Bayesian factorizations of big sparse tensors.

Area of Science:

• Statistics
• Data Science
• Multivariate Analysis

Background:

• Multiway array (tensor) data collection is routine.
• Existing low-rank matrix factorization methods have limited extensions to tensors.
• Parallel Factor Analysis (PARAFAC) is common but performs poorly on incomplete tensor data.

Purpose of the Study:

• To develop an improved tensor factorization method for sparse, high-dimensional data.
• To address the limitations of PARAFAC in scenarios with incomplete observations.
• To introduce a Bayesian approach for tensor decomposition with enhanced dimension reduction.

Main Methods:

• Developed a novel tensor factorization by allowing effective rank to vary across dimensions.
• Employed a Bayesian framework with priors on factorization terms.
• Implemented an efficient Gibbs sampler for posterior computation.
• Focused on contingency table applications for demonstration.

Main Results:

• The proposed method demonstrates superior performance compared to standard PARAFAC on incomplete tensor data.
• Theoretical analysis shows favorable posterior concentration rates in high-dimensional settings.
• Simulations and real-data applications confirm the method's effectiveness.

Conclusions:

• The novel tensor factorization method effectively handles sparse, high-dimensional data.
• The Bayesian approach with a varying effective rank offers a robust alternative to traditional PARAFAC.
• The method shows significant promise for analyzing complex, incomplete tensor datasets across various applications.