  We prove structure theorems for measures on the discrete cube and on Gaussian
space, which provide sufficient conditions for mean-field behavior. These
conditions rely on a new notion of complexity for such measures, namely the
Gaussian-width of the gradient of the log-density. On the cube $\{-1,1\}^n$, we
show that a measure $\nu$ which exhibits low complexity can be written as a
mixture of measures $\{\nu_\theta\}_{\theta \in \mathcal{I}}$ such that: i. for
each $\theta$, the measure $\nu_\theta$ is a small perturbation of $\nu$ such
that $\log \tfrac{d \nu_\theta}{d \nu}$ is a linear function whose gradient is
small and, ii. $\nu_\theta$ is close to some product measure, in Wasserstein
distance, for most $\theta$. Thus, our framework can be used to study the
behavior of low-complexity measures beyond approximation of the partition
function, showing that those measures are roughly mixtures of product measures
whose entropy is close to that of the original measure. In particular, as a
corollary of our theorems, we derive a bound for the na\"ive mean-field
approximation of the log-partition function which improves the nonlinear large
deviation framework of Chatterjee and Dembo in several ways: 1. It does not
require any bounds on second derivatives. 2. The covering number is replaced by
the weaker notion of Gaussian-width 3. We obtain stronger asymptotics with
respect to the dimension. Two other corollaries are decomposition theorems for
exponential random graphs and large-degree Ising models. In the Gaussian case,
we show that measures of low-complexity exhibit an almost-tight reverse
Log-Sobolev inequality.
