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Training primal twin support vector regression via unconstrained convex minimization

APPLIED INTELLIGENCE 2016年第4期44卷 931-955页

作者： Balasundaram, S. Meena, Yogendra Jawaharlal Nehru Univ Sch Comp & Syst Sci New Delhi 110067 India

In this paper, we propose a new unconstrained twin support vector regression model in the primal space (UPTSVR). With the addition of a regularization term in the formulation of the problem, the structural risk is minimized. The proposed formulation solves two smaller sized unconstrained minimization problems having continues, piece-wise quadratic objective functions by gradient based iterative methods. However, since their objective functions contain the non-smooth 'plus' function, two approaches are taken: (i) replace the non-smooth 'plus' function with their smooth approximate functions;(ii) apply a generalized derivative of the non-smooth 'plus' function. They lead to five algorithms whose pseudo-codes are also given. Experimental results obtained on a number of interesting synthetic and real-world benchmark datasets using these algorithms in comparison with the standard support vector regression (SVR) and twin SVR (TSVR) clearly demonstrates the effectiveness of the proposed method.

关键词： Gradient based iterative methods Smooth approximation Support vector regression Twin support vector regression unconstrained convex minimization

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Training Lagrangian twin support vector regression via unconstrained convex minimization

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KNOWLEDGE-BASED SYSTEMS 2014年 59卷 85-96页

作者： Balasundaram, S. Gupta, Deepak Jawaharlal Nehru Univ Sch Comp & Syst Sci New Delhi 110067 India

In this paper, a new unconstrained convex minimization problem formulation is proposed as the Lagrangian dual of the 2-norm twin support vector regression (TSVR). The proposed formulation leads to two smaller sized unconstrained minimization problems having their objective functions piece-wise quadratic and differentiable. It is further proposed to apply gradient based iterative method for solving them. However, since their objective functions contain the non-smooth 'plus' function, two approaches are taken: (i) either considering their generalized Hessian or introducing a smooth function in place of the 'plus' function, and applying Newton-Armijo algorithm;(ii) obtaining their critical points by functional iterative algorithm. Computational results obtained on a number of synthetic and real-world benchmark datasets clearly illustrate the superiority of the proposed unconstrained Lagrangian twin support vector regression formulation as comparable generalization performance is achieved with much faster learning speed in accordance with the classical support vector regression and TSVR. (C) 2014 Elsevier B.V. All rights reserved.

关键词： Generalized Hessian Gradient based iterative methods Smooth approximation Support vector regression Twin support vector regression unconstrained convex minimization

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Lagrangian support vector regression via unconstrained convex minimization

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NEURAL NETWORKS 2014年 51卷 67-79页

作者： Balasundaram, S. Gupta, Deepak Kapil Jawaharlal Nehru Univ Sch Comp & Syst Sci New Delhi 110067 India

In this paper, a simple reformulation of the Lagrangian dual of the 2-norm support vector regression (SVR) is proposed as an unconstrained minimization problem. This formulation has the advantage that its objective function is strongly convex and further having only m variables, where m is the number of input data points. The proposed unconstrained Lagrangian SVR (ULSVR) is solvable by computing the zeros of its gradient. However, since its objective function contains the non-smooth 'plus' function, two approaches are followed to solve the proposed optimization problem: (i) by introducing a smooth approximation, generate a slightly modified unconstrained minimization problem and solve it;(ii) solve the problem directly by applying generalized derivative. Computational results obtained on a number of synthetic and real-world benchmark datasets showing similar generalization performance with much faster learning speed in accordance with the conventional SVR and training time very close to least squares SVR clearly indicate the superiority of ULSVR solved by smooth and generalized derivative approaches. (C) 2013 Elsevier Ltd. All rights reserved.

关键词： Generalized derivative approach Smooth approximation Support vector regression unconstrained convex minimization

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Efficient implicit Lagrangian twin parametric insensitive support vector regression via unconstrained minimization problems

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ANNALS OF MATHEMATICS AND ARTIFICIAL INTELLIGENCE 2021年第3-4期89卷 301-332页

作者： Gupta, Deepak Richhariya, Bharat Natl Inst Technol Dept Comp Sci & Engn Yupia 791112 Arunachal Prade India

