Measuring Correlation Scaling in Images via Logistic Regression

It is well known in quantum many-body physics that local Hamiltonians often have ground states whose entanglement entropies scale with the boundary of the bipartition (known as an "area law"), allowing them to be characterized efficiently using tensor networks. Given recent interest in using tensor network models for machine learning, we explore whether correlations in classical image data may possess analogous scaling behavior in the mutual information (MI) between bipartitions of the pixels. Building on techniques developed in the context of generative machine learning, we recast the MI estimation problem as logistic regression on samples drawn from the joint and marginal distributions of the image partitions. We test the accuracy of our model using Gaussian Markov random fields designed to have analytic boundary law and volume law scaling patterns, and perform regression on the well-known MNIST and CIFAR image datasets to evaluate the MI scaling in real-world images. We find that our model can capture the scaling behavior of the Markov random fields even with hundreds of pixels, while the large MI values found in CIFAR and MNIST remain challenging to reproduce.