Matrix Completion via Non-Convex Relaxation and Adaptive Correlation Learning.

Area of Science:

• Machine Learning
• Data Science
• Numerical Analysis

Background:

• Existing matrix completion methods rely on rank function relaxations (e.g., nuclear norm), requiring many iterations.
• Current models primarily utilize low-rank properties, and incorporating additional knowledge is computationally expensive.
• There's a need for faster and more efficient matrix completion techniques.

Purpose of the Study:

• To develop a novel non-convex surrogate for matrix completion with fast convergence.
• To introduce an adaptive correlation learning method for enhanced matrix completion.
• To theoretically validate the proposed surrogate and its equivalence to existing models.

Main Methods:

• Proposed a novel non-convex surrogate function for optimizing matrix rank.
• Developed a parameter-free optimization approach with closed-form solutions.
• Introduced a scaling-invariant adaptive correlation learning technique.
• Theoretically proved convergence and equivalence to existing matrix completion models.

Main Results:

• The proposed non-convex surrogate converges empirically within dozens of iterations.
• The optimization process is parameter-free and has proven convergence.
• Adaptive correlation learning enhances matrix completion performance.
• The combined model maintains fast convergence through closed-form solutions.

Conclusions:

• The novel non-convex surrogate and adaptive correlation learning significantly improve matrix completion efficiency.
• The proposed methods offer faster convergence and are computationally less demanding.
• This work provides a more effective approach to matrix completion by leveraging both low-rank properties and column-wise correlations.