Statistics and Data Science, Nankai University（南开大学统计与数据科学学院NITFID）；College of Mathematics, Sichuan University（四川大学数学学院）
