Decentralized Nonconvex Low-rank Matrix Recovery

Area of Science:

• Machine Learning
• Optimization
• Distributed Systems

Background:

• Traditional low-rank matrix recovery is computationally intensive due to singular value decomposition.
• Matrix factorization offers an efficient, albeit nonconvex, alternative for matrix recovery.
• The performance of factorization-based methods in decentralized environments remains underexplored.

Purpose of the Study:

• To investigate the convergence properties of factorization-based low-rank matrix recovery in a decentralized setting.
• To analyze the distributed gradient descent algorithm for this problem.
• To demonstrate the algorithm's effectiveness on general networks.

Main Methods:

• Utilizing a distributed gradient descent algorithm for matrix factorization.
• Establishing theoretical convergence rates (local linear convergence).
• Conducting numerical experiments to validate convergence behavior.

Main Results:

• The distributed gradient descent algorithm achieves local linear convergence up to the approximation error.
• Numerical results confirm the algorithm's convergence over general network topologies.
• The study provides theoretical insights into decentralized matrix recovery.

Conclusions:

• Factorization-based distributed gradient descent is a viable and efficient method for decentralized low-rank matrix recovery.
• The algorithm's linear convergence is theoretically established.
• This work bridges the gap in understanding matrix recovery within distributed systems.