Admissibility of fuzzy support vector machine through loss function

In statistical decision theory, the admissibility is the first issue to fulfill the feasibility of a decision rule. Without the admissibility, the decision rule is impractical for discriminations. The study decomposes first the fuzzy support vector machine (fuzzy SVM), which is a crucial innovation due to its robust capability to resist the input contaminated noise, into a regularized optimization expression arg min
_f∈H
Ω[f]+λ
_R
R
_emp
[f] and exploits the regularization of loss function from the expression mathematically. The decomposition is beneficial to the programming of empirical risk minimization which uses the empirical risk instead of the true expected risk to learn a hypothesis. The empirical risk, composed elementally by the loss function, here indeed is the key for achieving the success of the fuzzy SVM. Because of the important causality, the study examines preliminarily the admissibility of loss functions which is recruited to form the fuzzy SVM. The examination is issued first by a loss function associated risk, called □-risk. By a step-by-step derivation of a sufficient and necessary condition for the □-risk to agree equivalently an unbiased Bayes risk, the admissibility of the loss function can then be confirmed and abbreviated as a simple rule in the study. Experimental chart examination is also issued simultaneously for an easy and clear observation to validate the admissibility of the loss function regularized fuzzy SVM.