Deep Isolation Forest Anomaly Detection

• Deep Isolation Forest (DIF) is a robust anomaly detection framework that leverages random non-linear deep mappings with axis-aligned isolation trees.
• It employs ensembles of untrained neural networks to create diverse representation spaces, enabling curved splits that overcome traditional iForest limitations.
• DIF achieves state-of-the-art performance with linear scalability across various data domains and shows promise in out-of-distribution detection in medical imaging.

Deep Isolation Forest (DIF) is a framework for anomaly detection that integrates random, untrained deep neural feature mappings with the axis-parallel partitioning strategy of Isolation Forests. The methodology overcomes key limitations of traditional iForest, including poor performance on high-dimensional or non-linearly separable data and inherent algorithmic bias, by introducing a synergy between ensembles of random, non-linear representations and the efficient isolation mechanisms native to partition-based methods. DIF achieves state-of-the-art anomaly detection performance and linear scalability across tabular, graph, and time series domains, and its principles have been applied to out-of-distribution detection in medical imaging (Xu et al., 2022, Li et al., 2020).

1. Rationale and Core Innovations

The original Isolation Forest (iForest) separates anomalies by recursively applying axis-aligned splits in the input feature space. While its computational cost and simplicity are advantageous, this mechanism is inherently limited in complex data regimes. Specifically, iForest often fails to efficiently isolate “hard” anomalies that do not differ markedly along any single feature direction, especially in high-dimensional settings. Further, iForest is subject to the “ghost region” bias, whereby empty or low-density regions—arising solely from how axis-parallel splits are arranged—lead to anomalously low scores for points not truly anomalous.

DIF addresses these issues by first mapping data points $\bm{x}\in\mathbb{R}^d$ into a collection of $r$ random, non-linear representation spaces using untrained neural networks. In each representation, classic axis-parallel trees are grown as in iForest. Critically, axis-aligned cuts in the learned representations can correspond to highly non-linear, possibly curved partitions in the original space, enabling richer isolation boundaries (Xu et al., 2022).

2. Random Non-Linear Representation Ensemble

For each representation, DIF utilizes a randomly initialized neural network—typically an $L$-layer MLP for tabular data, a GNN for graphs, or a dilated CNN for time series. For input $\bm{x}$, the $u$-th random network computes

$$z^{(u)} = f_{\theta_u}(\bm{x}), \quad f_{\theta_u}:\mathbb{R}^d \rightarrow \mathbb{R}^p,$$

where $p$ is the representation dimension, typically $p\ll d$. All network weights $\theta_u$ are independently drawn (e.g., $\theta_u\sim\mathcal{N}(0,\sigma^2)$) and fixed—no gradient-based training occurs.

The ensemble comprises $r$ such mappings,

$$\mathscr{G}(\bm{x}) = \{\, z^{(1)}, z^{(2)}, ..., z^{(r)} \,\},$$

with $r$ commonly set near 50. This ensemble confers substantial diversity in the resultant representations, capturing many directions for isolation unavailable to axis-parallel splits in the original feature space (Xu et al., 2022).

3. Isolation Tree Construction and Partition Mechanism

Within each representation space $\mathbb{R}^p$, DIF constructs $t$ isolation trees. The partitioning procedure at each node executes the following:

1. Select dimension $d^*\sim\mathrm{Uniform}\{1,\dots,p\}$.
2. Sample split threshold $$\eta\sim\mathrm{Uniform}(\min_{\bm z\in\mathcal{P}}z_{d^*},\,\max_{\bm z\in\mathcal{P}}z_{d^*})$$.
3. Partition pool $\mathcal{P}$ into

$$\mathcal{P}_\mathrm{left} = \{\bm z\mid z_{d^*}\leq\eta\},\quad \mathcal{P}_\mathrm{right} = \{\bm z\mid z_{d^*}>\eta\}.$$

1. Recursively repeat the procedure until singleton leaves or a maximum depth $J$ is reached.

Because each neural network is random and untrained, tree construction requires no data-dependent optimization beyond classic random partitioning. For graph or time-series data, the same construction applies—only the representation backbone differs (Xu et al., 2022).

