  Spin-glass systems are universal models for representing many-body phenomena
in statistical physics and computer science. High quality solutions of NP-hard
combinatorial optimization problems can be encoded into low energy states of
spin-glass systems. In general, evaluating the relevant physical and
computational properties of such models is difficult due to critical slowing
down near a phase transition. Ideally, one could use recent advances in deep
learning for characterizing the low-energy properties of these complex systems.
Unfortunately, many of the most promising machine learning approaches are only
valid for distributions over continuous variables and thus cannot be directly
applied to discrete spin-glass models. To this end, we develop a continuous
probability density theory for spin-glass systems with arbitrary dimensions,
interactions, and local fields. We show how our formulation geometrically
encodes key physical and computational properties of the spin-glass in an
instance-wise fashion without the need for quenched disorder averaging. We show
that our approach is beyond the mean-field theory and identify a transition
from a convex to non-convex energy landscape as the temperature is lowered past
a critical temperature. We apply our formalism to a number of spin-glass models
including the Sherrington-Kirkpatrick (SK) model, spins on random
Erd\H{o}s-R\'enyi graphs, and random restricted Boltzmann machines.
