Beyond Low Rank + Sparse: Multi-scale Low Rank Matrix Decomposition.

Area of Science:

• Matrix Decomposition
• Data Analysis
• Computer Vision

Background:

• Data matrices often exhibit local correlations across multiple scales.
• Recent advances include low rank plus sparse matrix decomposition.
• Handling multi-scale correlations is crucial for accurate data interpretation.

Purpose of the Study:

• To generalize low rank matrix decomposition to handle multiple scales.
• To develop a convex formulation for multi-scale low rank decomposition.
• To demonstrate the practical applications and advantages of the proposed method.

Main Methods:

• Proposed a multi-scale low rank modeling approach using block-wise low rank matrices.
• Developed a convex formulation to solve the inverse decomposition problem.
• Incorporated regularization parameter selection guidance and cycle spinning for artifact reduction.

Main Results:

• The convex program accurately recovers multi-scale low rank components under incoherence conditions.
• The multi-scale decomposition offers more intuitive results than conventional methods.
• Demonstrated effectiveness in face illumination normalization, video motion separation, MRI analysis, and collaborative filtering.

Conclusions:

• Multi-scale low rank decomposition is a powerful technique for data with multi-scale correlations.
• The proposed convex approach provides a robust and effective method for component separation.
• The technique shows significant promise across diverse applications in data analysis and computer vision.