We utilize the PageRank vector to generalize the $k$-means clustering algorithm to directed and undirected graphs. We demonstrate that PageRank and other centrality measures can be used in our setting to robustly compute centrality of nodes in a given graph. Furthermore, we show how our method can be generalized to metric spaces and apply it to other domains such as point clouds and triangulated meshes
RIKEN AIP（日本理化学研究所革新智能综合研究中心）；University of Tokyo（东京大学）；National Institute of Informatics（国立信息学研究所）
A hypergraph is a useful combinatorial object to model ternary or higher-order relations among entities. Clustering hypergraphs is a fundamental task in network analysis. In this study, we develop two clustering algorithms based on personalized PageRank on hypergraphs. The first one is local in the sense that its goal is to find a tightly connected vertex set with a bounded volume including a specified vertex. The second one is global in the sense that its goal is to find a tightly connected vertex set. For both algorithms, we discuss theoretical guarantees on the conductance of the output ver…
University of California at Berkeley（加州大学伯克利分校）；University of Waterloo（滑铁卢大学）；ICSI（国际计算机科学研究所）
Local graph clustering methods aim to find small clusters in very large graphs. These methods take as input a graph and a seed node, and they return as output a good cluster in a running time that depends on the size of the output cluster but that is independent of the size of the input graph. In this paper, we adopt a statistical perspective on local graph clustering, and we analyze the performance of the l1-regularized PageRank method~(Fountoulakis et. al.) for the recovery of a single target cluster, given a seed node inside the cluster. Assuming the target cluster has been generated by a r…
Univ Lyon（里昂大学）；Inria（法国国家信息与自动化研究所）；CNRS（法国国家科学研究中心）；ENS de Lyon（里昂高等师范学院）；UCB Lyon 1（里昂第一大学）；Univ Claude Bernard（克劳德·贝尔纳大学）；Laboratoire de Physique（物理实验室）
PageRank for Semi-Supervised Learning has shown to leverage data structures and limited tagged examples to yield meaningful classification. Despite successes, classification performance can still be improved, particularly in cases of fuzzy graphs or unbalanced labeled data. To address such limitations, a novel approach based on powers of the Laplacian matrix $L^γ$ ($γ> 0$), referred to as $L^γ$-PageRank, is proposed. Its theoretical study shows that it operates on signed graphs, where nodes belonging to one same class are more likely to share positive edges while nodes from different classes a…
Landing probabilities (LP) of random walks (RW) over graphs encode rich information regarding graph topology. Generalized PageRanks (GPR), which represent weighted sums of LPs of RWs, utilize the discriminative power of LP features to enable many graph-based learning studies. Previous work in the area has mostly focused on evaluating suitable weights for GPRs, and only a few studies so far have attempted to derive the optimal weights of GRPs for a given application. We take a fundamental step forward in this direction by using random graph models to better our understanding of the behavior of …
We study in this paper the problem of jointly clustering and learning representations. As several previous studies have shown, learning representations that are both faithful to the data to be clustered and adapted to the clustering algorithm can lead to better clustering performance, all the more so that the two tasks are performed jointly. We propose here such an approach for $k$-Means clustering based on a continuous reparametrization of the objective function that leads to a truly joint solution. The behavior of our approach is illustrated on various datasets showing its efficacy in learni…
In this work we propose a clustering framework based on the paradigm of transform learning. In simple terms the representation from transform learning is used for K-means clustering; however, the problem is not solved in such a naïve piecemeal fashion. The K-means clustering loss is embedded into the transform learning framework and the joint problem is solved using the alternating direction method of multipliers. Results on document clustering show that our proposed approach improves over the state-of-the-art.
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…
University of Edinburgh（爱丁堡大学）；University of Surrey（萨里大学）；Samsung AI Centre（三星人工智能中心）
We address the problem of simultaneously learning a k-means clustering and deep feature representation from unlabelled data, which is of interest due to the potential of deep k-means to outperform traditional two-step feature extraction and shallow-clustering strategies. We achieve this by developing a gradient-estimator for the non-differentiable k-means objective via the Gumbel-Softmax reparameterisation trick. In contrast to previous attempts at deep clustering, our concrete k-means model can be optimised with respect to the canonical k-means objective and is easily trained end-to-end witho…
We present a study on how to effectively reduce the dimensions of the $k$-means clustering problem, so that provably accurate approximations are obtained. Four algorithms are presented, two \textit{feature selection} and two \textit{feature extraction} based algorithms, all of which are randomized.
