Problem
Research questions and friction points this paper is trying to address.
Iteratively Reweighted Least Squares (IRLS)
Nuclear Norm Minimization
Convergence Rates
Low-Rank Recovery
Weight Operator
Tight Majorizations and Convergence Rates of Nuclear Norm Minimization IRLS
🤖 AI Summary
本文研究了迭代重加权最小二乘法在低秩恢复中的收敛速度，通过新的主次化分析证明了调和平均权重算子的有效性，并展示了其优于单边加权方案。
📝 Abstract
Iteratively reweighted least squares (IRLS) methods constitute a natural approach to nuclear norm minimization, but their convergence rates and the role of the weight operator have remained poorly understood. This paper establishes sharp convergence rates for IRLS methods for constrained nuclear norm minimization in low-rank recovery. A central ingredient is a new majorization analysis for the smoothed nuclear norm: we prove that the harmonic-mean weight operator defines a valid global quadratic majorizer. Furthermore, we show that this weight operator is optimal within the family of power-mean weights, clarifying why it improves over classical one-sided reweighting schemes that use only row- or column-space information. Under a Schatten-1 null space property, we prove global linear convergence of IRLS algorithms using a variety of weight operators, including the harmonic-mean weights. For IRLS with harmonic-mean weights, we prove a dimension-independent, locally linear convergence rate. We provide a counterexample showing that this dimension-independent local rate cannot in general be obtained for IRLS algorithms using one-sided weight operators, which predominate in the literature. Numerical experiments corroborate the theoretical results and illustrate the practical advantage of harmonic-mean reweighting across square, rectangular, and adversarially initialized recovery problems.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Iteratively Reweighted Least Squares (IRLS)
Nuclear Norm Minimization
Harmonic-Mean Weight Operator
Convergence Rates
Low-Rank Recovery
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