Tensor Robust Principal Component Analysis with a New Tensor Nuclear Norm.

Area of Science:

• Multivariate statistics
• Tensor decomposition
• Machine learning

Background:

• Robust Principal Component Analysis (RPCA) is crucial for separating low-rank and sparse components from data.
• Existing matrix-based RPCA methods face limitations with higher-order data structures.
• Tensor analysis offers a powerful framework for multi-dimensional data, necessitating new decomposition techniques.

Purpose of the Study:

• To extend Robust Principal Component Analysis to the tensor domain, addressing the Tensor Robust Principal Component Analysis (TRPCA) problem.
• To rigorously define and analyze tensor norms (spectral and nuclear) analogous to their matrix counterparts.
• To establish theoretical guarantees for the exact recovery of low-rank and sparse components in tensor data.

Main Methods:

• Development of tensor norms (spectral and nuclear) based on the tensor-tensor product (t-product).
• Formulation of TRPCA as a convex program utilizing the newly defined tensor nuclear norm.
• Theoretical analysis to prove the convex envelope property of the tensor nuclear norm and guarantee exact recovery.

Main Results:

• Rigorous definitions and properties of tensor spectral norm and tensor nuclear norm are established.
• The tensor nuclear norm is proven to be the convex envelope of the tensor average rank.
• A convex program for TRPCA is proposed, with theoretical guarantees for exact component recovery.

Conclusions:

• The proposed TRPCA framework accurately recovers low-rank and sparse components from tensor data.
• The method generalizes matrix RPCA and demonstrates effectiveness in image recovery and background modeling.
• This work provides a robust theoretical and practical foundation for tensor-based data decomposition.