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.
Keywords
Cite
@article{arxiv.2511.23268,
title = {Avoidance of non-strict saddle points by blow-up},
author = {El Mehdi Achour and Umberto L. Hryniewicz and Michael Westdickenberg},
journal= {arXiv preprint arXiv:2511.23268},
year = {2026}
}
Comments
25 pages, major revision including results for general Riemannian manifolds and criteria for detecting weakly strict saddle-points in linear neural networks