Low-Rank High-Order Tensor Completion With Applications in Visual Data.

Area of Science:

• Computer Science
• Data Science
• Applied Mathematics

Background:

• Tensor Singular Value Decomposition (t-SVD) is effective for low-rank tensor completion (LRTC) in pattern analysis.
• Existing LRTC methods primarily focus on third-order tensors, limiting applications for higher-order tensors (d ≥ 4).
• Real-world data like videos and images often involve tensors of order four or higher.

Purpose of the Study:

• To develop a generalized order-d tensor recovery framework for high-order tensors.
• To establish a novel algebraic foundation for order-d t-SVD.
• To enable accurate completion of low t-SVD rank tensors with missing values.

Main Methods:

• Proposed a novel algebraic foundation for order-d t-SVD.
• Developed a comprehensive order-d tensor recovery framework, including model, algorithm, and theoretical guarantees.
• Implemented and evaluated the framework on synthetic and real-world visual data.

Main Results:

• The proposed framework achieves exact completion for order-d low t-SVD rank tensors with high probability.
• Empirical studies demonstrate highly competitive performance compared to state-of-the-art methods.
• The framework shows significant superiority over peers, particularly with low observed data density (around 10%).

Conclusions:

• The developed order-d tensor recovery framework effectively addresses limitations of existing methods.
• The novel algebraic foundation for order-d t-SVD enables robust completion of high-order tensors.
• This work provides a powerful tool for pattern analysis in complex, high-dimensional data.