Local minima are rare in high dimensions because a strict local minimum has to curve upward in every direction, so all Hessian eigenvalues must be positive.
In a D-dimensional toy model where eigenvalue signs are independent, that’s a 2^(-D) event. In GOE-like random matrix models, positive definiteness is even rarer, roughly exp(-cD^2).
So as dimension grows, random critical points are much more likely to be saddles than minima. This is one reason high-dimensional optimization is often a saddle-escape problem, not a bad-local-minimum problem.
Wrote up some of the math here:
grantstenger.com/local-minim…