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 two-dimensional 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.