Convex Coupled Matrix and Tensor Completion.

Area of Science:

• Machine Learning
• Data Science
• Tensor Analysis

Background:

• Coupled learning involves matrices and tensors sharing information via common modes.
• Existing coupled learning methods often rely on non-convex optimization, limiting global optimality.
• Imputing missing values in coupled tensors is crucial for many data analysis tasks.

Purpose of the Study:

• To propose a novel set of convex, low-rank-inducing norms for coupled tensors.
• To develop a tensor completion model regularized by these new norms.
• To demonstrate the advantages of convex norms over non-convex ones in coupled learning.

Main Methods:

• Formulation of a mixture norm combining overlapped trace norm, latent norms, and matrix trace norm.
• Development of a tensor completion model utilizing these convex norms for regularization.
• Theoretical analysis of excess risk bounds for the proposed completion model.
• Empirical validation using synthetic and real-world datasets.

Main Results:

• The proposed norms are convex, ensuring a globally optimal solution for coupled tensor imputation.
• The developed completion model effectively exploits the low-rank structure of coupled tensors.
• Theoretical analysis yields improved excess risk bounds compared to uncoupled norms.
• Experimental results show superior performance of the proposed model over existing methods.

Conclusions:

• The novel convex norms provide a more robust and efficient approach to coupled tensor completion.
• The proposed method offers significant improvements in imputation accuracy and theoretical guarantees.
• This work advances the field of tensor analysis and coupled learning with practical implications.