  An image pattern can be represented by a probability distribution whose
density is concentrated on different low-dimensional subspaces in the
high-dimensional image space. Such probability densities have an astronomical
number of local modes corresponding to typical pattern appearances. Related
groups of modes can join to form macroscopic image basins that represent
pattern concepts. Recent works use neural networks that capture high-order
image statistics to learn Gibbs models capable of synthesizing realistic images
of many patterns. However, characterizing a learned probability density to
uncover the Hopfield memories of the model, encoded by the structure of the
local modes, remains an open challenge. In this work, we present novel
computational experiments that map and visualize the local mode structure of
Gibbs densities. Efficient mapping requires identifying the global basins
without enumerating the countless modes. Inspired by Grenander's jump-diffusion
method, we propose a new MCMC tool called Attraction-Diffusion (AD) that can
capture the macroscopic structure of highly non-convex densities by measuring
metastability of local modes. AD involves altering the target density with a
magnetization potential penalizing distance from a known mode and running an
MCMC sample of the altered density to measure the stability of the initial
chain state. Using a low-dimensional generator network to facilitate
exploration, we map image spaces with up to 12,288 dimensions (64 $\times$ 64
pixels in RGB). Our work shows: (1) AD can efficiently map highly non-convex
probability densities, (2) metastable regions of pattern probability densities
contain coherent groups of images, and (3) the perceptibility of differences
between training images influences the metastability of image basins.
