Low-rank structure learning via nonconvex heuristic recovery

Area of Science:

• Data Science
• Machine Learning
• Signal Processing

Background:

• Traditional methods use convex norms for sparse data, which may not fully capture essential structures.
• Learning low-rank structures from corrupted data is crucial for many applications.

Purpose of the Study:

• To propose a nonconvex framework for enhanced low-rank structure learning from corrupted data.
• To introduce nonconvex measurements (ℓp norm and log-sum) for improved sparsity.
• To develop an efficient algorithm for solving the nonconvex optimization problem.

Main Methods:

• A nonconvex framework incorporating ℓp norm (0 < p < 1) and log-sum terms.
• A majorization-minimization (MM)-type algorithm to solve the nonconvex optimization.
• Iterative replacement of the nonconvex objective with a convex surrogate.

Main Results:

• The MM-type algorithm converges to a stationary point.
• Nonconvex methods, especially log-sum, show superior performance over convex methods.
• Demonstrated effectiveness in robust principal component analysis and low-rank representation.

Conclusions:

• The proposed nonconvex framework effectively learns low-rank structures from corrupted data.
• Nonconvex heuristics offer significant advantages over convex-norm-based methods for challenging datasets.
• The log-sum heuristic recovery algorithm is particularly effective for high-rank and densely corrupted data.