Exact higher-dimensional Isolation Forest solution

Derive the exact closed-form expected Isolation Forest depth and corresponding anomaly score for data in dimensions greater than one.

Background

The paper builds RIFT from a closed-form expression for the infinite-forest Isolation Forest limit in one dimension. It notes that this expression does not extend trivially to higher dimensions because the expected depth depends on the full geometric configuration of the data and on the changing coordinate-split structure produced by recursive partitioning.

The authors therefore use the one-dimensional geometric interpretation to motivate a Minimum Spanning Tree formulation rather than solving the higher-dimensional Isolation Forest process itself. An exact higher-dimensional solution would clarify the theoretical relationship between Isolation Forest and geometric methods such as RIFT.

References

Although the exact solution of $\bar{h}(x)$ is not known for $d>1$, we can develop new anomaly detection methods inspired by the geometric interpretation of the one-dimensional result.

— RIFT: Relative Isolation From Trees For Anomaly Detection  (2610.12244 - Szalai et al., 8 Oct 2026) in Section 3.1, “Concept of the algorithm”