In this paper, an efficient implicit Lagrangian twin parametric insensitive support vector regression is proposed which leads to a pair of unconstrained minimization problems, motivated by the works on twin parametric insensitive support vector regression (Peng: Neurocomputing. 79, 26-38, 2012), and Lagrangian twin support vector regression (Balasundaram and Tanveer: Neural Comput. Applic. 22(1), 257-267, 2013). Since its objective function is strongly convex, piece-wise quadratic and differentiable, it can be solved by gradient-based iterative methods. Notice that its objective function having non-smooth 'plus' function, so one can consider either generalized Hessian, or smooth approximation function to replace the 'plus' function and further apply the simple Newton-Armijo step size algorithm. These algorithms can be easily implemented in MATLAB and do not require any optimization toolbox. The advantage of this method is that proposed algorithms take less training time and can deal with data having heteroscedastic noise structure. To demonstrate the effectiveness of the proposed method, computational results are obtained on synthetic and real-world datasets which clearly show comparable generalization performance and improved learning speed in accordance with support vector regression, twin support vector regression, and twin parametric insensitive support vector regression.

关键词： Support vector regression Twin support vector regression Regression estimation Parametric insensitive model unconstrained convex minimization

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On robust asymmetric Lagrangian ν-twin support vector regression using pinball loss function

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APPLIED SOFT COMPUTING 2021年 102卷

作者： Gupta, Deepak Gupta, Umesh Natl Inst Technol Arunachal Pradesh Dept Comp Sci & Engn Yupia Arunachal Prade India

The main objective of twin support vector regression (TSVR) is to find the optimum regression function based on the epsilon-insensitive up- and down-bound with equal influences on the regression function where all the data points have a different location above the up-bound points and below the down-bound points. However, the effects of all data points must be distinct based on their distribution in the regression function. Recently, asymmetric nu-twin support vector regression (Asy-nu-TSVR) is encouraged on the same subject but still, the present matrices in the mathematical formulation have faced the problem of semi-definite. In order to handle this problem effectively, a new regressor model named as robust asymmetric Lagrangian nu-twin support vector regression using pinball loss function (URALTSVR) proposes as a pair of the unconstrained minimization problem to handle not only the noise sensitivity and instability of re-sampling but also consist positive definite matrices. Here, we suggest the proposed model URALTSVR in such a way where the pinball loss function is playing a vital role to control the fitting error inside the asymmetric tube. One of the advantages is that unlike TSVR and Asy-nu-TSVR, it considers the concept of structural risk minimization principle through the inclusion of regularization term as well as change the one-norm of the vector of the slack variable by two-norm, which yields the dual problem to be strongly convex, stable and well-posed. Aforementioned, the proposed formulation has a continuous and piecewise quadratic problem that is solved by their gradients based iterative approaches. Specifically, we analyze the three implementations of URALTSVR with the baselines approaches support vector regression (SVR), TSVR and Asy-nu-TSVR, which discard the dependencies to solve a pair of quadratic programming problem (QPP) for obtaining the unique global solution. Overall, SRALTSVR1 based on smooth approximation function performs outstanding fo

关键词： Support vector regression Asymmetric loss function Pinball loss unconstrained convex minimization Twin support vector regression

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One norm linear programming support vector regression

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NEUROCOMPUTING 2016年 173卷 1508-1518页

作者： Tanveer, Mohammad Mangal, Mohit Ahmad, Izhar Shao, Yuan-Hai Nanyang Technol Univ Sch Comp Engn Singapore 639798 Singapore LNM Inst Informat Technol Dept Commun & Comp Engn Jaipur 302031 Rajasthan India King Fahd Univ Petr & Minerals Dept Math & Stat Dhahran 31261 Saudi Arabia Zhejiang Univ Technol Zhijiang Coll Hangzhou 310024 Zhejiang Peoples R China

In this paper, a new linear programming formulation of a 1-norm support vector regression (SVR) is proposed whose solution is obtained by solving an exterior penalty problem in the dual space as an unconstrained minimization problem using Newton method. The solution of modified unconstrained minimization problem reduces to solving just system of linear equations as opposed to solving quadratic programming problem in SVR, which leads to extremely simple and fast algorithm. The algorithm converges from any starting point and can be easily implemented in MATLAB without using any optimization packages. The main advantage of the proposed approach is that it leads to a robust and sparse model representation meaning that many components of the optimal solution vector will become zero and therefore the decision function can be determined using much less number of support vectors in comparison to SVR, smooth SVR (SSVR) and weighted SVR (WSVR). To demonstrate its effectiveness, experiments were performed on well-known synthetic and real-world benchmark datasets. Similar or better generalization performance of the proposed method in less training time in comparison with SVR, SSVR and WSVR clearly exhibits its suitability and applicability. (C) 2015 Elsevier B.V. All rights reserved.

