Balanced Unfolding Induced Tensor Nuclear Norms for High-Order Tensor Completion.

Area of Science:

• Multidimensional data analysis
• Tensor decomposition
• Machine learning

Background:

• Tensor data completion is crucial for real-world applications.
• Existing tensor tubal rank methods have limitations in capturing multidimensional structures.
• The third mode is often the fixed transform orientation in current approaches.

Purpose of the Study:

• To introduce novel unfolding-induced tensor nuclear norms (TNNs) for tensor completion.
• To extend the concept of tensor tubal rank to high-order data.
• To address the limitations of existing methods in capturing multidimensional low-tubal-rank structures.

Main Methods:

• Development of two TNNs: overlapped TNN (OTNN) and latent TNN (LTNN).
• Utilizing a novel balanced unfolding strategy to capture multidimensional low-tubal-rank structure.
• Proposing two efficient tensor completion models with theoretical guarantees using a unified nonasymptotic upper bound.
• Employing alternating direction methods of multipliers (ADMM) based algorithms for optimization.

Main Results:

• Demonstrated relationship between unfolding tensor tubal rank and existing tensor network ranks (CP, Tucker, TR).
• Proposed models show superior performance in tensor completion tasks.
• Experimental validation on synthetic and real-world tensor data (facial images, light field images, video sequences).

Conclusions:

• The proposed OTNN and LTNN effectively capture multidimensional low-tubal-rank structures.
• The novel TNNs offer an effective extension of tensor tubal rank for high-order data.
• The developed tensor completion models provide a promising approach for handling complex tensor data.