A Continuous Formulation of Discrete Spin-Glass Systems

We introduce a new, continuous formulation of discrete spin-glasses in which the discrete Boltzmann distribution is replaced by a continuous probability density over the real numbers. This formulation applies for any discrete spin-glass with Ising spins coupled through 2-body interaction terms. A major benefit of working with such a continuous formulation is that the energy landscape may be studied directly using tools from differential geometry and topology. In particular, we show that for a given set of couplings there is a critical temperature above which the energy landscape is convex. Below this temperature the landscape becomes non-convex due to the appearance of multiple critical points. In general, this convex/non-convex transition is distinct from phase transitions to the spin-glass or ferromagnetic phases. In this talk, we introduce our general formalism and theoretically establish the similarities and differences with the mean-field models and the Thouless-Anderson-Palmer equation. We then provide details for a few specific cases including the Sherrington-Kirkpatrick model and random restricted Boltzmann machines.