  Tensor networks have emerged as promising tools for machine learning,
inspired by their widespread use as variational ansatzes in quantum many-body
physics. It is well known that the success of a given tensor network ansatz
depends in part on how well it can reproduce the underlying entanglement
structure of the target state, with different network designs favoring
different scaling patterns. We demonstrate here how a related correlation
analysis can be applied to tensor network machine learning, and explore whether
classical data possess correlation scaling patterns similar to those found in
quantum states. We utilize mutual information as a natural analog to
entanglement for classical data, and show that it can serve as a lower-bound on
the network entanglement needed for probabilistic classification. We then
develop a logistic regression algorithm to estimate the mutual information
between bipartitions of data features, and verify its accuracy on a set of
Gaussian distributions designed to mimic different correlation patterns. Using
this algorithm, we characterize the scaling patterns in the MNIST and Tiny
Images datasets, and find clear evidence of boundary-law scaling in the latter.
