Noisy Tensor Completion via Low-Rank Tensor Ring.

Area of Science:

• Data Science
• Machine Learning
• Applied Mathematics

Background:

• Tensor completion is vital for analyzing incomplete datasets.
• Existing methods struggle with noisy or high-order observations, limiting practical application.
• There's a need for robust tensor completion models that handle real-world data imperfections.

Purpose of the Study:

• To propose a novel noisy tensor completion model.
• To address the limitations of existing methods in handling noisy and high-order tensor data.
• To provide a statistically sound and computationally efficient solution for incomplete tensor analysis.

Main Methods:

• Utilized tensor ring nuclear norm (TRNN) for underlying tensor regularization.
• Employed a least-squares estimator for observed entries.
• Developed two efficient algorithms for optimization, including a large-scale tensor approach using heterogeneous tensor decomposition.

Main Results:

• Provided a non-asymptotic upper bound for estimation error, ensuring statistical performance.
• Demonstrated the model's effectiveness and efficiency on synthetic and real-world datasets.
• Showcased superior performance in recovering noisy incomplete tensor data compared to state-of-the-art methods.

Conclusions:

• The proposed noisy tensor completion model effectively recovers missing data from imperfect observations.
• The developed algorithms are efficient and suitable for both small and large-scale tensor completion tasks.
• This work advances the field of incomplete data analysis by providing a robust solution for noisy tensor data.