A Novel Hierarchical High‐Dimensional Unsupervised Active Learning Method.pdf
Vol.:(0123456789) International Journal of Computational Intelligence Systems          (2024) 17:193   https://doi.org/10.1007/s44196-024-00601-w RESEARCH ARTICLE A Novel Hierarchical High‑Dimensional Unsupervised Active Learning  Method Sajad Haghzad Klidbary1   · Mohammad Javadian2 Received: 11 March 2024 / Accepted: 10 July 2024  © The Author(s) 2024 Abstract This paper processes a novel hierarchical high-dimensional clustering algorithm based on the Active Learning Method  (ALM), which is a fuzzy-learning algorithm. The hierarchical part of the algorithm is composed of two phases: divisible and  agglomerative. The divisible phase, a zooming-in-process, searches for sub-clusters in already-found clusters hierarchically.  At each level of the hierarchy, the clusters are found by an ensemble clustering method based on the density of data. This part  of the algorithm blurs each data point as multiple one-dimensional fuzzy membership functions called ink-drop patterns;  then, it accumulates the ink-drop patterns of all data points on every dimension separately. Next, it performs one-dimensional  density partitioning to produce an ensemble of clustering solutions; after that, combining the results is done based on a novel  consensus method with the aid of prime numbers. An agglomerative phase is a bottom-up approach that merges clusters  based on a novel distance metric, named K2-nearest neighbor. The algorithm is named as the Hierarchical High-Dimensional  Unsupervised Active Learning Method (HiDUALM) and is explained in more detail throughout this paper. Although the  classical clustering methods are not suitable for high-dimensional data clustering, the proposed method solves the problems  related to speed and memory using ensemble learning, while due to its hierarchy and the use of different distance criteria,  different levels of the cluster provide the clause. Experiments on synthetic and real-world datasets are presented to show the  effectiveness of the proposed-clustering algorithm. Keywords  Machine learning (ML) · Ensemble clustering · High-dimensional clustering · Hierarchical clustering ·  Unsupervised active learning method (ALM) · Noise elimination Abbreviations 1-D-MF	 One-dimensional-membership function ALM	 Active learning method AMI	 Adjusted mutual information APCGR​	 Adaptive projected clustering with graph  regularization ARI	 Adjusted rand index CLIQUE	 Clustering in QUEst DBSCAN	 Density-based spatial clustering of  applications with noise DOC	 Density-based optimal projective clustering DPC	 Discriminative projected clustering EPCH	 Efficient projective clustering by histograms FCM	 Fuzzy C-means FUALM	 Fuzzy unsupervised active learning method HiDUALM	 Hierarchical high-dimensional unsupervised  active learning method IDS	 Ink drop spread Ir	 Ink radius IRFLLRR	 Iterative reweighted Frobenius norm  regularized latent low rank representation IRFN	 Iterative reweighted Frobenius norm LatLRR	 Latent LRR (low rank representation) LPFCM	 Locality preserving based fuzzy C-means MAFIA	 Merging of adaptive finite intervals MV-RTSC	 Multi-view robust tensor-based subspace  clustering NP	 Narrow path PROCLUS	 PROjected CLUstering RI	 Rand index SP	 Spread UALM	 Unsupervised active learning method  *	 Sajad Haghzad Klidbary  	 s.haghzad@znu.ac.ir 1	 Faculty of Engineering, Department of Electrical  and Computer Engineering, University of Zanjan, Zanjan,  Iran 2	 School of Electrical Engineering, Shahid Beheshti  University, Tehran, Iran 	 International Journal of Computational Intelligence Systems          (2024) 17:193    193   Page 2 of 26 1  Introduction Clustering is unsupervised learning that categorizes similar  data into the same category. Various clustering algorithms  with different capabilities and structures have been proposed  [1–6]. In high-dimensional data, each object has a large num- ber of features. Examples of high-dimensional data types can  be found in computer vision applications, pattern recognition  [7], and molecular biology [8]. Scalability is one of the major issues in clustering  algorithms which often cause difficulties when facing  problems with high-dimensional datasets [3, 5, 9]. High- dimensional data, suffers from the curse of dimension- ality. In high-dimensional datasets, the distance between  data points become practically indifferentiable. Therefore,  it is hard to separate similar data points from dissimilar  ones. Some clustering algorithms suffer from high time  and space complexity to cluster high-dimensional datasets,  which usually results in malfunctioning of the algorithms.  In addition to these problems, high-dimensional data has  many irrelevant features, meaning clusters are embedded  in the subspaces of the entire feature space. Traditional  clustering algorithms, such as K-means, hierarchical clus- tering, and DBSCAN, UALM [10], and FUALM [11]are  not originally designed for high-dimensional data, and  they often failed when applied to such datasets due to  the “curse of dimensionality”. Therefore, many concepts  are proposed for clustering high-dimensional data, such  as: projected clustering, subspace clustering, multi-view  clustering (MVC), ensemble clustering and hierarchical  clustering for high-dimensional data. Subspace-clustering algorithms, projected clustering  algorithms [12–15] and MVC are introduced to tackle with  high-dimensional clustering difficulties. The main goal  of these algorithms is to find clusters within subspaces  of the entire feature space. The subspace-clustering goal  is to find all clusters in all subspaces, which means that  a data point may belong to multiple clusters [16]. This  phenomenon results in overlapping clusters. On the  other hand, projected clustering, allocates each point  to a unique cluster, thus resulting in non-overlapping  clusters. Subspace and projected clustering algorithms  have their drawbacks. As subspace- clustering-algorithms  produce many large numbers of overlapping clusters,  the interpretation of the result is complicated. Despite  producing non-overlapping clusters, the projected  clustering algorithms still have two limitations. First, the  resulting subspace clusters have different dimensionality.  Second, they are vulnerable to finding clusters of  different shapes and densities [17]. MVC is a subspace  clustering that aims to group similar subjects and separate  dissimilar subjects by utilizing multiple sources of feature  information. The goal is to find consistent clustering  across different views. There are two main categories  of existing MVC methods: generative (or model-based)  approaches and discriminative (or similarity-based)  approaches. Generative approaches focus on learning  the data distribution and exploit generative models to  represent each cluster. Discriminative approaches optimize  an objective function that involves pairwise similarities  to maximize the average similarity between clusters and  minimize the average similarity within clusters [18]. In  addition, some of these algorithms have growing time  complexity as the dimensions of the datasets increase.  Hence, subspace, projected, and MVC clustering have  some restrictions in dealing with high-dimensional data.  These restrictions of subspace-clustering algorithms show  that a flexible high-dimensional clustering algorithm with  a reasonable degree of generality is needed. Ensemble clustering is developed as an essential expansion  of the classical clustering problem. It overcomes the challenges  faced by high-dimensional data and obtains high performance on  various datasets. In the face of high-dimensional data; it divides  the space into a series of subspaces and clusters each subspace  separately. Clustering on the subspace reduces the complexity of  clustering. Ensemble clustering combines the results of different  clustering on a particular dataset, and finds a single (consensus)  clustering result that is better in some sense than existing cluster- ing. Therefore, ensemble clustering integrates clustering results  on the same dataset from different sources. In ensemble cluster- ing, finding the final cluster for each point is an NP-complete  problem [16]. Hierarchical clustering is a method of clustering high- dimensional data that seeks to build a hierarchy of clus- ters. There are two strategies for hierarchical clustering:  agglomerative and divisive. Agglomerative strategy is a  “bottom-up” approach, in which each data start in its clus- ter, and clusters are merged hierarchically until all data  fall into one cluster. Divisive strategy is a “top-down”  approach, where all data start in one cluster, and the clus- ter is split hierarchically until each data fall into a separate  cluster. The results of hierarchical clustering are usually  presented in a dendrogram. Majority of the mentioned clustering algorithms require  the number of clusters to be known in advance while this  is not compatible with the nature of many clustering prob- lems. In this paper, we have used ensemble clustering  along with hierarchical clustering to overcome problems  associated with high-dimensional datasets. The proposed  algorithm does not need the number of clusters as an input  parameter. The algorithm is developed based on the con- cepts of the Active Learning Method (ALM) [19], while  improves the UALM and FUALM-clustering algorithms  developed for low-dimensional datasets. 	 International Journal of Computational Intelligence Systems          (2024) 17:193    193   Page 24 of 26 not unique prime numbers.” We provide a counterexample  to show that the statement is not correct. Counterexample: For a two-dimensional dataset, if  the labels of one-dimension are 1, 2, 3 and the labels of  another dimension are 4, 5, 6, the results of multiplying  the labels will not be unique for each cluster. As shown in  Fig. 17a, the labels of two separate partitions are the same  and are wrongly considered as one cluster. Therefore, if we want a unique label (multiplication  product) for each cluster, the initial labels must be unique  prime numbers. Figure 17b shows the same example, while  the initial labels are unique prime numbers. As it is shown,  the multiplication product for each partition is unique. Acknowledgements  All the experiments and ideas of this research  work have been developed in Computer Engineering Department,  University of Zanjan, IRAN. Authors Contributions  All authors have accepted responsibility for the  entire content of this manuscript and approved its submission. Funding  The author(s) received no financial support for the research,  authorship, and/or publication of this paper. Data Availability  The data that support the findings of this study are  available from the corresponding author, upon reasonable request. Declarations  Conflict of interest  The authors declare that they have no conflicts of  interest with respect to the research, authorship, contribution, and/or  publication of this paper. Ethical Approval  This manuscript has not been published nor is it  currently under consideration for publication elsewhere. 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Title: A Novel Hierarchical High-Dimensional Unsupervised Active Learning Method ​

