In this work, two new initialization methods for K-means clustering are proposed. Both proposals are based on applying a divide-and-conquer approach for the K-means|| type of an initialization strategy. The second proposal also utilizes multiple lower-dimensional subspaces produced by the random projection method for the initialization. The proposed methods are scalable and can be run in parallel, which make them suitable for initializing large-scale problems. In the experiments, comparison of the proposed methods to the K-means++ and K-means|| methods is conducted using an extensive set of reference and synthetic large-scale datasets. Concerning the latter, a novel high-dimensional clustering data generation algorithm is given. The experiments show that the proposed methods compare favorably to the state-of-the-art. We also observe that the currently most popular K-means++ initialization behaves like the random one in the very high-dimensional cases.
Distributed Gradient Clustering: Convergence and the Effect of Initialization
Carnegie Mellon University（卡内基梅隆大学）；Indian Institute of Technology Kharagpur（印度理工学院克勒格布尔分校）；Faculty of Technical Sciences, University of Novi Sad（诺维萨德大学技术科学学院）；Faculty of Sciences, University of Novi Sad（诺维萨德大学理学院）
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…
Google Research（谷歌研究院）；Uber Inc（优步公司）；Princeton University（普林斯顿大学）
We provide initial seedings to the Quick Shift clustering algorithm, which approximate the locally high-density regions of the data. Such seedings act as more stable and expressive cluster-cores than the singleton modes found by Quick Shift. We establish statistical consistency guarantees for this modification. We then show strong clustering performance on real datasets as well as promising applications to image segmentation.
Metric Embedding Initialization-Based Differentially Private and Explainable Graph Clustering
Graduate School of Arts and Sciences（艺术与科学研究生院）；Columbia University, New York, USA（哥伦比亚大学）；School of Artificial Intelligence（人工智能学院）；Hebei Institute of Communications, Shijiazhuang, PR China（河北通信学院）
Big Data is a massive volume of both structured and unstructured data that is too large and it also difficult to process using traditional techniques. Clustering algorithms have developed as a powerful learning tool that can exactly analyze the volume of data that produced by modern applications. Clustering in data mining is the grouping of a particular set of objects based on their characteristics. The main aim of clustering is to classified data into clusters such that objects are grouped in the same clusters when they are corresponding according to similarities and features mainly. Till now…
A unified clustering approach that can estimate number of clusters and produce clustering against this number simultaneously is proposed. Average silhouette width (ASW) is a widely used standard cluster quality index. A distance based objective function that optimizes ASW for clustering is defined. The proposed algorithm named as OSil, only, needs data observations as an input without any prior knowledge of the number of clusters. This work is about thorough investigation of the proposed methodology, its usefulness and limitations. A vast spectrum of clustering structures were generated, and s…
Louisiana State University（路易斯安那州立大学）；Georgia Institute of Technology（佐治亚理工学院）
K-means is undoubtedly the most widely used partitional clustering algorithm. Unfortunately, due to its gradient descent nature, this algorithm is highly sensitive to the initial placement of the cluster centers. Numerous initialization methods have been proposed to address this problem. In this paper, we first present an overview of these methods with an emphasis on their computational efficiency. We then compare eight commonly used linear time complexity initialization methods on a large and diverse collection of data sets using various performance criteria. Finally, we analyze the experimen…
Careful Seeding for k-Medois Clustering with Incremental k-Means++ Initialization
Institute of Automation, Chinese Academy of Sciences（中国科学院自动化研究所）；Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；School of Data Science, The Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳）数据科学学院）
We present a meta-method for initializing (seeding) the $k$-means clustering algorithm called PNN-smoothing. It consists in splitting a given dataset into $J$ random subsets, clustering each of them individually, and merging the resulting clusterings with the pairwise-nearest-neighbor (PNN) method. It is a meta-method in the sense that when clustering the individual subsets any seeding algorithm can be used. If the computational complexity of that seeding algorithm is linear in the size of the data $N$ and the number of clusters $k$, PNN-smoothing is also almost linear with an appropriate choi…
