AI 中文总结
研究中心化高斯分布样本协方差误差，基于随机对偶理论开发通用框架，确定误差谱范数精确极限值，通过闭式上界、新颖下界机制及双副本策略使上下界匹配，理论与数值模拟相符。
AI 中文摘要
我们研究了中心化高斯分布的样本协方差误差。一项显著突破确定了正确的误差缩放阶数，并明确揭示了有效秩和真实协方差谱的关键作用。在本工作中，我们超越了缩放特征描述，确定了误差谱范数的精确极限值。为此，我们基于随机对偶理论（RDT）开发了一个通用框架。在此框架内，首先确定基于RDT的闭式显式上界，然后通过引入一种新颖的双线性二次RDT下界机制建立互补下界。通过将该机制与双副本系统界定策略相结合，表明在大维度情况下上下界匹配。理论结果辅以数值评估和模拟，显示对于数千量级的问题规模已有良好一致性。
英文摘要
We study the sample covariance error of centered Gaussians. A remarkable breakthrough [66] established the correct error scaling order and explicitly revealed the critical role of both the effective rank and the true covariance spectrum. In this work, we move beyond scaling characterizations and determine the precise limiting value of the error's spectral norm. To do so, we develop a generic framework based on Random Duality Theory (RDT). Within this framework, we first determine closed-form, explicit RDT-based upper bounds. We then establish complementary lower bounds by introducing a novel bilinear-quadratic RDT lower-bounding mechanism. By combining this mechanism with a two-replica systems bounding strategy, we show that our lower and upper bounds match in large-dimensional contexts. Our theoretical results are supplemented with numerical evaluations and simulations, demonstrating an excellent agreement already for problem sizes on the order of thousands.