Kernel-based clustering algorithm can identify and capture the non-linear structure in datasets, and thereby it can achieve better performance than linear clustering. However, computing and storing the entire kernel matrix occupy so large memory that it is difficult for kernel-based clustering to deal with large-scale datasets. In this paper, we employ incomplete Cholesky factorization to accelerate kernel clustering and save memory space. The key idea of the proposed kernel $k$-means clustering using incomplete Cholesky factorization is that we approximate the entire kernel matrix by the prod…
Institute of Information Engineering, Chinese Academy of Sciences（中国科学院信息工程研究所）；Inception Institute of Artificial Intelligence（启智人工智能研究院）
In this paper, we study the statistical properties of kernel $k$-means and obtain a nearly optimal excess clustering risk bound, substantially improving the state-of-art bounds in the existing clustering risk analyses. We further analyze the statistical effect of computational approximations of the Nyström kernel $k$-means, and prove that it achieves the same statistical accuracy as the exact kernel $k$-means considering only $Ω(\sqrt{nk})$ Nyström landmark points. To the best of our knowledge, such sharp excess clustering risk bounds for kernel (or approximate kernel) $k$-means have never bee…
Kleinberg's axioms for distance based clustering proved to be contradictory. Various efforts have been made to overcome this problem. Here we make an attempt to handle the issue by embedding in high-dimensional space and granting wide gaps between clusters.
The present work proposes hybridization of Expectation-Maximization (EM) and K-Means techniques as an attempt to speed-up the clustering process. Though both K-Means and EM techniques look into different areas, K-means can be viewed as an approximate way to obtain maximum likelihood estimates for the means. Along with the proposed algorithm for hybridization, the present work also experiments with the Standard EM algorithm. Six different datasets are used for the experiments of which three are synthetic datasets. Clustering fitness and Sum of Squared Errors (SSE) are computed for measuring the…
Graph based semi-supervised learning is the problem of learning a labeling function for the graph nodes given a few example nodes, often called seeds, usually under the assumption that the graph's edges indicate similarity of labels. This is closely related to the local graph clustering or community detection problem of finding a cluster or community of nodes around a given seed. For this problem, we propose a novel generalization of random walk, diffusion, or smooth function methods in the literature to a convex p-norm cut function. The need for our p-norm methods is that, in our study of exi…
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…
Multiple kernel methods based on k-means aims to integrate a group of kernels to improve the performance of kernel k-means clustering. However, we observe that most existing multiple kernel k-means methods exploit the nonlinear relationship within kernels, whereas the local manifold structure among multiple kernel space is not sufficiently considered. In this paper, we adopt the manifold adaptive kernel, instead of the original kernel, to integrate the local manifold structure of kernels. Thus, the induced multiple manifold adaptive kernels not only reflect the nonlinear relationship but also …
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…
We propose the Lasso Weighted $k$-means ($LW$-$k$-means) algorithm as a simple yet efficient sparse clustering procedure for high-dimensional data where the number of features ($p$) can be much larger compared to the number of observations ($n$). In the $LW$-$k$-means algorithm, we introduce a lasso-based penalty term, directly on the feature weights to incorporate feature selection in the framework of sparse clustering. $LW$-$k$-means does not make any distributional assumption of the given dataset and thus, induces a non-parametric method for feature selection. We also analytically investiga…
University of Science and Technology of China（中国科学技术大学）
To cluster data that are not linearly separable in the original feature space, $k$-means clustering was extended to the kernel version. However, the performance of kernel $k$-means clustering largely depends on the choice of kernel function. To mitigate this problem, multiple kernel learning has been introduced into the $k$-means clustering to obtain an optimal kernel combination for clustering. Despite the success of multiple kernel $k$-means clustering in various scenarios, few of the existing work update the combination coefficients based on the diversity of kernels, which leads to the resu…
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…
In this work we build a unifying framework to interpolate between density-driven and geometry-based algorithms for data clustering, and specifically, to connect the mean shift algorithm with spectral clustering at discrete and continuum levels. We seek this connection through the introduction of Fokker-Planck equations on data graphs. Besides introducing new forms of mean shift algorithms on graphs, we provide new theoretical insights on the behavior of the family of diffusion maps in the large sample limit as well as provide new connections between diffusion maps and mean shift dynamics on a …
Tepper School of Business, Carnegie Mellon University（卡内基梅隆大学泰珀商学院）
This paper explores hierarchical clustering in the case where pairs of points have dissimilarity scores (e.g. distances) as a part of the input. The recently introduced objective for points with dissimilarity scores results in every tree being a 1/2 approximation if the distances form a metric. This shows the objective does not make a significant distinction between a good and poor hierarchical clustering in metric spaces. Motivated by this, the paper develops a new global objective for hierarchical clustering in Euclidean space. The objective captures the criterion that has motivated the use …