Université de Lyon（里昂大学）；ERIC EA 3083；ArAr UMR 5138
Clustering is an extensive research area in data science. The aim of clustering is to discover groups and to identify interesting patterns in datasets. Crisp (hard) clustering considers that each data point belongs to one and only one cluster. However, it is inadequate as some data points may belong to several clusters, as is the case in text categorization. Thus, we need more flexible clustering. Fuzzy clustering methods, where each data point can belong to several clusters, are an interesting alternative. Yet, seeding iterative fuzzy algorithms to achieve high quality clustering is an issue.…
University of Technology Sydney（悉尼科技大学）；Southern University of Science and Technology（南方科技大学）；University of Birmingham（伯明翰大学）
The K-means algorithm is a widely used clustering algorithm that offers simplicity and efficiency. However, the traditional K-means algorithm uses the random method to determine the initial cluster centers, which make clustering results prone to local optima and then result in worse clustering performance. Many initialization methods have been proposed, but none of them can dynamically adapt to datasets with various characteristics. In our previous research, an initialization method for K-means based on hybrid distance was proposed, and this algorithm can adapt to datasets with different chara…
The main contribution of the paper is a new approach to subspace clustering that is significantly more computationally efficient and scalable than existing state-of-the-art methods. The central idea is to modify the regression technique in sparse subspace clustering (SSC) by replacing the $\ell_1$ minimization with a generalization called Ordered Weighted $\ell_1$ (OWL) minimization which performs simultaneous regression and clustering of correlated variables. Using random geometric graph theory, we prove that OWL regression selects more points within each subspace, resulting in better cluster…
Louisiana State University（路易斯安那州立大学）；Georgia Institute of Technology（佐治亚理工学院）
K-means is undoubtedly the most widely used partitional clustering algorithm. Unfortunately, due to its gradient descent nature, this algorithm is highly sensitive to the initial placement of the cluster centers. Numerous initialization methods have been proposed to address this problem. Many of these methods, however, have superlinear complexity in the number of data points, making them impractical for large data sets. On the other hand, linear methods are often random and/or order-sensitive, which renders their results unrepeatable. Recently, Su and Dy proposed two highly successful hierarch…
South China Agricultural University（华南农业大学）；Nanyang Technological University（南洋理工大学）；Sun Yat-sen University（中山大学）；Nanchang University（南昌大学）
This paper focuses on scalability and robustness of spectral clustering for extremely large-scale datasets with limited resources. Two novel algorithms are proposed, namely, ultra-scalable spectral clustering (U-SPEC) and ultra-scalable ensemble clustering (U-SENC). In U-SPEC, a hybrid representative selection strategy and a fast approximation method for K-nearest representatives are proposed for the construction of a sparse affinity sub-matrix. By interpreting the sparse sub-matrix as a bipartite graph, the transfer cut is then utilized to efficiently partition the graph and obtain the cluste…
German Aerospace Center (DLR)（德国航空航天中心）；Technical University of Munich（慕尼黑工业大学）
Lots of effort in neural architecture search (NAS) research has been dedicated to algorithmic development, aiming at designing more efficient and less costly methods. Nonetheless, the investigation of the initialization of these techniques remain scare, and currently most NAS methodologies rely on stochastic initialization procedures, because acquiring information prior to search is costly. However, the recent availability of NAS benchmarks have enabled low computational resources prototyping. In this study, we propose to accelerate a NAS algorithm using a data-driven initialization technique,…
Amirkabir University of Technology（阿米拉比尔理工大学）；University of Twente（特温特大学）；ICAR-CNR（意大利国家研究委员会计算与高性能网络研究所）
This paper presents a batch-wise density-based clustering approach for local outlier detection in massive-scale datasets. Unlike the well-known traditional algorithms, which assume that all the data is memory-resident, our proposed method is scalable and processes the input data chunk-by-chunk within the confines of a limited memory buffer. A temporary clustering model is built at the first phase; then, it is gradually updated by analyzing consecutive memory loads of points. Subsequently, at the end of scalable clustering, the approximate structure of the original clusters is obtained. Finally…
