Efficient Tensor Completion for Color Image and Video Recovery: Low-Rank Tensor Train

Area of Science:

• Data Science
• Numerical Analysis
• Machine Learning

Background:

• Tensor completion is crucial for recovering missing data in multi-dimensional arrays.
• Existing methods often struggle with capturing complex correlations within tensor data.
• The tensor train (TT) rank offers a promising framework for low-rank tensor approximation.

Purpose of the Study:

• To propose a novel tensor completion approach leveraging the tensor train (TT) rank.
• To develop new optimization formulations and algorithms for efficient tensor completion.
• To enhance tensor completion effectiveness through a tensor augmentation strategy.

Main Methods:

• Utilizing the tensor train (TT) rank, derived from a balanced matricization scheme, to capture latent tensor information.
• Introducing two novel algorithms: Simple Low-Rank Tensor Completion via TT (SiLRTC-TT) and Tensor Completion by Parallel Matrix Factorization via TT (TMac-TT).
• Implementing a tensor augmentation scheme to transform low-order tensors into higher orders, boosting algorithm performance.

Main Results:

• SiLRTC-TT minimizes a nuclear norm based on the TT rank.
• TMac-TT employs a multilinear matrix factorization model to approximate the TT rank.
• Both methods, especially with tensor augmentation, show significant advantages over existing techniques in simulations.

Conclusions:

• The proposed TT-rank-based tensor completion methods offer a powerful new approach for data recovery.
• SiLRTC-TT and TMac-TT provide effective solutions for reconstructing missing tensor entries.
• The tensor augmentation scheme further improves the accuracy and robustness of these novel completion techniques.