University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
Clustering is an important exploratory data analysis technique to group objects based on their similarity. The widely used $K$-means clustering method relies on some notion of distance to partition data into a fewer number of groups. In the Euclidean space, centroid-based and distance-based formulations of the $K$-means are equivalent. In modern machine learning applications, data often arise as probability distributions and a natural generalization to handle measure-valued data is to use the optimal transport metric. Due to non-negative Alexandrov curvature of the Wasserstein space, barycente…
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…
Lehigh University（里海大学）；University of Coimbra（科英布拉大学）
In the application of data clustering to human-centric decision-making systems, such as loan applications and advertisement recommendations, the clustering outcome might discriminate against people across different demographic groups, leading to unfairness. A natural conflict occurs between the cost of clustering (in terms of distance to cluster centers) and the balance representation of all demographic groups across the clusters, leading to a bi-objective optimization problem that is nonconvex and nonsmooth. To determine the complete trade-off between these two competing goals, we design a no…
Shahid Bahonar University of Kerman（克尔曼沙希德·巴霍纳尔大学）
One of the applications of center-based clustering algorithms such as K-Means is partitioning data points into K clusters. In some examples, the feature space relates to the underlying problem we are trying to solve, and sometimes we can obtain a suitable feature space. Nevertheless, while K-Means is one of the most efficient offline clustering algorithms, it is not equipped to estimate the number of clusters, which is useful in some practical cases. Other practical methods which do are simply too complex, as they require at least one run of K-Means for each possible K. In order to address thi…
This paper investigates the model degrees of freedom in k-means clustering. An extension of Stein's lemma provides an expression for the effective degrees of freedom in the k-means model. Approximating the degrees of freedom in practice requires simplifications of this expression, however empirical studies evince the appropriateness of our proposed approach. The practical relevance of this new degrees of freedom formulation for k-means is demonstrated through model selection using the Bayesian Information Criterion. The reliability of this method is validated through experiments on simulated d…
To devise efficient solutions for approximating a mean partition in consensus clustering, Dimitriadou et al. [3] presented a necessary condition of optimality for a consensus function based on least square distances. We show that their result is pivotal for deriving interesting properties of consensus clustering beyond optimization. For this, we present the necessary condition of optimality in a slightly stronger form in terms of the Mean Partition Theorem and extend it to the Expected Partition Theorem. To underpin its versatility, we show three examples that apply the Mean Partition Theorem:…
This paper provides new algorithms for distributed clustering for two popular center-based objectives, k-median and k-means. These algorithms have provable guarantees and improve communication complexity over existing approaches. Following a classic approach in clustering by \cite{har2004coresets}, we reduce the problem of finding a clustering with low cost to the problem of finding a coreset of small size. We provide a distributed method for constructing a global coreset which improves over the previous methods by reducing the communication complexity, and which works over general communicati…
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.
Comparison of three kind of the clustering and find cost function and loss function and calculate them. Error rate of the clustering methods and how to calculate the error percentage always be one on the important factor for evaluating the clustering methods, so this paper introduce one way to calculate the error rate of clustering methods. Clustering algorithms can be divided into several categories including partitioning clustering algorithms, hierarchical algorithms and density based algorithms. Generally speaking we should compare clustering algorithms by Scalability, Ability to work with …
Bhagwan Parshuram Institute of Technology（帕舒拉姆理工学院）
K-means is an effective clustering technique used to separate similar data into groups based on initial centroids of clusters. In this paper, Normalization based K-means clustering algorithm(N-K means) is proposed. Proposed N-K means clustering algorithm applies normalization prior to clustering on the available data as well as the proposed approach calculates initial centroids based on weights. Experimental results prove the betterment of proposed N-K means clustering algorithm over existing K-means clustering algorithm in terms of complexity and overall performance.
We consider the $k$-means clustering problem in the dynamic streaming setting, where points from a discrete Euclidean space $\{1, 2, \ldots, Δ\}^d$ can be dynamically inserted to or deleted from the dataset. For this problem, we provide a one-pass coreset construction algorithm using space $\tilde{O}(k\cdot \mathrm{poly}(d, \logΔ))$, where $k$ is the target number of centers. To our knowledge, this is the first dynamic geometric data stream algorithm for $k$-means using space polynomial in dimension and nearly optimal (linear) in $k$.