Discrete random variables are natural components of probabilistic clustering models. A number of VAE variants with discrete latent variables have been developed. Training such methods requires marginalizing over the discrete latent variables, causing training time complexity to be linear in the number clusters. By applying a continuous relaxation to the discrete variables in these methods we can achieve a reduction in the training time complexity to be constant in the number of clusters used. We demonstrate that in practice for one such method, the Gaussian Mixture VAE, the use of a continuous…
We study in this paper a variant of Wasserstein barycenter problem, which we refer to as tree-Wasserstein barycenter, by leveraging a specific class of ground metrics, namely tree metrics, for Wasserstein distance. Drawing on the tree structure, we propose an efficient algorithmic approach to solve the tree-Wasserstein barycenter and its variants. The proposed approach is not only fast for computation but also efficient for memory usage. Exploiting the tree-Wasserstein barycenter and its variants, we scale up multi-level clustering and scalable Bayes, especially for large-scale applications wh…
Columbia University（哥伦比亚大学）；Courant Institute of Mathematical Sciences（库朗数学科学研究所）；Google Research（谷歌研究院）
In this paper, we compare three initialization schemes for the KMEANS clustering algorithm: 1) random initialization (KMEANSRAND), 2) KMEANS++, and 3) KMEANSD++. Both KMEANSRAND and KMEANS++ have a major that the value of k needs to be set by the user of the algorithms. (Kang 2013) recently proposed a novel use of determinantal point processes for sampling the initial centroids for the KMEANS algorithm (we call it KMEANSD++). They, however, do not provide any evaluation establishing that KMEANSD++ is better than other algorithms. In this paper, we show that the performance of KMEANSD++ is comp…
We propose k^2-means, a new clustering method which efficiently copes with large numbers of clusters and achieves low energy solutions. k^2-means builds upon the standard k-means (Lloyd's algorithm) and combines a new strategy to accelerate the convergence with a new low time complexity divisive initialization. The accelerated convergence is achieved through only looking at k_n nearest clusters and using triangle inequality bounds in the assignment step while the divisive initialization employs an optimal 2-clustering along a direction. The worst-case time complexity per iteration of our k^2-m…
We propose a simple and easy to implement neural network compression algorithm that achieves results competitive with more complicated state-of-the-art methods. The key idea is to modify the original optimization problem by adding K independent Gaussian priors (corresponding to the k-means objective) over the network parameters to achieve parameter quantization, as well as an L1 penalty to achieve pruning. Unlike many existing quantization-based methods, our method uses hard clustering assignments of network parameters, which adds minimal change or overhead to standard network training. We als…
Louisiana State University（路易斯安那州立大学）；Georgia Institute of Technology（佐治亚理工学院）
Over the past five decades, k-means has become the clustering algorithm of choice in many application domains primarily due to its simplicity, time/space efficiency, and invariance to the ordering of the data points. Unfortunately, the algorithm's sensitivity to the initial selection of the cluster centers remains to be its most serious drawback. Numerous initialization methods have been proposed to address this drawback. Many of these methods, however, have time complexity superlinear in the number of data points, which makes them impractical for large data sets. On the other hand, linear met…
Department of Statistical Science, Duke University（杜克大学统计科学系）；Departments of Computational Medicine, Statistics, and Human Genetics, UCLA（加州大学洛杉矶分校计算医学、统计与人类遗传学系）
Clustering, a fundamental activity in unsupervised learning, is notoriously difficult when the feature space is high-dimensional. Fortunately, in many realistic scenarios, only a handful of features are relevant in distinguishing clusters. This has motivated the development of sparse clustering techniques that typically rely on k-means within outer algorithms of high computational complexity. Current techniques also require careful tuning of shrinkage parameters, further limiting their scalability. In this paper, we propose a novel framework for sparse k-means clustering that is intuitive, sim…
King Digital Entertainment, Activision Blizzard Group（动视暴雪集团King数字娱乐公司）；Uppsala University（乌普萨拉大学）
