比较三种矩阵求逆更新方法，找出不同场景下的最优选择。
Cost Trade-offs in Matrix Inversion Updates for Streaming Outlier Detection
- 用DI、ISM、WMI三种方法更新矩阵逆
- ISM适合秩1更新，WMI适合小更新，其余情况选DI
- 结果可直接用于在线异常检测系统优化
异常检测识别显著偏离预期模式的数据点，揭示可能需要特别关注的异常。引入在线学习可提升精度，通过持续更新模型以反映最新数据。当使用Christoffel函数作为异常分数时，需在初始逆矩阵基础上对秩-k更新后的矩阵进行逆更新。然而，目前尚无共识确定最优方法。本文对比了三种方法：直接求逆（DI）、迭代Sherman-Morrison（ISM）和Woodbury矩阵恒等式（WMI），推导其理论计算成本，并通过在CPU上运行的完整Python仿真验证。结果表明：对于秩-1更新，ISM最优；小更新相对于矩阵尺寸时，WMI表现最佳；其他情况则建议使用DI。该结论适用于任何涉及矩阵逆更新的问题，尤其推动高效在线异常检测技术的发展。
原文摘要 · Abstract (English)
Outlier detection identifies data points that deviate significantly from expected patterns, revealing anomalies that may require special attention. Incorporating online learning further improves accuracy by continuously updating the model to reflect the most recent data. When employing the Christoffel function as an outlier score, online learning requires updating the inverse of a matrix following a rank-k update, given the initial inverse. Surprisingly, there is no consensus on the optimal method for this task. This technical note aims to compare three different updating methods: Direct Inversion (DI), Iterative Sherman-Morrison (ISM), and Woodbury Matrix Identity (WMI), to identify the most suitable approach for different scenarios. We first derive the theoretical computational costs of each method and then validate these findings through comprehensive Python simulations run on a CPU. These results allow us to propose a simple, quantitative, and easy-to-remember rule that can be stated qualitatively as follows: ISM is optimal for rank-1 updates, WMI excels for small updates relative to matrix size, and DI is preferable otherwise. This technical note produces a general result for any problem involving a matrix inversion update. In particular, it contributes to the ongoing development of efficient online outlier detection techniques.
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