4. Anomaly Scoring: Path Length and Deviation Enhancement

DIF extends the classic iForest scoring mechanism. For sample $\bm{x}$, map into each representation, traverse every tree $\tau$ in the forest, and accumulate:

• Path length $h_\tau(\bm{z})$: number of edges from root to isolation leaf.
• At each split $k$, compute absolute deviation $\delta_k = |z_{d_k} - \eta_k|$.

For normalizing path lengths, the standard harmonic mean is used: $$c(n) = 2H_{n-1} - \frac{2(n-1)}{n},\quad H_m = \sum_{k=1}^m\frac{1}{k}.$$ Average path length and average split deviation are computed over all $T=rt$ trees. The deviation-enhanced score (DEAS) further refines detection power: $$g_\tau(\bm z) = \frac{1}{h_\tau(\bm z)}\sum_{k=1}^{h_\tau(\bm z)} \delta_k,$$

$$s_{\mathrm{DIF}}(\bm{x}) = 2^{- \bigl( \bar h(\bm{x}) / c(n) \bigr) \times \frac{1}{T} \sum_\tau g_\tau(f_{\theta_\tau}(\bm x)) }.$$

Empirically, DEAS leads to an $\sim$11% increase in AUC-PR compared to depth-only scoring (Xu et al., 2022).

5. Algorithmic Workflow: Training and Inference

Training:

Given dataset $\mathcal{D}$, choose $r$, $t$, subsample size $n$, and tree depth $J$.

• For $u=1$ to $r$: randomly initialize $f_{\theta_u}$, compute $$\mathcal{X}_u = \{f_{\theta_u}(\bm{x}):\bm{x}\in\mathcal{D}\}$$.
• For $i=1$ to $t$, grow an iTree on $n$ samples from $\mathcal{X}_u$.
• Aggregate all trees into the forest $\mathcal{T}$.

Inference:

For new point $\bm{x}$:

• For each tree $\tau$, compute $f_{\theta_\tau}(\bm{x})$, traverse to leaf, and record $h_\tau(\bm z)$ and deviations.
• Compute $s_{\mathrm{DIF}}(\bm{x})$ as above (Xu et al., 2022).

A related application is Deep Isolation Forest for out-of-distribution image detection (“DeepIF”), in which the untrained mappings are replaced by a pretrained CNN, and class-wise forests are trained on deep features per label (Li et al., 2020).

6. Empirical Results and Comparative Evaluation

Experiments demonstrate DIF's strong performance across modalities.

• Tabular: On 10 datasets, DIF achieves mean AUC-PR $\approx$ 0.35 (EIF: 0.22, iForest: 0.14). Mean AUC-ROC $\approx$ 0.93 (EIF: 0.89).
• Graphs: On Tox21 with GIN-based random maps, DIF outperforms or matches deep graph anomaly detectors.
• Time Series: On UCR datasets using dilated CNN backbones, DIF attains top ranks (e.g. ECG-wandering: AUC-ROC=1.0).
• Speed: Retains linear time, executes orders of magnitude faster than learned deep ensembles—training on $10^5$ samples with $10^3$ features completes in seconds to minutes.
• Removing DEAS reduces AUC-PR by $\sim$11%; replacing random maps with data-learned ones degrades AUC-ROC by 5–15%; results are robust to up to 10% anomaly contamination (Xu et al., 2022).

In medical OOD detection, DeepIF yields AUROC=0.7136 on HAM10000 (vs. Mahalanobis: 0.5771, VAE: 0.5315), preserving lesion classification accuracy (90.3%) (Li et al., 2020).

7. Limitations, Open Questions, and Future Directions

• DIF's random mappings are not data-adaptive. Incorporating lightweight self-supervised tuning could further focus partitions on informative structure.
• The current tree splitting is fully independent across representation spaces; correlated split strategies could reduce ensemble variance.
• Theoretical understanding of anomaly isolation with random deep mappings remains to be developed.
• In the DeepIF variant, TNR@95%TPR is still below 30% in OOD detection, and rare or very subtle anomalies may not be perfectly detected. Further, only single-class holdout OOD has been evaluated; broader settings are a target for future exploration (Xu et al., 2022, Li et al., 2020).

DIF represents a non-parametric, data-type-agnostic anomaly detection mechanism with strong performance, scalability, and extensibility, generalizing the iForest principle through random deep projection and maintaining applicability across structured and unstructured data spaces (Xu et al., 2022, Li et al., 2020).