Tensor ring rank determination using odd-dimensional unfolding

Area of Science:

• Multilinear algebra
• Data analysis
• Numerical methods

Background:

• Tensor ring (TR) decomposition is vital for high-dimensional data analysis.
• Determining TR-ranks is difficult, with existing methods sensitive to initial ranks and lacking adaptivity for noisy or incomplete data.
• Computational inefficiency is a concern for high-dimensional TR decomposition.

Purpose of the Study:

• To propose an effective method for determining tensor ring ranks.
• To address the limitations of existing TR-rank determination techniques, including sensitivity, adaptivity, and computational efficiency.
• To enhance the applicability of TR decomposition in real-world scenarios.

Main Methods:

• An odd-dimensional unfolding method leveraging TR model symmetry and bound rank relationships.
• Singular value thresholding for adaptive TR-rank determination.
• Randomized sketching techniques for improved computational efficiency and scalability.

Main Results:

• The proposed method effectively determines TR-ranks.
• Demonstrated adaptivity in the presence of noisy and incomplete data.
• Enhanced computational efficiency and scalability for high-dimensional data.

Conclusions:

• The novel odd-dimensional unfolding method offers a robust solution for TR-rank determination.
• The approach shows significant potential for applications in rank identification, data denoising, and data completion.
• This work advances the practical utility of tensor ring decomposition.