Nonconvex Robust High-Order Tensor Completion Using Randomized Low-Rank Approximation.

Area of Science:

• Data Science
• Applied Mathematics
• Scientific Computing

Background:

• Tensor completion is crucial for large-scale data analysis in science and engineering.
• Existing robust low-rank tensor completion methods, based on tensor singular value decomposition (T-SVD), face computational challenges and are limited to third-order tensors.

Purpose of the Study:

• To develop efficient low-rank tensor approximation methods for higher-order tensors (order-d, d ≥ 3) using random projection techniques.
• To address the limitations of high computational costs and order restrictions in existing T-SVD-based tensor completion.

Main Methods:

• Proposed two efficient low-rank tensor approximation algorithms incorporating random projection techniques within the order-d T-SVD framework.
• Developed a double nonconvex model and fast optimization algorithms with convergence guarantees for robust high-order tensor completion.

Main Results:

• Provided theoretical error bounds for the proposed randomized tensor approximation algorithms.
• Demonstrated superior performance of the developed method over state-of-the-art approaches on large-scale synthetic and real tensor data.
• Achieved significant improvements in both computational efficiency and estimation precision.

Conclusions:

• The proposed randomized tensor approximation and completion methods offer efficient and accurate solutions for high-order tensor data.
• These advancements overcome the computational bottlenecks and order limitations of traditional T-SVD-based methods, enabling broader applications in data science and engineering.