Learning from Binary Multiway Data: Probabilistic Tensor Decomposition and its Statistical Optimality.

Area of Science:

• Multilinear algebra
• Statistical modeling
• Machine learning

Background:

• Higher-order tensor decomposition is vital for analyzing complex datasets in various fields.
• Binary tensor data presents unique challenges compared to continuous-valued data.
• Applications include neuroimaging, recommendation systems, topic modeling, and sensor network localization.

Purpose of the Study:

• To develop a novel method for decomposing higher-order tensors with binary entries.
• To establish theoretical accuracy guarantees for the proposed decomposition method.
• To address the unique characteristics of binary tensor data, including phase transitions.

Main Methods:

• Proposed a multilinear Bernoulli model for binary tensor data.
• Developed a rank-constrained likelihood-based estimation method.
• Designed an alternating optimization algorithm with convergence guarantees.

Main Results:

• Established theoretical error bounds for parameter tensor estimation.
• Demonstrated a phase transition phenomenon based on signal-to-noise ratio.
• Showcased that the obtained estimation rate is minimax optimal.
• Validated the approach through simulations and real-world datasets.

Conclusions:

• The proposed multilinear Bernoulli model and estimation method are effective for binary tensor decomposition.
• The method achieves theoretical accuracy and minimax optimality.
• The developed algorithm is efficient and demonstrates strong performance in tensor completion and clustering.