College of Computer Studies De La Salle University（德拉萨大学计算机研究学院）
Deep clustering algorithms combine representation learning and clustering by jointly optimizing a clustering loss and a non-clustering loss. In such methods, a deep neural network is used for representation learning together with a clustering network. Instead of following this framework to improve clustering performance, we propose a simpler approach of optimizing the entanglement of the learned latent code representation of an autoencoder. We define entanglement as how close pairs of points from the same class or structure are, relative to pairs of points from different classes or structures.…
In this paper we target the class of modal clustering methods where clusters are defined in terms of the local modes of the probability density function which generates the data. The most well-known modal clustering method is the k-means clustering. Mean Shift clustering is a generalization of the k-means clustering which computes arbitrarily shaped clusters as defined as the basins of attraction to the local modes created by the density gradient ascent paths. Despite its potential, the Mean Shift approach is a computationally expensive method for unsupervised learning. Thus, we introduce two …
A natural way to characterize the cluster structure of a dataset is by finding regions containing a high density of data. This can be done in a nonparametric way with a kernel density estimate, whose modes and hence clusters can be found using mean-shift algorithms. We describe the theory and practice behind clustering based on kernel density estimates and mean-shift algorithms. We discuss the blurring and non-blurring versions of mean-shift; theoretical results about mean-shift algorithms and Gaussian mixtures; relations with scale-space theory, spectral clustering and other algorithms; exten…
Georgia Tech（佐治亚理工学院）；MPI for Intelligent Systems（马克斯·普朗克智能系统研究所）
We show that the popular k-means clustering algorithm (Lloyd's heuristic), used for a variety of scientific data, can result in outcomes that are unfavorable to subgroups of data (e.g., demographic groups). Such biased clusterings can have deleterious implications for human-centric applications such as resource allocation. We present a fair k-means objective and algorithm to choose cluster centers that provide equitable costs for different groups. The algorithm, Fair-Lloyd, is a modification of Lloyd's heuristic for k-means, inheriting its simplicity, efficiency, and stability. In comparison w…
The University of Queensland（昆士兰大学）；University of Texas（德克萨斯大学）；Carnegie Mellon University（卡内基梅隆大学）
Clustering is an effective technique in data mining to generate groups that are the matter of interest. Among various clustering approaches, the family of k-means algorithms and min-cut algorithms gain most popularity due to their simplicity and efficacy. The classical k-means algorithm partitions a number of data points into several subsets by iteratively updating the clustering centers and the associated data points. By contrast, a weighted undirected graph is constructed in min-cut algorithms which partition the vertices of the graph into two sets. However, existing clustering algorithms te…
We present a new clustering algorithm called k-means-u* which in many cases is able to significantly improve the clusterings found by k-means++, the current de-facto standard for clustering in Euclidean spaces. First we introduce the k-means-u algorithm which starts from a result of k-means++ and attempts to improve it with a sequence of non-local "jumps" alternated by runs of standard k-means. Each jump transfers the "least useful" center towards the center with the largest local error, offset by a small random vector. This is continued as long as the error decreases and often leads to an imp…
Tel Aviv University（特拉维夫大学）；University of California, San Diego（加利福尼亚大学圣迭戈分校）
Despite the popularity of explainable AI, there is limited work on effective methods for unsupervised learning. We study algorithms for $k$-means clustering, focusing on a trade-off between explainability and accuracy. Following prior work, we use a small decision tree to partition a dataset into $k$ clusters. This enables us to explain each cluster assignment by a short sequence of single-feature thresholds. While larger trees produce more accurate clusterings, they also require more complex explanations. To allow flexibility, we develop a new explainable $k$-means clustering algorithm, ExKMC…
The Chinese University of Hong Kong（香港中文大学）；Microsoft Research Asia（微软亚洲研究院）
Recent spectral clustering methods are a propular and powerful technique for data clustering. These methods need to solve the eigenproblem whose computational complexity is $O(n^3)$, where $n$ is the number of data samples. In this paper, a non-eigenproblem based clustering method is proposed to deal with the clustering problem. Its performance is comparable to the spectral clustering algorithms but it is more efficient with computational complexity $O(n^2)$. We show that with a transitive distance and an observed property, called K-means duality, our algorithm can be used to handle data sets …
