Current Trends in Advanced Tensor Processing

Recent developments in the field of tensor processing have seen a significant shift towards nonconvex and implicit regularization techniques, particularly in the context of low-rank tensor factorizations and robust matrix completions. The focus has been on enhancing the efficiency and accuracy of data recovery and denoising methods by leveraging novel nonconvex functions and advanced optimization algorithms. These advancements are particularly relevant for applications in imaging processing, hyperspectral image representation, and large-scale data handling. The integration of nonconvex regularization not only improves the low-rankness and sparseness of the data but also addresses the irregularities and non-convexities inherent in the data structures. This trend underscores a move towards more sophisticated and adaptive models that can better capture the complexities of real-world data.

Noteworthy Developments

• Nonconvex Robust Quaternion Matrix Completion: Introduces a nonconvex model using MCP function and quaternion $L_p$-norm, enhancing both low-rankness and sparseness. The method outperforms existing techniques in color image and video processing.
• Implicit Regularization for Tubal Tensor Factorizations: Provides a rigorous analysis of implicit regularization in tensor factorizations via gradient descent, moving beyond the lazy training regime. This work is the first to establish implicit bias towards low tubal rank solutions with small random initialization.
• Irregular Tensor Low-Rank Representation: Proposes a novel model for irregular HSI data, using non-convex nuclear norm and a negative global low-rank term to improve global consistency. The method significantly outperforms existing low-rank HSI methods.
• Low-Rank Tensor Learning by Generalized Nonconvex Regularization: Introduces a nonconvex model based on transformed tensor nuclear norm, with an error bound established under restricted strong convexity. The PMM algorithm demonstrates superior performance in tensor completion and binary classification tasks.