Large-Scale Affine Matrix Rank Minimization With a Novel Nonconvex Regularizer

Area of Science:

• Machine Learning
• Data Science
• Optimization Theory

Background:

• Low-rank minimization is crucial for data analysis tasks like recommender systems and signal processing.
• Existing nuclear norm minimization methods overlook singular value differences, while nonconvex regularizers face efficiency and accuracy challenges.
• Addressing these limitations is vital for advancing matrix recovery techniques.

Purpose of the Study:

• To propose a flexible low-rank minimization model with a novel nonconvex regularizer.
• To develop an efficient and accurate algorithm for solving the proposed optimization problem.
• To demonstrate the model's effectiveness across various low-rank matrix problems.

Main Methods:

• Developed a novel nonconvex regularizer for low-rank minimization.
• Transformed the low-rank problem into an equivalent optimization problem under the rank-RIP condition.
• Employed Nesterov's rule and inexact proximal strategies for an efficient algorithm with O(1/K) convergence.
• Analyzed asymptotic convergence using the Kurdyka-ojasiewicz (KL) inequality.

Main Results:

• The proposed model promotes low rankness and achieves faster, more accurate solutions compared to existing methods.
• The algorithm demonstrates a convergence rate of O(1/K).
• Empirical studies on matrix completion, RPCA, and tensor completion show superior performance.

Conclusions:

• The novel nonconvex regularizer effectively addresses limitations of existing low-rank minimization techniques.
• The proposed model and algorithm offer significant improvements in both accuracy and computational efficiency.
• The approach is validated across diverse data analysis tasks, including image recovery and personalized recommendation.