Deep clustering (DC) has become the state-of-the-art for unsupervised clustering. In principle, DC represents a variety of unsupervised methods that jointly learn the underlying clusters and the latent representation directly from unstructured datasets. However, DC methods are generally poorly applied due to high operational costs, low scalability, and unstable results. In this paper, we first evaluate several popular DC variants in the context of industrial applicability using eight empirical criteria. We then choose to focus on variational deep clustering (VDC) methods, since they mostly mee…
FISHDBC is a flexible, incremental, scalable, and hierarchical density-based clustering algorithm. It is flexible because it empowers users to work on arbitrary data, skipping the feature extraction step that usually transforms raw data in numeric arrays letting users define an arbitrary distance function instead. It is incremental and scalable: it avoids the $\mathcal O(n^2)$ performance of other approaches in non-metric spaces and requires only lightweight computation to update the clustering when few items are added. It is hierarchical: it produces a "flat" clustering which can be expanded …
IBM Research AI（IBM研究院人工智能部门）；Carnegie Mellon University（卡内基梅隆大学）；College of William and Mary（威廉与玛丽学院）；Huawei Research（华为研究院）
Spectral clustering is one of the most effective clustering approaches that capture hidden cluster structures in the data. However, it does not scale well to large-scale problems due to its quadratic complexity in constructing similarity graphs and computing subsequent eigendecomposition. Although a number of methods have been proposed to accelerate spectral clustering, most of them compromise considerable information loss in the original data for reducing computational bottlenecks. In this paper, we present a novel scalable spectral clustering method using Random Binning features (RB) to simu…
Information Sciences and Systems Laboratory (LSIS), UMR CNRS 7296（信息科学与系统实验室（LSIS），法国国家科学研究中心 UMR 7296）；University of the South Toulon-Var (USTV)（南土伦瓦尔大学（USTV））
Model-based clustering approaches concern the paradigm of exploratory data analysis relying on the finite mixture model to automatically find a latent structure governing observed data. They are one of the most popular and successful approaches in cluster analysis. The mixture density estimation is generally performed by maximizing the observed-data log-likelihood by using the expectation-maximization (EM) algorithm. However, it is well-known that the EM algorithm initialization is crucial. In addition, the standard EM algorithm requires the number of clusters to be known a priori. Some soluti…
DBSCAN is a classical density-based clustering procedure with tremendous practical relevance. However, DBSCAN implicitly needs to compute the empirical density for each sample point, leading to a quadratic worst-case time complexity, which is too slow on large datasets. We propose DBSCAN++, a simple modification of DBSCAN which only requires computing the densities for a chosen subset of points. We show empirically that, compared to traditional DBSCAN, DBSCAN++ can provide not only competitive performance but also added robustness in the bandwidth hyperparameter while taking a fraction of the …
University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
Semidefinite programming (SDP) is a powerful tool for tackling a wide range of computationally hard problems such as clustering. Despite the high accuracy, semidefinite programs are often too slow in practice with poor scalability on large (or even moderate) datasets. In this paper, we introduce a linear time complexity algorithm for approximating an SDP relaxed $K$-means clustering. The proposed sketch-and-lift (SL) approach solves an SDP on a subsampled dataset and then propagates the solution to all data points by a nearest-centroid rounding procedure. It is shown that the SL approach enjoy…
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…
The performance of EM in learning mixtures of product distributions often depends on the initialization. This can be problematic in crowdsourcing and other applications, e.g. when a small number of 'experts' are diluted by a large number of noisy, unreliable participants. We develop a new EM algorithm that is driven by these experts. In a manner that differs from other approaches, we start from a single mixture class. The algorithm then develops the set of 'experts' in a stagewise fashion based on a mutual information criterion. At each stage EM operates on this subset of the players, effectiv…
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 …
The University of Melbourne（墨尔本大学）；Deakin University（迪肯大学）；Data61, CSIRO（联邦科学与工业研究组织Data61）
Subspace clustering aims to find groups of similar objects (clusters) that exist in lower dimensional subspaces from a high dimensional dataset. It has a wide range of applications, such as analysing high dimensional sensor data or DNA sequences. However, existing algorithms have limitations in finding clusters in non-disjoint subspaces and scaling to large data, which impinge their applicability in areas such as bioinformatics and the Internet of Things. We aim to address such limitations by proposing a subspace clustering algorithm using a bottom-up strategy. Our algorithm first searches for…
