Learning with uncertain kernel matrix set

We study support vector machines (SVM) for which the kernel matrix is not specified exactly and it is only known to belong to a given uncertainty set. We consider uncertainties that arise from two sources: (i) data measurement uncertainty, which stems from the statistical errors of input samples; (ii) kernel combination uncertainty, which stems from the weight of individual kernel that needs to be optimized in multiple kernel learning (MKL) problem. Much work has been studied, such as uncertainty sets that allow the corresponding SVMs to be reformulated as semi-definite programs (SDPs), which is very computationally expensive however. Our focus in this paper is to identify uncertainty sets that allow the corresponding SVMs to be reformulated as second-order cone programs (SOCPs), since both the worst case complexity and practical computational effort required to solve SOCPs is at least an order of magnitude less than that needed to solve SDPs of comparable size. In the main part of the paper we propose four uncertainty sets that meet this criterion. Experimental results are presented to confirm the validity of these SOCP reformulations.