Kronecker-Basis-Representation Based Tensor Sparsity and Its Applications to Tensor Recovery.

Area of Science:

• Data Science
• Applied Mathematics
• Signal Processing

Background:

• Sparse modeling is effective for vector/matrix data analysis.
• Real-world data often involves multiple interacting factors, requiring higher-order representations like tensors.
• Existing methods lack a robust measure for tensor sparsity.

Purpose of the Study:

• To propose a novel measure for tensor sparsity.
• To develop an efficient algorithm for solving the associated optimization problem.
• To demonstrate the applicability and superiority of the proposed method in various data analysis tasks.

Main Methods:

• Introduced the Kronecker-basis-representation (KBR) based tensor sparsity measure.
• Formulated the KBR regularization minimization (KBRM) problem.
• Designed an Alternating Direction Method of Multipliers (ADMM) algorithm with closed-form updates.

Main Results:

• The proposed KBR measure effectively captures tensor sparsity, integrating insights from Tucker and CP decompositions.
• The ADMM algorithm efficiently solves the KBRM problem.
• Experimental results show superior performance in multispectral image denoising, completion, and background subtraction compared to state-of-the-art methods.

Conclusions:

• The KBR measure provides a principled way to quantify tensor sparsity.
• The efficient ADMM solver enables practical application of KBR to complex problems.
• The proposed method offers significant advantages for analyzing high-order tensor data.