Cost Trade-offs in Matrix Inversion Updates for Streaming Outlier Detection

2026-03-17

AI Summary

This work addresses the computational inefficiency in online anomaly detection when frequently updating the inverse of matrices derived from Christoffel functions, where no clear criterion exists for selecting optimal update strategies across varying rank-update scenarios. The study systematically compares three approaches—direct inversion (DI), iterative Sherman–Morrison (ISM), and the Woodbury matrix identity (WMI)—through theoretical complexity analysis and Python-based simulations. It establishes a concise quantitative selection rule: ISM is optimal for rank-1 updates, WMI is most efficient when the update rank is substantially smaller than the matrix dimension, and DI is preferred otherwise. This rule significantly enhances computational efficiency in online anomaly detection and offers general guidance for streaming matrix inversion problems.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Data Mining & Knowledge Management: Anomaly/Outlier Detection
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

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.

Problem

• matrix inversion update
• online outlier detection
• Christoffel function
• rank-k update
• computational cost

Innovation

• matrix inversion update
• online outlier detection
• Christoffel function
• Sherman-Morrison
• Woodbury identity