A Nonconvex Relaxation Approach to Low-Rank Tensor Completion

Area of Science:

• Applied Mathematics
• Computer Vision
• Machine Learning

Background:

• Low-rank tensor completion is crucial for image processing, computer vision, and machine learning.
• Existing convex relaxation methods, like nuclear norm minimization, can yield suboptimal solutions.
• The square deal matrix reshaping is a common technique in tensor completion.

Purpose of the Study:

• To develop a more accurate method for low-rank tensor completion.
• To approximate tensor rank using nonconvex functions applied to singular values.
• To introduce and analyze a proximal linearized minimization (PLM) algorithm for the proposed model.

Main Methods:

• Utilized a family of nonconvex functions to approximate the rank of a tensor via its square deal matrix singular values.
• Developed a proximal linearized minimization (PLM) algorithm to solve the resulting nonconvex optimization problem.
• Leveraged the Kurdyka-Łojasiewicz property to prove global convergence of the PLM algorithm to a critical point.

Main Results:

• The proposed nonconvex model demonstrates improved accuracy compared to existing state-of-the-art methods.
• The PLM algorithm successfully converges to a critical point of the objective function.
• Numerical experiments on synthetic, video, and face recognition data validate the model's effectiveness.

Conclusions:

• The proposed nonconvex approach offers a more accurate solution for low-rank tensor completion.
• The PLM algorithm provides a robust and convergent method for solving the nonconvex model.
• The method shows significant potential for applications in image processing, computer vision, and machine learning.