When is the average number of saddle points typical?

Abstract A common measure of a function's complexity is the count its stationary points. For complicated functions, this grows exponentially with volume and dimension their domain. In practice, averaged over class functions (the annealed average), but large numbers involved can produce averages biased by extremely rare samples. Typical counts are reliably found taking average logarithm quenched which more difficult not often done in practice. When most points uncorrelated each other, equal. Equilibrium heuristics guarantee when lowest minima will be uncorrelated. We show that these equilibrium cannot used to draw conclusions about other saddles producing examples among Gaussian-correlated on hypersphere where certain has different averages, despite being guaranteed “safe” setting. determine conditions for emergence non-trivial correlations between saddles, discuss implications geometry those what out-of-equilibrium settings might affected.