ORIGINAL ARTICLE Fuzzy least squares twin support vector clustering Reshma Khemchandani 1 • Aman Pal 1 • Suresh Chandra 2 Received: 12 September 2015 / Accepted: 6 July 2016 Ó The Natural Computing Applications Forum 2016 Abstract In this paper, we have formulated a fuzzy least squares version of recently proposed clustering method, namely twin support vector clustering (TWSVC). Here, a fuzzy membership value of each data pattern to different cluster is optimized and is further used for assigning each data pattern to one or other cluster. The formulation leads to finding k cluster center planes by solving modified pri- mal problem of TWSVC, instead of the dual problem usually solved. We show that the solution of the proposed algorithm reduces to solving a series of system of linear equations as opposed to solving series of quadratic pro- gramming problems along with system of linear equations as in TWSVC. The experimental results on several publicly available datasets show that the proposed fuzzy least squares twin support vector clustering (F-LS-TWSVC) achieves comparable clustering accuracy to that of TWSVC with comparatively lesser computational time. Further, we have given an application of F-LS-TWSVC for segmentation of color images. Keywords Machine learning  Twin support vector clustering  Plane-based clustering  Fuzzy clustering 1 Introduction Clustering is a powerful tool which aims at grouping similar objects into the same cluster and dissimilar objects into different clusters by identifying dominant structures in the data. It has remained a widely studied research area in machine learning [1, 2] and has applications in diverse domains such as computer vision, text mining, bioinfor- matics and signal processing [3–6]. Traditional point-based clustering methods such as k- means [1] and k-median [7] work by partitioning the data into clusters based on the cluster prototype points. These methods perform poorly in case when data are not dis- tributed around several cluster points. In contrast to these, plane-based clustering methods such as k-plane cluster- ing [8], proximal plane clustering [9] and local k-proximal plane clustering [10] have been proposed in the literature. These methods calculate k cluster center planes and parti- tion the data into k clusters according to the proximity of the data points with these k planes. Jayadeva et al. [11] have proposed twin support vector machine (TWSVM) classifier for binary data classification where the two hyperplanes are obtained by solving two related smaller-sized quadratic programming problems (QPPs) as compare to single large-sized QPP in conven- tional support vector machine (SVM). Mehrkanoon et al. [12] introduced a general framework of non-parallel support vector machines, which involves a regularization term, a scatter loss and a misclassification loss. Taking motivation from Xie and Sun [11, 13], they have proposed multi-view twin support vector machines in the semi-su- pervised learning framework which combines two views by introducing the constraint of similarity between two one- dimensional projections identifying two distinct TWSVMs from two feature spaces. An inherent shortcoming of twin & Reshma Khemchandani reshma.khemchandani@sau.ac.in Aman Pal aman.pal@students.sau.ac.in Suresh Chandra chandra@maths.iitd.ac.in 1 South Asian University, New Delhi, India 2 Indian Institute of Technology, New Delhi, India 123 Neural Comput & Applic DOI 10.1007/s00521-016-2468-4