Authors: Sajad Haghzad Klidbary, Mohammad Javadian ​
Publisher: 
International Journal of Computational Intelligence Systems ​

Date: 2024 ​

Abstract: This paper processes a novel hierarchical high-dimensional clustering algorithm based on the Active Learning Method (ALM), which is a fuzzy-learning algorithm. ​ The hierarchical part of the algorithm is composed of two phases: divisible and agglomerative. ​ The divisible phase, a zooming-in-process, searches for sub-clusters in already-found clusters hierarchically. ​ At each level of the hierarchy, the clusters are found by an ensemble clustering method based on the density of data. ​ This part of the algorithm blurs each data point as multiple one-dimensional fuzzy membership functions called ink-drop patterns; then, it accumulates the ink-drop patterns of all data points on every dimension separately. ​ Next, it performs one-dimensional density partitioning to produce an ensemble of clustering solutions; after that, combining the results is done based on a novel consensus method with the aid of prime numbers. ​ An agglomerative phase is a bottom-up approach that merges clusters based on a novel distance metric, named K 2-nearest neighbor. ​ The algorithm is named as the Hierarchical High-Dimensional Unsupervised Active Learning Method (HiDUALM) and is explained in more detail throughout this paper. ​ Although the classical clustering methods are not suitable for high-dimensional data clustering, the proposed method solves the problems related to speed and memory using ensemble learning, while due to its hierarchy and the use of different distance criteria, different levels of the cluster provide the clause. ​ Experiments on synthetic and real-world datasets are presented to show the effectiveness of the proposed-clustering algorithm. ​
