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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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