Yahoo Labs（雅虎实验室）；IBM Zürich Research Lab（IBM苏黎世研究实验室）；UC Berkeley（加州大学伯克利分校）；Rensselaer Polytechnic Institute（伦斯勒理工学院）
We study the topic of dimensionality reduction for $k$-means clustering. Dimensionality reduction encompasses the union of two approaches: \emph{feature selection} and \emph{feature extraction}. A feature selection based algorithm for $k$-means clustering selects a small subset of the input features and then applies $k$-means clustering on the selected features. A feature extraction based algorithm for $k$-means clustering constructs a small set of new artificial features and then applies $k$-means clustering on the constructed features. Despite the significance of $k$-means clustering as well…
International University of Business Agriculture and Technology（国际商业农业与技术大学）；Mississippi State University（密西西比州立大学）；Rajshahi College（拉杰沙希学院）
In this paper, several two-dimensional clustering scenarios are given. In those scenarios, soft partitioning clustering algorithms (Fuzzy C-means (FCM) and Possibilistic c-means (PCM)) are applied. Afterward, VAT is used to investigate the clustering tendency visually, and then in order of checking cluster validation, three types of indices (e.g., PC, DI, and DBI) were used. After observing the clustering algorithms, it was evident that each of them has its limitations; however, PCM is more robust to noise than FCM as in case of FCM a noise point has to be considered as a member of any of the …
Among many clustering algorithms, the K-means clustering algorithm is widely used because of its simple algorithm and fast convergence. However, this algorithm suffers from incomplete data, where some samples have missed some of their attributes. To solve this problem, we mainly apply MM principles to restore the symmetry of the data, so that K-means could work well. We give the pseudo-code of the algorithm and use the standard datasets for experimental verification. The source code for the experiments is publicly available in the following link: \url{https://github.com/AliBeikmohammadi/MM-Opt…
Quantitative Clustering in Mean-Field Transformer Models
Department of Mathematics, Massachusetts Institute of Technology（数学系，麻省理工学院）；Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology（电气工程与计算机科学系，麻省理工学院）
K-means Clustering is the most well-known partitioning algorithm among all clustering, by which we can partition the data objects very easily in to more than one clusters. However, for K-means to choose an appropriate number of clusters without any prior domain knowledge about the dataset is challenging, especially in high-dimensional data objects. Hence, we have implemented the Silhouette and Elbow methods with PCA to find an optimal number of clusters. Also, previously, so many meta-heuristic swarm intelligence algorithms inspired by nature have been employed to handle the automatic data clu…
This note introduces a novel clustering preserving transformation of cluster sets obtained from $k$-means algorithm. This transformation may be used to generate new labeled data{}sets from existent ones. It is more flexible that Kleinberg axiom based consistency transformation because data points in a cluster can be moved away and datapoints between clusters may come closer together.
The Ohio State University（俄亥俄州立大学）；Johns Hopkins University（约翰斯·霍普金斯大学）
We introduce a sketch-and-solve approach to speed up the Peng-Wei semidefinite relaxation of k-means clustering. When the data is appropriately separated we identify the k-means optimal clustering. Otherwise, our approach provides a high-confidence lower bound on the optimal k-means value. This lower bound is data-driven; it does not make any assumption on the data nor how it is generated. We provide code and an extensive set of numerical experiments where we use this approach to certify approximate optimality of clustering solutions obtained by k-means++.
Stefan Cardinal Wyszyński University in Warsaw（华沙斯特凡·维辛斯基大学）；Polish Academy of Sciences（波兰科学院）
This paper investigates the validity of Kleinberg's axioms for clustering functions with respect to the quite popular clustering algorithm called $k$-means. While Kleinberg's axioms have been discussed heavily in the past, we concentrate here on the case predominantly relevant for $k$-means algorithm, that is behavior embedded in Euclidean space. We point at some contradictions and counter intuitiveness aspects of this axiomatic set within $\mathbb{R}^m$ that were evidently not discussed so far. Our results suggest that apparently without defining clearly what kind of clusters we expect we wil…
Afeka Tel-Aviv Academic College of Engineering, School of Electrical Engineering, Israel（阿费卡特拉维理工学院电子工程学院，以色列）；Avignon University, LIA, France（阿维尼翁大学，法国）；Sapir Academic College, Department of Engineering, Sderot, Israel（萨皮尔学术学院工程系，以色列）；Shamoon College of Engineering, Department of Electrical Engineering, Ashdod, Israel（沙蒙理工学院电子工程系，以色列）
