相关矩阵广义Fisher变换的变分方法
Fast inversion of the generalized Fisher transformation of correlation matrices
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中文总结 AI 辅助
本文提出一种变分方法证明广义Fisher变换的逆变换为唯一极小值点，并基于此开发GFT-FP+N算法，结合不动点迭代与共轭梯度牛顿步，在基准测试中实现更快且更稳健的收敛。
中文摘要 AI 辅助
广义Fisher变换通过矩阵对数的非对角元素将非奇异相关矩阵映射到无约束实向量。我们证明逆变换是矩阵对数对角元素的某个光滑、严格凸且强制的函数的唯一极小值点，这为变换是一一且满射的提供了新的简短证明。该函数的Hessian矩阵在每一点都介于相关矩阵指数的极端特征值之间，并且被该指数的对角元素所控制，且该界可达。这些界将逆变换的标准不动点迭代识别为准牛顿方法，其最坏情况局部因子在近奇异相关矩阵时可接近1，并保证牛顿系统的条件数不差于矩阵指数。我们提出GFT-FP+N，该方法通过共轭梯度计算无矩阵牛顿步来增强不动点迭代，无需显式雅可比矩阵。在基准实验中，GFT-FP+N在所有测试实例中均收敛，包括Broyden方法失败的实例，并且相对于不动点迭代将计算时间减少了最多三十倍。
英文摘要
The generalized Fisher transformation maps a non-singular correlation matrix to an unconstrained real vector through the off-diagonal elements of its matrix logarithm. Evaluating its inverse is a computational bottleneck in dynamic correlation and multivariate volatility models. We develop a fast inversion algorithm by characterizing the unknown diagonal as the minimizer of a smooth, strictly convex, and coercive objective. An explicit Hessian and global spectral bounds identify the standard fixed-point iteration as a quasi-Newton method and explain why it can converge slowly near singularity. Every fixed-point step decreases the objective, and the iteration converges from every starting point. These results motivate GFT-FP+N, a hybrid of fixed-point and matrix-free Newton steps that never forms the Jacobian. In benchmarks with up to 1,000 replications per design and dimensions up to 800, GFT-FP+N reduces computation time by up to a factor of forty-five relative to the fixed-point iteration and converged in every replication, including on designs where Broyden's method almost always fails. Julia and R packages are provided.
发表机构
- York University（约克大学）
- University of North Carolina at Chapel Hill（北卡罗来纳大学教堂山分校）
机构由 AI 辅助整理，请以论文原文为准。
