New Analysis Improves Nuclear Norm Minimization Convergence

Iteratively Reweighted Least Squares (IRLS) methods are a common approach for nuclear norm minimization, a technique widely used in low-rank matrix recovery problems. However, the precise convergence rates and the optimal design of the weight operator within these methods have not been fully understood. This research addresses these gaps by providing a detailed majorization analysis. The paper proves that the harmonic-mean weight operator acts as a valid and optimal global quadratic majorizer within the family of power-mean weights. This finding clarifies why it outperforms traditional one-sided reweighting schemes that only use partial information. Under specific conditions (Schatten-1 null space property), the study demonstrates global linear convergence for IRLS algorithms using various weight operators, and notably, a dimension-independent, locally linear convergence rate for methods employing harmonic-mean weights. This theoretical advantage is also supported by numerical experiments across different recovery problems.