  The support vector machine (SVM) and minimum Euclidean norm least squares
regression are two fundamentally different approaches to fitting linear models,
but they have recently been connected in models for very high-dimensional data
through a phenomenon of support vector proliferation, where every training
example used to fit an SVM becomes a support vector. In this paper, we explore
the generality of this phenomenon and make the following contributions. First,
we prove a super-linear lower bound on the dimension (in terms of sample size)
required for support vector proliferation in independent feature models,
matching the upper bounds from previous works. We further identify a sharp
phase transition in Gaussian feature models, bound the width of this
transition, and give experimental support for its universality. Finally, we
hypothesize that this phase transition occurs only in much higher-dimensional
settings in the $\ell_1$ variant of the SVM, and we present a new geometric
characterization of the problem that may elucidate this phenomenon for the
general $\ell_p$ case.
