Extended Isolation Forest with feature sensitivities

Compared to theoretical frameworks that assume equal sensitivity to deviations in all features of data, the theory of anomaly detection allowing for variable sensitivity across features is less developed. To the best of our knowledge, this issue has not yet been addressed in the context of isolation-based methods, and this paper represents the first attempt to do so. This paper introduces an Extended Isolation Forest with feature sensitivities, which we refer to as the Anisotropic Isolation Forest (AIF). In contrast to the standard EIF, the AIF enables anomaly detection with controllable sensitivity to deviations in different features or directions in the feature space. The paper also introduces novel measures of directional sensitivity, which allow quantification of AIF’s sensitivity in different directions in the feature space. These measures enable adjustment of the AIF’s sensitivity to task-specific requirements. We demonstrate the performance of the algorithm by applying it to synthetic and real-world datasets. The results show that the AIF enables anomaly detection that focuses on directions in the feature space where deviations from typical behavior are more important.

💡 Research Summary

The paper addresses a notable gap in the literature on isolation‑based anomaly detection: existing methods such as Isolation Forest (IF) and its extension, Extended Isolation Forest (EIF), assume isotropic sensitivity, treating deviations in all features or directions equally. In many real‑world scenarios, however, certain variables are more critical than others, and a detector that can be tuned to be more or less sensitive along specific axes would be highly valuable.

To fill this gap, the authors propose the Anisotropic Isolation Forest (AIF). The core idea is to replace the isotropic sampling of hyperplane normal vectors (normally drawn from a standard multivariate Gaussian N(0, I)) with sampling from a non‑isotropic distribution. Two families of distributions are considered: (i) a single multivariate Gaussian with a general covariance matrix A (not necessarily the identity) and (ii) a mixture of Gaussians Σπ_i N(0, A_i). By shaping A (or the mixture components) the orientation of the random hyperplanes can be biased toward or away from particular coordinate axes or arbitrary directions defined by the eigenvectors of A.

The authors formalize a directional‑sensitivity measure α(n)=√(nᵀAn) for any unit direction n∈S^{d‑1}. For a diagonal A, α(e_i)=√a_i directly quantifies sensitivity to the i‑th feature; for a general A, α(g_i)=√λ_i where g_i and λ_i are eigenvectors and eigenvalues, respectively. They also define an average‑over‑region sensitivity τ(B)=|B|^{-1}∫_B α(n) dn, useful when a whole sector of directions should be emphasized. When a mixture of Gaussians is used, analogous measures eα(n)=∑π_i √(nᵀA_i n) and eτ(B) are introduced. The authors enforce unit spectral radius on A to keep α(·) bounded in