Stevens Institute of Technology（斯蒂文斯理工学院）；Rensselaer Polytechnic Institute（伦斯勒理工学院）；International Computer Science Institute（国际计算机科学研究所）；University of California at Berkeley（加州大学伯克利分校）
Kernel $k$-means clustering can correctly identify and extract a far more varied collection of cluster structures than the linear $k$-means clustering algorithm. However, kernel $k$-means clustering is computationally expensive when the non-linear feature map is high-dimensional and there are many input points. Kernel approximation, e.g., the Nyström method, has been applied in previous works to approximately solve kernel learning problems when both of the above conditions are present. This work analyzes the application of this paradigm to kernel $k$-means clustering, and shows that applying t…
Chongqing University（重庆大学）；Rice University（莱斯大学）；University of Electronic Science and Technology of China（电子科技大学）；The Chinese University of Hong Kong-Shenzhen（香港中文大学（深圳））
The K-means algorithm is arguably the most popular data clustering method, commonly applied to processed datasets in some "feature spaces", as is in spectral clustering. Highly sensitive to initializations, however, K-means encounters a scalability bottleneck with respect to the number of clusters K as this number grows in big data applications. In this work, we promote a closely related model called K-indicators model and construct an efficient, semi-convex-relaxation algorithm that requires no randomized initializations. We present extensive empirical results to show advantages of the new al…
Self-Supervised Weight Templates for Scalable Vision Model Initialization
School of Computer Science and Engineering, Southeast University, Nanjing, China（计算机科学与工程学院，东南大学，南京，中国）；Key Laboratory of New Generation Artificial Intelligence Technology and Its Interdisciplinary Applications (Southeast University), Ministry of Education, China（新一代人工智能技术及其交叉应用重点实验室（东南大学），教育部，中国）
The eigendeomposition of nearest-neighbor (NN) graph Laplacian matrices is the main computational bottleneck in spectral clustering. In this work, we introduce a highly-scalable, spectrum-preserving graph sparsification algorithm that enables to build ultra-sparse NN (u-NN) graphs with guaranteed preservation of the original graph spectrums, such as the first few eigenvectors of the original graph Laplacian. Our approach can immediately lead to scalable spectral clustering of large data networks without sacrificing solution quality. The proposed method starts from constructing low-stretch span…
Beijing University of Technology（北京工业大学）；Guilin University of Electronic Technology（桂林电子科技大学）
K-means is a classical clustering algorithm with wide applications. However, soft K-means, or fuzzy c-means at m=1, remains unsolved since 1981. To address this challenging open problem, we propose a novel clustering model, i.e. Probabilistic K-Means (PKM), which is also a nonlinear programming model constrained on linear equalities and linear inequalities. In theory, we can solve the model by active gradient projection, while inefficiently. Thus, we further propose maximum-step active gradient projection and fast maximum-step active gradient projection to solve it more efficiently. By experim…
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…
This article explores and analyzes the unsupervised clustering of large partially observed graphs. We propose a scalable and provable randomized framework for clustering graphs generated from the stochastic block model. The clustering is first applied to a sub-matrix of the graph's adjacency matrix associated with a reduced graph sketch constructed using random sampling. Then, the clusters of the full graph are inferred based on the clusters extracted from the sketch using a correlation-based retrieval step. Uniform random node sampling is shown to improve the computational complexity over clu…
School of Computer Science and Engineering, Southeast University（东南大学计算机科学与工程学院）；City University of Hong Kong（香港城市大学）；City University of Hong Kong Shenzhen Research Institute（香港城市大学深圳研究院）
Symmetric nonnegative matrix factorization (SNMF) has demonstrated to be a powerful method for data clustering. However, SNMF is mathematically formulated as a non-convex optimization problem, making it sensitive to the initialization of variables. Inspired by ensemble clustering that aims to seek a better clustering result from a set of clustering results, we propose self-supervised SNMF (S$^3$NMF), which is capable of boosting clustering performance progressively by taking advantage of the sensitivity to initialization characteristic of SNMF, without relying on any additional information. Sp…
