Bipartite spin glasses -- a variant of usual spin glasses, in which spins are grouped into two species -- are a classical testing ground for new approaches in spin glasses. The annealed complexity of bipartite spherical models was initially considered by Auffinger and Chen, who gave upper and lower bounds. We give an exact variational formula for this complexity, both for pure spin glasses and for mixtures. We also find connections between this model and the usual spherical spin glasses, which were studied by Auffinger, Ben Arous, and Černý.

Bipartite spin glass models have been gaining popularity in the study of glassy systems with distinct interacting species. Recently, the annealed complexity of the pure spherical bipartite model was obtained by B. McKenna. In this talk, I will explain how to show that the low-lying complexity actually concentrates around this value, and how from this one can obtain a formula for the ground-state energy.

In this talk, I will introduce how we can use the Kac-Rice method to obtain the average and the typical value of the number of critical points of the empirical risk landscape for generalized linear estimation problems and variants. This represents the first application of Kac-Rice to obtain exact complexity asymptotics for non-Gaussian random functions. We obtain a rigorous explicit variational formula for the average number of critical points. This result is then extended using the non-rigorous Kac-Rice replicated method from theoretical physics. In this way we derive a second variational formula for the number of critical points for typical instances up to exponential accuracy. Finally, I will discuss some of the challenges ahead, in the numerical evaluation and extension of these formulas. This talk is based on the results of the paper https://arxiv.org/abs/1912.02143.

TBA