  Recent advance on linear support vector machine with the 0-1 soft margin loss
($L_{0/1}$-SVM) shows that the 0-1 loss problem can be solved directly.
However, its theoretical and algorithmic requirements restrict us extending the
linear solving framework to its nonlinear kernel form directly, the absence of
explicit expression of Lagrangian dual function of $L_{0/1}$-SVM is one big
deficiency among of them. In this paper, by applying the nonparametric
representation theorem, we propose a nonlinear model for support vector machine
with 0-1 soft margin loss, called $L_{0/1}$-KSVM, which cunningly involves the
kernel technique into it and more importantly, follows the success on
systematically solving its linear task. Its optimal condition is explored
theoretically and a working set selection alternating direction method of
multipliers (ADMM) algorithm is introduced to acquire its numerical solution.
Moreover, we firstly present a closed-form definition to the support vector
(SV) of $L_{0/1}$-KSVM. Theoretically, we prove that all SVs of $L_{0/1}$-KSVM
are only located on the parallel decision surfaces. The experiment part also
shows that $L_{0/1}$-KSVM has much fewer SVs, simultaneously with a decent
predicting accuracy, when comparing to its linear peer $L_{0/1}$-SVM and the
other six nonlinear benchmark SVM classifiers.
