Bayesian CP Factorization of Incomplete Tensors with Automatic Rank Determination

Area of Science:

• Multilinear data analysis
• Machine learning
• Probabilistic modeling

Background:

• CANDECOMP/PARAFAC (CP) tensor factorization is vital for incomplete data analysis.
• Existing CP methods require manual rank specification, a significant challenge.
• Current approaches often neglect uncertainty in latent factors and missing data.

Purpose of the Study:

• To develop a CP factorization method with automatic rank determination.
• To incorporate uncertainty information for latent factors and missing entries.
• To provide a tuning parameter-free approach for robust tensor completion.

Main Methods:

• Formulated CP factorization using a hierarchical probabilistic model.
• Employed a fully Bayesian treatment with sparsity-inducing priors and hyperpriors.
• Developed an efficient, linearly scaling deterministic Bayesian inference algorithm.

Main Results:

• Achieved automatic determination of the CP tensor rank, preventing overfitting.
• Demonstrated effective inference of multilinear latent factors with low-rank constraints.
• Provided predictive distributions for missing entries, outperforming state-of-the-art methods.

Conclusions:

• The proposed Bayesian CP factorization method offers automatic rank selection and robust tensor completion.
• The approach effectively handles incomplete data and uncertainty, outperforming existing methods in simulations and real-world applications like image inpainting.