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 Show more
the greatest whitepill of all is that local minima are rare in high dimensional spaces
The Penrose triangle is a useful intuition pump for a nontrivial Čech cohomology class: each local patch looks coherent but the patches can't be glued into a consistent global object. The obstruction appears only around the loop as an accumulated contradiction in depth ordering.
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