Bouligand Derivatives and Robustness of Support Vector Machines for Regression

We investigate robustness properties for a broad class of support vector machines with non-smooth loss functions. These kernel methods are inspired by convex risk minimization in infinite dimensional Hilbert spaces. Leading examples are the support vector machine based on the ε-insensitive loss function, and kernel based quantile regression based on the pinball loss function. Firstly, we propose with the Bouligand influence function (BIF) a modification of F.R. Hampel’s influence function. The BIF has the advantage of being positive homogeneous which is in general not true for Hampel’s influence function. Secondly, we show that many support vector machines based on a Lipschitz continuous loss function and a bounded kernel have a bounded BIF and are thus robust in the sense of robust statistics based on influence functions.