Abstract
Recently, soft margin smooth support vector machine with 1-norm penalty term (SSVM1) is discovered to possess better outlier resistance than soft margin smooth support vector machine with 2-norm penalty term (SSVM2). One of the most important steps in the framework of SSVMs is to replace the x+ by a differential function in the primal model, and get an approximate solution. This study proposes one function constructed by Padé approximant via the formal orthogonal polynomials as the smoothing technique, and a new 1-norm SSVM, Padé SSVM1, is represented. A method for outlier filtering is proposed to improve the ability of outlier resistance. The experimental results show that Padé SSVM1, even without outlier filtering, performs better than the previous SSVM2 and SSVM1 on the polluted synthetic datasets.
Highlights
Support vector machines (SVMs) have been proven to be one of the promising learning algorithms for classification [1]
The experiments in [2] showed that Sigmoid SSVM1 can remedy the drawback of 2-norm soft margin smooth support vector machine (SSVM2) [3] for outlier effect and get outlier resistance
SVMs have the advantage of being robust for outlier effect [4], there are still some violent cases that will mislead SVM classifiers to lose their generalization ability for prediction, even the good sigmoid SSVM1 became powerless at this time
Summary
Support vector machines (SVMs) have been proven to be one of the promising learning algorithms for classification [1]. In an analogous manner as in [3], it is easy to be proved that the solution of (5) converges to the unique solution of the primal problem when the smoothing parameter η in the SSVM1 approaches infinity It is just because of the truth: if the value of η increases, the ρ(x,η) will approximate the plus function more accurately. Where 1 denotes a column vector of ones for arbitrary dimension, and function P has an effect on all components of a matrix or a vector in (21), i.e., P(1D(Aw+1b),η)∈Rm,(P(1-D(Aw+1b),η))i=P(1-Di(Aiw+b), η), and η whose value is not a main factor for the final SSVM1 is called smoothing parameter. The Newton-Armijo algorithm with respect to SSVM1 is omitted here because it is running the same procedure as that in 2-norm problem
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