Who is your neighbor? Inferring locality from pairwise correlations

Modeling multivariate biological systems, such as multielectrode neural recordings, genetic sequences, or gene expression patterns, requires identification of combinatorial interaction coefficients coupling the measured variables. This is impossible to do from data sets of realistic sizes. One could hope to regularize the inference by imposing the constraint that interactions must be local. However, whether two variables are neighbors and thus can interact is unknown for many biological data sets. Here we explore the possibility that neighborhood relations can be inferred from the pairwise correlation matrix, even in the undersampled data limit. Our toy model consists of a set of images whose pixels are shuffled randomly, but in the same way for all images, such that spatial information is lost, but pixel-to-pixel correlations are preserved. We use t-SNE, a dimensionality reduction and visualization technique, to embed the shuffled pixels in space, such that strongly correlated pixels end up next to each other. We observe that embedding the data in 2D space produces images nearly identical to the originals, save for global transformations. This shows that analysis of the covariance matrix correctly identifies local neighborhoods, as well as the global dimensionality of the data.