  To construct models of large, multivariate complex systems, such as those in
biology, one needs to constrain which variables are allowed to interact. This
can be viewed as detecting "local" structures among the variables. In the
context of a simple toy model of 2D natural and synthetic images, we show that
pairwise correlations between the variables -- even when severely undersampled
-- provide enough information to recover local relations, including the
dimensionality of the data, and to reconstruct arrangement of pixels in fully
scrambled images. This proves to be successful even though higher order
interaction structures are present in our data. We build intuition behind the
success, which we hope might contribute to modeling complex, multivariate
systems and to explaining the success of modern attention-based machine
learning approaches.
