Inferring Local Structure from Pairwise Correlations
This work addresses the challenge of modeling complex systems like those in biology by constraining variable interactions, though it is incremental as it builds on existing correlation-based methods in a toy model context.
The authors tackled the problem of inferring local structure in multivariate systems from pairwise correlations, demonstrating that even with severely undersampled data, they could recover local relations, dimensionality, and reconstruct scrambled pixel arrangements in a 2D image toy model, despite the presence of higher-order interactions.
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.