Phase transition in the hard-margin support vector machines

This paper establishes a phase transition for convergence of the hard-margin support vector machines (SVM) in high dimensional and numerous data regime, drawn from a Gaussian mixture distribution. Particularly, we characterize the maximum number of training samples that the hard-margin SVM is capable of perfectly separating. Under the assumption that the number of training samples is less than this threshold, we provide a sharp characterization of the margin parameter and the classification error performance of the hard-margin SVM classifier. Our analysis, validated through a set of numerical experiments, is based on the convex Gaussian min-max framework.