Classical multivariate statistical methods such as covariance estimation and principal component analysis are well understood mathematically, yet their application at extreme data scales remains challenging. When the number of observations reaches billions, performance is limited by data movement, input-output bottlenecks, and numerical stability rather than arithmetic complexity. This work presents a case study of scaling classical multivariate statistics on a single multi-GPU node. Using C++ and CUDA, a GPU-accelerated workflow was developed to compute sufficient statistics in a single pass over a 10-billion-row dataset. Column sums and cross-product matrices are used to enable downstream computation of means, covariance, correlation, and principal component analysis without revisiting the raw data. The results highlight the importance of data representation, validation using known invariants, and careful numerical treatment when applying established statistical methods at large scale.

翻译：经典多变量统计方法（如协方差估计与主成分分析）在数学上已被充分理解，但当观测数据规模达到极端尺度时，其应用仍面临严峻挑战。当观测数量达到数十亿级别时，性能瓶颈主要源自数据迁移、输入输出阻塞以及数值稳定性问题，而非算术复杂度本身。本工作展示了在单节点多GPU系统上扩展经典多变量统计方法的案例研究。通过采用C++与CUDA技术，我们开发了GPU加速工作流，可在单次遍历中完成百亿行数据集充分统计量的计算。利用列和与叉积矩阵，无需重复访问原始数据即可实现均值、协方差、相关系数及主成分分析的下游计算。研究结果凸显了在超大规模场景下应用成熟统计方法时，数据表示设计、基于已知不变量的验证机制以及精细数值处理的重要价值。