Model Selection in Kernel Based Regression using the Influence Function

Michiel Debruyne, Mia Hubert, Johan A.K. Suykens.

Year: 2008, Volume: 9, Issue: 78, Pages: 2377−2400


Recent results about the robustness of kernel methods involve the analysis of influence functions. By definition the influence function is closely related to leave-one-out criteria. In statistical learning, the latter is often used to assess the generalization of a method. In statistics, the influence function is used in a similar way to analyze the statistical efficiency of a method. Links between both worlds are explored. The influence function is related to the first term of a Taylor expansion. Higher order influence functions are calculated. A recursive relation between these terms is found characterizing the full Taylor expansion. It is shown how to evaluate influence functions at a specific sample distribution to obtain an approximation of the leave-one-out error. A specific implementation is proposed using a L1 loss in the selection of the hyperparameters and a Huber loss in the estimation procedure. The parameter in the Huber loss controlling the degree of robustness is optimized as well. The resulting procedure gives good results, even when outliers are present in the data.