An invariance property of kernel based predictors

We consider kernel based learning methods for regression and analyze what happens to the risk minimizer when new variables, statistically independent of input and target variables, are added to the set of input variables; this problem arises, for example, in the detection of causality relations between two time series. We find that the risk minimizer remains unchanged if we constrain the risk minimization to hypothesis spaces induced by suitable kernel functions. We show that not all kernel induced hypothesis spaces enjoy this property. We present sufficient conditions ensuring that the risk minimizer does not change, and show that they hold for inhomogeneous polynomial and Gaussian RBF kernels. We also provide examples of kernel induced hypothesis spaces whose risk minimizer changes if independent variables are added as input.