发表机构
University of Central Florida; Wirtschaftsuniversität Wien; KU Eichstätt-Ingolstadt（中佛罗里达大学; 维也纳经济大学; 艾希施泰特-英戈尔施塔特大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对低秩恢复的约束核范数最小化问题，通过新的上界分析证明调和均值权重算子的最优性，给出IRLS算法的收敛速率，并经数值实验验证其优于单侧重加权方案。
AI 中文摘要
迭代重加权最小二乘（IRLS）方法是求解核范数最小化问题的自然途径，但其收敛速率及权重算子的作用仍未得到充分理解。本文针对低秩恢复中的约束核范数最小化问题，建立了IRLS方法的精确收敛速率。核心内容是对平滑核范数开展新的上界分析：证明调和均值权重算子定义了有效的全局二次上界。此外，本文表明该权重算子在幂均值权重族中是最优的，阐明了其为何优于仅使用行或列空间信息的经典单侧重加权方案。在Schatten-1零空间性质下，本文证明了使用包括调和均值权重在内的多种权重算子的IRLS算法具有全局线性收敛性；对于采用调和均值权重的IRLS，本文证明了其具有与维度无关的局部线性收敛速率。本文提供了一个反例，表明采用单侧权重算子的IRLS算法通常无法获得这种与维度无关的局部速率，而单侧权重算子在现有文献中占主导地位。数值实验验证了理论结果，并展示了调和均值重加权在方阵、长方阵及对抗初始化的恢复问题中的实用优势。
英文摘要
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.
Comments97 pages, 9 figures, 2 tables
