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A comparative study for error approximation of some kernel functions in Smooth Support Vector Machines

机译:光滑支持向量机中一些内核功能误差近似的比较研究

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Support Vector Machine (SVM) as one of the most popular machine learning methods is playing a significant role in statistical learning theory. Smooth Support Vector Machine (SSVM) is one of new formulation to improve the SVM. In SSVM, smoothing method is used to optimize the unconstrained model. Smoothing function can be used to replace plus function in SVM. In this paper we evaluate eight smoothing functions including quadratic polynomial, fourth order polynomial, piecewise polynomial, spline function, sixth order polynomial, advanced fourth order polynomial function, quadratic Bezier function, third order Bezier function, and fourth order Bezier function. Some of those functions have been studied previously in order to find better accuracy in SVM. In this research, we evaluate and analyze the performance of all those eight smoothing functions based on their infinity-norm values. We compare the error approximation between smoothing function and plus function (as the standard kernel function in SVM) where the best smoothing function has the minimum error of its infinity-norm. Based on theoretical analysis and numerical approximation, our results show that piecewise polynomial function is better than quadratic polynomial function, fourth polynomial function, and third order spline function. While the advanced fourth order polynomial function and sixth order polynomial function have error approximation value almost the same as that of plus function. However, since the piecewise polynomial function has less control parameters than those of the advanced fourth order polynomial function and the sixth order polynomial function, we could not conclude which one is the best. Furthermore, based on values of infinity-norm of all the smoothing functions, we found that the quadratic Bezier and quadratic polynomial show the same error values. Meanwhile the forth order Bezier function shows the smallest error approximation value among the other functions which have been tested. In conclusion, based on our results in this study we found that the forth order Bezier function is the best choice among the eight smoothing function for SSVM.
机译:支持向量机(SVM)作为最受欢迎的机器学习方法之一在统计学习理论中发挥着重要作用。光滑的支持向量机(SSVM)是改善SVM的新配方之一。在SSVM中,使用平滑方法来优化不受约束的模型。平滑功能可用于在SVM中替换加函数。在本文中,我们评估了八个平滑功能,包括二次多项式,四阶多项式,分段多项式,样条函数,第六级多项式,前进的四阶多项式,二次贝塞函数,三阶Bezier函数和四阶Bezier函数。以前研究了其中一些功能,以便在SVM中找到更好的准确性。在这项研究中,我们根据其无穷大值评估和分析所有这八个平滑功能的性能。我们将平滑函数和加函数(作为SVM中的标准内核函数的函数之间的错误近似进行比较,其中最佳平滑功能具有其无限常态的最小误差。基于理论分析和数值近似,我们的结果表明,分段多项式函数优于二次多项式函数,第四多项式函数和三阶样条函数。虽然高级的四阶多项式函数和第六阶多项式函数具有误差近似值与Plus函数的误差近似值几乎相同。然而,由于分段多项式函数的控制参数较少,而不是先进的四阶多项式函数和第六阶多项式函数,因此我们无法得出结论是哪一个是最好的。此外,基于所有平滑函数的无穷大的值,我们发现二次贝塞尔和二次多项式显示相同的误差值。同时,第四级逐令函数显示了已经测试的其他功能中的最小误差近似值。总之,根据我们的研究结果,我们发现,第四次偏标功能是SSVM八个平滑功能中的最佳选择。

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