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Let $K_1$, $\dots$, $K_n$ be bounded, complete, separable metric spaces. Let$\lambda_i$ be a Borel probability measure on $K_i$ for each $i$. Let$f:\prod_i K_i \to \mathbb{R}$ be a bounded and continuous potential function,and let $$\mu(d \mathbf{x})\ \propto\ e^{f(\mathbf{x})}\lambda_1(d x_1)\cdots\lambda_n(d x_n)$$ be the associated Gibbs distribution. At each point$\mathbf{x} \in \prod_i K_i$, one can define a `discrete gradient' $\nablaf(\mathbf{x},\,\cdot\,)$ by comparing the values of $f$ at all points whichdiffer from $\mathbf{x}$ in at most one coordinate. In case $\prod_i K_i =\{-1,1\}^n \subset \mathbb{R}^n$, the discrete gradient $\nablaf(\mathbf{x},\,\cdot\,)$ is naturally identified with a vector in$\mathbb{R}^n$.  This paper shows that a `low-complexity' assumption on $\nabla f$ impliesthat $\mu$ can be approximated by a mixture of other measures, relatively fewin number, and most of them close to product measures in the sense of optimaltransport. This implies also an approximation to the partition function of $f$in terms of product measures, along the lines of Chatterjee and Dembo's theoryof `nonlinear large deviations'.  An important precedent for this work is a result of Eldan in the case$\prod_i K_i = \{-1,1\}^n$. Eldan's assumption is that the discrete gradients$\nabla f(\mathbf{x},\,\cdot\,)$ all lie in a subset of $\mathbb{R}^n$ that hassmall Gaussian width. His proof is based on the careful construction of adiffusion in $\mathbb{R}^n$ which starts at the origin and ends with thedesired distribution on the subset $\{-1,1\}^n$. Here our assumption is a morenaive covering-number bound on the set of gradients $\{\nablaf(\mathbf{x},\,\cdot\,):\ \mathbf{x} \in \prod_i K_i\}$, and our proof reliesonly on basic inequalities of information theory. As a result, it is shorter,and applies to Gibbs measures on arbitrary product spaces.

The structure of low-complexity Gibbs measures on product spaces