Fast and accurate matrix completion via truncated nuclear norm regularization.

Area of Science:

• Applied Mathematics
• Computer Vision
• Machine Learning

Background:

• Matrix completion is crucial for applications like image inpainting and recommender systems.
• Existing methods often use nuclear norm minimization, which can poorly approximate matrix rank.
• A limitation is the simultaneous minimization of all singular values, hindering accurate rank approximation.

Purpose of the Study:

• To propose a novel matrix completion approach using truncated nuclear norm for improved rank approximation.
• To develop efficient algorithms for solving the truncated nuclear norm minimization problem.
• To evaluate the performance of the proposed algorithms against state-of-the-art methods.

Main Methods:

• Introduced the concept of truncated nuclear norm for matrix completion.
• Developed three iterative algorithms: TNNR-ADMM, TNNR-APGL, and TNNR-ADMMAP.
• TNNR-ADMM uses alternating direction method of multipliers (ADMM).
• TNNR-APGL employs accelerated proximal gradient line search (APGL).
• TNNR-ADMMAP incorporates an adaptive penalty for faster convergence.

Main Results:

• The proposed truncated nuclear norm approach achieves better rank approximation.
• Empirical studies demonstrate encouraging performance of the developed algorithms.
• The algorithms outperform current state-of-the-art matrix completion techniques on synthetic and real datasets.

Conclusions:

• Truncated nuclear norm offers a superior alternative to standard nuclear norm minimization for matrix completion.
• The developed TNNR algorithms are efficient and effective for matrix recovery.
• This work advances the field of matrix completion with practical implications for various applications.