Kavli Affiliate: Birgitta Whaley
| First 5 Authors: Ian Convy, William Huggins, Haoran Liao, K. Birgitta Whaley,
| Summary:
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.
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