Avoidance of non-strict saddle points by blow-up

Abstract

It is an old idea to use gradient flows or time-discretized variants thereof as methods for solving minimization problems. In some applications, for example in machine learning contexts, it is important to know that for generic initial data, gradient flow trajectories do not get stuck at saddle points. There are classical results concerned with the non-degenerate situation. But if the Hessian of the objective function has a non-trivial kernel at the critical point, then these results are inconclusive in general. In this paper, we show how relevant information can be extracted by ``blowing up'' the objective function around the non-strict saddle point, i.e., by a suitable non-linear rescaling that makes the higher order geometry visible.