Degeneracy in model selection for SVMs with radial Gaussian kernel

We consider the model selection problem for support vector machines applied to binary classification. As the data generating process is unknown, we have to rely on heuristics as model section criteria. In this study, we analyze the behavior of two criteria, radius margin quotient and kernel polarization, applied to SVMs with radial Gaussian kernel. We proof necessary and sufficient conditions for local optima at the boundary of the kernel parameter space in the limit of arbitrarily narrow kernels. The theorems show that multi-modality of the model selection objectives can arise due to insignificant properties of the training dataset.