Low Tensor-Ring Rank Completion by Parallel Matrix Factorization.

Area of Science:

• Applied Mathematics
• Data Science
• Numerical Analysis

Background:

• Tensor-ring (TR) decomposition is a key technique for low-rank tensor completion (LRTC).
• Conventional TR-based methods suffer from high computational costs due to unbalanced unfolding schemes.
• Achieving optimal performance often requires a large TR rank, exacerbating computational burdens.

Purpose of the Study:

• To develop a more efficient TR-based LRTC method by exploiting low TR-rank structures.
• To address the limitations of unbalanced unfolding in existing TR decomposition algorithms.
• To reduce the computational cost associated with TR-based tensor completion.

Main Methods:

• Introduced a balanced unfolding operation termed tensor circular unfolding.
• Established a theoretical link between TR rank and the ranks of tensor unfoldings.
• Developed an algorithm performing parallel low-rank matrix factorizations on circularly unfolded matrices.
• Applied a row weighting trick to handle nonuniform missing data patterns.

Main Results:

• The proposed algorithm achieves outstanding performance with a significantly smaller TR rank.
• Demonstrated substantial reduction in computational cost compared to conventional TR-based methods.
• Showcased improved adaptive ability to diverse missing data patterns through row weighting.

Conclusions:

• The novel tensor circular unfolding and parallel factorization approach efficiently exploits low TR-rank structures.
• This method offers a computationally efficient and high-performance solution for low-rank tensor completion.
• The proposed technique effectively handles nonuniform missing patterns, enhancing its practical applicability.