The estimation of large covariance matrices has a high dimensional bias. Correcting for this bias can be reformulated via the tool of Free Probability Theory as a free deconvolution. The goal of this work is a computational and statistical resolution of this problem. Our approach is based on complex-analytic methods methods to invert $S$-transforms. In particular, one needs a theoretical understanding of the Riemann surfaces where multivalued $S$ transforms live and an efficient computational scheme.

翻译：大协方差矩阵的估计存在高维偏差，通过自由概率论工具可将此偏差校正问题重构为自由反卷积过程。本研究的核心目标是从计算与统计两个维度解决该问题。我们提出基于复分析方法来求逆$S$-变换的技术路线，其中关键环节包括：从理论层面理解多值$S$变换所依存的黎曼曲面结构，以及构建高效的计算方案。