Low-Rank Tensor Completion Based on Self-Adaptive Learnable Transforms.

Area of Science:

• Tensor decomposition and analysis
• Machine learning for data recovery
• Signal and image processing

Background:

• Tensor nuclear norm (TNN) methods are useful for low-rank tensor recovery but rely on fixed or data-independent transformations.
• These limitations prevent adaptive exploitation of inherent low-rank structures within tensor data.

Purpose of the Study:

• To propose a novel framework, Self-Adaptive Learnable Transform (SALT), for learning optimal transformations from tensor data.
• To generalize SALT to SALTS, enabling adaptive exploitation of low-rank structures across all tensor dimensions.
• To enhance tensor completion accuracy by adaptively uncovering underlying tensor structures.

Main Methods:

• Developed the Self-Adaptive Learnable Transform (SALT) framework to learn a transformation matrix from tensor data.
• Utilized the Schatten-p quasi-norm as a rank proxy to induce a lower average-rank tensor via a lossless transformation.
• Generalized SALT to SALTS, learning three simultaneous self-adaptive transformation matrices for multi-directional low-rank structure exploitation.
• Proposed a unified optimization framework using the alternating direction multiplier method and proved its weak convergence.

Main Results:

• SALTS demonstrates superior accuracy in tensor completion compared to existing methods across diverse datasets (HSI, color video, MRI, COIL-20).
• The adaptive nature of SALTS effectively exploits potential low-rank structures in multiple directions.
• The proposed optimization algorithm ensures theoretical convergence properties.

Conclusions:

• SALTS provides a powerful and adaptive approach for low-rank tensor recovery, outperforming traditional TNN-based methods.
• The framework's ability to learn transformations tailored to specific tensor data unlocks new potential in various applications.
• SALTS represents a significant advancement in adaptive tensor decomposition and completion techniques.