Binary Classification plays an important role in machine learning. For linear
classification, SVM is the optimal binary classification method. For nonlinear
classification, the SVM algorithm needs to complete the classification task by
using the kernel function. Although the SVM algorithm with kernel function is
very effective, the selection of kernel function is empirical, which means that
the kernel function may not be optimal. Therefore, it is worth studying how to
obtain an optimal binary classifier.
In this paper, the problem of finding the optimal binary classifier is
considered as a variational problem. We design the objective function of this
variational problem through the max-min problem of the (Euclidean) distance
between two classes. For linear classification, it can be deduced that SVM is a
special case of this variational problem framework. For Euclidean distance, it
is proved that the proposed variational problem has some limitations for
nonlinear classification. Therefore, how to design a more appropriate objective
function to find the optimal binary classifier is still an open problem.
Further, it's discussed some challenges and problems in finding the optimal
classifier.

通过最大最小问题设计这个变分问题的目标函数，我们将二元分类器的最优问题视为一个变分问题，并推导出 SVM 是这个变分问题框架的一个特例。然而，对于非线性分类，提出的变分问题在欧氏距离方面存在一些限制，因此如何设计更合适的目标函数来寻找最优二元分类器仍然是一个开放问题。进一步讨论了寻找最优分类器中的一些挑战和问题。