Quantitative Robustness of Localized Support Vector Machines

There are many general introductions to these methods from the view of computer science and statistics. Summarizing textbooks are for example Cristianini & Shawe-Taylor (2000), Schölkopf & Smola (2001), Cucker & Zhou (2007), or Steinwart & Christmann (2008). These methods became pretty popular in many fields of science, see for example Ma & Guo (2014). The analysis provided by this paper refers to supervised learning, i. e. to classification or regression problems. Beyond this, support vector machines are a suitable method for unsupervised learning (e. g. novelty detection), too. The paper can be seen as a sequel to Dumpert & Christmann (2018) where universal consistency and robustness with respect to the maxbias of localized support vector machines have already been shown. This paper is dedicated to refine the robustness analysis. It is organized as follows: Section 2.1 gives a short overview on support vector machines, Section 2.2 introduces shortly the idea of local approaches. The results concerning the influence function of localized support vector machines are given in Section 3. Section 4 finally summarizes the paper.