Qualitative Robustness of Support Vector Machines

TL;DR

This paper demonstrates that support vector machines are qualitatively robust by proving the continuity of their defining functional with respect to weak convergence of probability measures, establishing their well-posedness.

Contribution

It shows that support vector machines are qualitatively robust and well-posed, linking their robustness to the continuity of a functional on probability measures.

Findings

Support vector machines are qualitatively robust.

The functional defining SVMs is continuous under weak convergence.

SVMs are solutions of a well-posed mathematical problem.

Abstract

Support vector machines have attracted much attention in theoretical and in applied statistics. Main topics of recent interest are consistency, learning rates and robustness. In this article, it is shown that support vector machines are qualitatively robust. Since support vector machines can be represented by a functional on the set of all probability measures, qualitative robustness is proven by showing that this functional is continuous with respect to the topology generated by weak convergence of probability measures. Combined with the existence and uniqueness of support vector machines, our results show that support vector machines are the solutions of a well-posed mathematical problem in Hadamard's sense.