Mutual Information Scaling for Tensor Network Machine Learning.

Area of Science:

• Machine Learning
• Quantum Many-Body Physics
• Data Analysis

Background:

• Tensor networks are increasingly used in machine learning, drawing inspiration from quantum physics.
• The effectiveness of tensor network ansatze depends on their ability to capture the entanglement structure of target states.
• Different tensor network architectures exhibit varying scaling patterns.

Purpose of the Study:

• To apply correlation analysis, similar to that used in quantum states, to classical data in machine learning.
• To investigate if classical data exhibit correlation scaling patterns that can inform tensor network design.
• To determine the most suitable tensor network for a given dataset based on its correlation structure.

Main Methods:

• Utilized mutual information as a measure of correlations in classical data.
• Developed a logistic regression algorithm to estimate mutual information between feature bipartitions.
• Verified the algorithm's accuracy using Gaussian distributions with designed correlation patterns.
• Analyzed correlation scaling in MNIST and Tiny Images datasets.

Main Results:

• Mutual information serves as a lower-bound for entanglement in probabilistic tensor network classifiers.
• Identified boundary-law scaling in the Tiny Images dataset.
• Demonstrated that classical data exhibit correlation scaling patterns.

Conclusions:

• Quantum-inspired correlation analysis can guide the selection of appropriate tensor networks for machine learning.
• Understanding data correlation scaling is crucial for optimizing tensor network-based machine learning models.
• This approach provides insights into designing effective tensor networks for specific learning tasks.