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Intuitionistic Fuzzy Proximal Support Vector Machines for Pattern Classification

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Abstract

Support vector machine is a powerful technique for classification and regression problems. In the binary data problems, it classifies the points by assigning them to one of the two disjoint halfspaces. However, this method fails to handle the noises and outliers present in the dataset and the solution of a large-sized quadratic programming problem is required to obtain the decision surface in input or in feature space. We propose the intuitionistic fuzzy proximal support vector machine (IFPSVM) which classifies the patterns according to its proximity with the two parallel planes that are kept as distant as possible from each other. These two parallel ‘proximal’ planes can be obtained by solving a system of linear equations only. There is an intuitionistic fuzzy number associated with each training point which is framed by its degree of membership and non-membership. The membership degree of a pattern considers its distance from the corresponding class center and the degree of non-membership of a pattern is given by the ratio of the number of heterogeneous points to the number of total points in its neighborhood. The proposed technique effectively reduces the impact of noises and distinguishes the edge support vectors and outliers. Computational simulations on an artificial and eleven UCI benchmark datasets using linear, polynomial and Gaussian kernel functions, show the effectiveness of the proposed IFPSVM method. The experiments prove that it can handle large datasets with less computational time and yields better accuracy.

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Acknowledgements

The authors sincerely thank the reviewers for the recommendation, valuable comments and the interesting suggestions which have considerably improved the presentation of the paper. The first author is grateful to the Ministry of Human Resource Development, India, for financial support, to carry out this work.

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Authors and Affiliations

  1. Department of Mathematics, IIT Roorkee, Roorkee, Uttarakhand, India

    Scindhiya Laxmi & Shiv Kumar Gupta

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  1. Scindhiya Laxmi
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  2. Shiv Kumar Gupta
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Correspondence to Shiv Kumar Gupta.

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Appendix A

Appendix A

Let H be any rectangular matrix of order \(m \times (n+1),~I\) be an identity matrix of order \((n+1)\times (n+1)\) and \(C>0.\) Then the matrix \(\displaystyle \left( H^TH+\frac{I}{C}\right) \) is always invertible.

Proof

Let x be a vector of order \((n+1) \times 1\). Then

$$\begin{aligned} x^TH^THx= & {} (Hx)^T(Hx)\\= & {} ||Hx||^2\\\ge & {} 0. \end{aligned}$$

This implies that \(H^TH\) is a positive semi definite matrix.

This further yields \(\displaystyle \bigg (H^TH+\frac{I}{C}\bigg ),\) where \(C>0\) is a positive real number, is positive definite matrix and hence it will be invertible. \(\square \)

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Laxmi, S., Gupta, S.K. Intuitionistic Fuzzy Proximal Support Vector Machines for Pattern Classification. Neural Process Lett 51, 2701–2735 (2020). https://doi.org/10.1007/s11063-020-10222-x

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