Mutual Information Scaling for Tensor Network Machine Learning

Abstract

Tensor networks have emerged as promising tools for machine learning, inspired by their widespread use as variational ansatze 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 which might indicate the best network to use for a given dataset. We utilize mutual information as measure of correlations in classical data, and show that it can serve as a lower-bound on the entanglement needed for a probabilistic tensor network classifier. 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. This quantum-inspired classical analysis offers insight into the design of tensor networks which are best suited for specific learning tasks.

29 pages, 10 figures

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• Machine learning with tree tensor networks, CP rank constraints, and tensor dropout
• Tensor network to learn the wavefunction of data
• Tensor networks and efficient descriptions of classical data
• Tensor tree learns hidden relational structures in data to construct generative models
• Deep tensor networks with matrix product operators
• Universal scaling laws in quantum-probabilistic machine learning by tensor network towards interpreting representation and generalization powers