解析孤立森林的归纳偏置，揭示其在异常检测中的高效原理
Theoretical Investigation on Inductive Bias of Isolation Forest
- 将隔离森林生长过程建模为随机游走，推导期望深度函数
- 发现其对中心异常不敏感且参数适应性优于kNN
- 为无监督异常检测提供理论基础，适合算法研究者参考
孤立森林（iForest）是一种广泛应用的无监督异常检测方法，因其出色的运行效率和在大规模任务中的优异表现而备受青睐。尽管应用广泛，其成功背后的理论基础仍不清晰。本文聚焦于iForest的归纳偏置，理论上阐明其在何种条件下表现良好。核心是建模iForest的生长过程——分裂维度与分裂值均随机选择，并将其视为随机游走过程，利用转移概率推导出期望深度函数（即iForest的输出）。案例研究揭示关键归纳偏置：iForest对中心异常不敏感，且相比k-近邻（k-Nearest Neighbor）具有更强的参数适应性。本研究为理解iForest的有效性提供了理论依据，并为后续理论探索奠定基础。
原文摘要 · Abstract (English)
Isolation Forest (iForest) stands out as a widely-used unsupervised anomaly detector, primarily owing to its remarkable runtime efficiency and superior performance in large-scale tasks. Despite its widespread adoption, a theoretical foundation explaining iForest's success remains unclear. This paper focuses on the inductive bias of iForest, which theoretically elucidates under what circumstances and to what extent iForest works well. The key is to formulate the growth process of iForest, where the split dimensions and split values are randomly selected. We model the growth process of iForest as a random walk, enabling us to derive the expected depth function, which is the outcome of iForest, using transition probabilities. The case studies reveal key inductive biases: iForest exhibits lower sensitivity to central anomalies while demonstrating greater parameter adaptability compared to $k$-Nearest Neighbor. Our study provides a theoretical understanding of the effectiveness of iForest and establishes a foundation for further theoretical exploration.
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