Tensor Wheel Decomposition: Theory and Application to Tensor Completion

Area of Science:

• Computer Vision
• Numerical Analysis
• Data Science

Background:

• Tensor network (TN) decompositions are crucial for high-order tensor representation in computer vision.
• Current TN methods often involve complex structures and high ranks, demanding extensive hyper-parameter tuning.

Purpose of the Study:

• Introduce a novel tensor wheel (TW) decomposition for efficient high-order tensor representation.
• Address the limitations of existing TN methods regarding complexity and hyper-parameter selection.

Main Methods:

• Developed a novel TW decomposition based on graph structure analysis and wheel topology.
• Implemented sequential Singular Value Decomposition (SVD)-based and Alternating Least Squares (ALS)-based algorithms for TW computation.
• Applied TW decomposition to tensor completion (TC) using a proximal alternating minimization algorithm.

Main Results:

• TW decomposition offers superior representation capabilities with lower hyper-parameter scales.
• Demonstrated flexibility in controlling parameter storage and computational costs.
• TW decomposition significantly outperformed state-of-the-art methods in tensor completion, especially with limited observations.

Conclusions:

• TW decomposition provides a more efficient and effective approach for high-order tensor representation and recovery.
• The method shows significant potential for applications like incomplete-tensor inference.
• TW decomposition is a reliable and superior alternative to existing tensor decomposition techniques.