关键词： Linear programming 1-Norm support vector machines Newton method unconstrained convex minimization

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Newton method for implicit Lagrangian twin support vector machines

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INTERNATIONAL JOURNAL OF MACHINE LEARNING AND CYBERNETICS 2015年第6期6卷 1029-1040页

作者： Tanveer, M. Nanyang Technol Univ Sch Comp Engn Singapore 639798 Singapore

In this paper, we proposed an implicit Lagrangian twin support vector machine (TWSVM) classifiers by formulating a pair of unconstrained minimization problems (UMPs) in dual variables whose solutions will be obtained using finite Newton method. The advantage of considering the generalized Hessian approach for our modified UMPs reduces to solving just two systems of linear equations as opposed to solving two quadratic programming problems in TWSVM and TBSVM, which leads to extremely simple and fast algorithm. Unlike the classical TWSVM and least square TWSVM (LSTWSVM), the structural risk minimization principle is implemented by adding regularization term in the primal problems of our proposed algorithm. This embodies the essence of statistical learning theory. Computational comparisons of our proposed method against GEPSVM, TWSVM, STWSVM and LSTWSVM have been made on both synthetic and well-known real world benchmark datasets. Experimental results show that our method yields significantly better generalization performance in both computational time and classification accuracy.

关键词： Machine learning Nonparallel planes Lagrangian support vector machines Twin support vector machines unconstrained convex minimization

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Application of smoothing techniques for linear programming twin support vector machines

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KNOWLEDGE AND INFORMATION SYSTEMS 2015年第1期45卷 191-214页

作者： Tanveer, M. LNM Inst Informat Technol Dept Comp Sci & Engn Jaipur 302031 Rajasthan India

In this paper, a new unconstrained minimization problem formulation is proposed for linear programming twin support vector machine (TWSVM) classifiers. The proposed formulation leads to two smaller-sized unconstrained minimization problems having their objective functions piecewise differentiable. However, since their objective functions contain the non-smooth "plus" function, two new smoothing approaches are assumed to solve the proposed formulation, and then apply Newton-Armijo algorithm. The idea of our formulation is to reformulate TWSVM as a strongly convex problem by incorporated regularization techniques and then derive smooth 1-norm linear programming formulation for TWSVM to improve robustness. One significant advantage of our proposed algorithm over TWSVM is that the structural risk minimization principle is implemented in the primal problems which embodies the marrow of statistical learning theory. In addition, the solution of two modified unconstrained minimization problems reduces to solving just two systems of linear equations as opposed to solving two quadratic programming problems in TWSVM and TBSVM, which leads to extremely simple and fast algorithm. Our approach has the advantage that a pair of matrix equation of order equals to the number of input examples is solved at each iteration of the algorithm. The algorithm converges from any starting point that can be easily implemented in MATLAB without using any optimization packages. The performance of our proposed method is verified experimentally on several benchmark and synthetic datasets. Experimental results show the effectiveness of our methods in both training time and classification accuracy.

关键词： Linear programming Twin support vector machines unconstrained convex minimization Smoothing techniques 1-Norm support vector machines

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Robust and Sparse Linear Programming Twin Support Vector Machines

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COGNITIVE COMPUTATION 2015年第1期7卷 137-149页

作者： Tanveer, M. LNM Inst Informat Technol Dept Comp Sci & Engn Jaipur 302031 Rajasthan India

In this paper, we propose a new linear programming formulation of exact 1-norm twin support vector machine (TWSVM) for classification whose solution is obtained by solving a pair of dual exterior penalty problems as unconstrained minimization problems using Newton-Armijo algorithm. The idea of our formulation is to reformulate TWSVM as a strongly convex problem by incorporated regularization techniques and then derive an exact 1-norm linear programming formulation for TWSVM to improve robustness and sparsity. The solution of two modified unconstrained minimization problems reduces to solving just two systems of linear equations as opposed to solving two quadratic programming problems in TWSVM and TBSVM, which leads to extremely simple and fast algorithm. One significant advantage of our proposed method is the implementation of structural risk minimization principle. However, only empirical risk is considered in the primal problems of TWSVM due to its complex structure and thus may incur overfitting and suboptimal in some cases. Our approach has the advantage that a pair of matrix equation of order equals to the number of input examples is solved at each iteration of the algorithm. The algorithm converges from any starting point that can be easily implemented in MATLAB without using any optimization packages. Computational comparisons of our proposed method against original TWSVM, GEPSVM and SVM have been made on both synthetic and benchmark datasets. Experimental results show that our method is better or comparable in both computation time and classification accuracy.

关键词： Generalized Hessian unconstrained convex minimization Twin support vector machines Linear programming 1-norm support vector machines

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