Bayesian tensor network structure search and its application to tensor completion.

Area of Science:

• Data Science
• Machine Learning
• Applied Mathematics

Background:

• Tensor networks (TN) excel at compact data representation.
• Automated tensor network structure search (TNSS) methods learn structures from data.
• Existing TNSS methods require manual tuning of complexity-controlling parameters, especially for noisy or incomplete data.

Purpose of the Study:

• To develop a parameter tuning-free TNSS algorithm for automated, data-driven structure discovery.
• To effectively handle missing or noisy data within the TNSS framework.
• To improve the performance and efficiency of TNSS for high-order data representation and completion.

Main Methods:

• Proposed a Bayesian modeling approach for parameter-free TNSS.
• Incorporated data corruption uncertainty into the probabilistic model's prior.
• Reframed TN structure determination as a rank learning problem using generalized inverse Gaussian (GIG) distribution.
• Employed a fully Bayesian approach with Markov chain Monte Carlo (MCMC) for posterior distribution sampling.

Main Results:

• The proposed algorithm effectively and efficiently identifies latent TN structures under various missing and noise conditions.
• Achieved superior data recovery results compared to previous TNSS methods.
• Demonstrated state-of-the-art performance in tensor completion with real-world data.

Conclusions:

• The developed Bayesian TNSS algorithm eliminates the need for manual hyperparameter tuning.
• It offers a robust and efficient solution for discovering compact tensor network structures.
• The method significantly advances tensor completion capabilities, particularly for imperfect real-world datasets.