Seminar by Brice Huang

Strong Topological Trivialization for Multi-Species Spin Glasses

This talk studies the random landscapes of spherical spin glasses. Recent works using the Kac-Rice formula identify the phase boundary for annealed topological trivialization, where the expected number of critical points drops from exponentially many to constant. We show this transition coincides with a quenched strong topological trivialization transition: in the "trivial" regime the number of critical points is constant, all are well-conditioned, and all approximate critical points are near a true one, and in the complementary regime there are exponentially many well-separated approximate critical points. This has direct consequences for optimization algorithms such as Langevin dynamics. We show analogous results for spin glasses with multiple species, where the threshold for annealed trivialization is also new. Based on joint work with Mark Sellke.