Anomaly Detection by an Ensemble of Random Pairs of Hyperspheres

We sincerely thank the reviewer for their perceptive review.

W1 and Q1 - Experimental Design

Each stochastic algorithm is executed using the random seeds ${0, 1, 2, 100, 1000}$—three consecutive PRNG states plus two distant ones—to probe robustness without excessive runtime. Furthermore, for every dataset, we pre-compute and cache three stratified 70/30 train–test splits using fixed split-seeds; these splits are reused verbatim by all methods.

Deterministic models (e.g., LOF, ECOD) run once per split; stochastic ones run for all seed–split combinations, yielding 15 scores each. Unless noted, all ROC-AUC and AP values are averages over these runs: 3 for deterministic, 15 for stochastic approaches.

To verify seed-independence, we reran ADERH, INNE, IForest, and EIF with seeds ${0,1,2,3,4}$—results deviated only slightly from the canonical set. This confirms that our findings are robust to random seed choice, ensuring fair and reproducible comparisons.

W2 - Strict $\delta$-separation

Empirically, ADERH is validated on several datasets—including large-scale benchmarks such as Census and high-dimensional feature embeddings like MNIST-Stripe—where it achieves strong AUC-ROC performance.

Furthermore, our standard deviation analysis confirms that ADERH exhibits substantially lower output variance compared to other isolation-based anomaly detection methods, indicating superior stability (please see our response to Reviewer TZiB regarding Q3).

W3 - Optimal Choice of Parameters $n$ and $\omega$

Please see our response in Q3 or Q4.

W4 - Comparison to INNE

ADERH is fundamentally novel, both in its design philosophy and operational execution.
Unlike traditional methods like INNE that rely on single-point statistics (e.g., radius ratios), ADERH introduces a synergistic dual-metric model capturing two orthogonal anomaly cues:

• Geometric alignment via Pitch
• Local sparsity via NDensity

---

Core Innovations

1. Multiplicative Dual-Factor Scoring

Rather than summing or tuning separate metrics, ADERH fuses Pitch and NDensity multiplicatively.
Anomalies only score high when both metrics indicate deviation, yielding precise, low-false-positive detection.
No existing method fuses geometry and density in this way with such semantic clarity.

$$\text{ADERH-Score}(x) = \text{Pitch}(x) \cdot \left(1 - \text{NDensity}(x)\right)$$

---

2. Adaptive Geometry via Pairwise Sphere Sampling

ADERH bypasses fixed-radius or $k$-NN constraints by randomly sampling point pairs within small subsets.
Each pair induces two adaptive spheres using half their distance. This enables local structure encoding without rigid thresholds, ideal for complex or nested geometries.

---

3. Built-in Robustness to Structural Anomalies

ADERH natively flags:

• Edge anomalies, often missed by INNE, due to high Pitch ($\approx 1$)
• Sparse-center anomalies, undetectable by density-only methods, via low NDensity
  (as formalized in Lemma 3.10)

This is design-level robustness, not a post-hoc adjustment.

---

4. Empirical & Theoretical Superiority

ADERH is not just conceptually superior—it outperforms INNE and others on 20+ datasets on AUC-ROC and AUC-AP.

Concretely, ADERH’s ROC-AUC std ≈ 0.0133 and AP std ≈ 0.0317, both much lower than INNE’s (≈ 0.0241 and ≈ 0.0515). This shows greater stability, validating ADERH's good performance.

---

No prior method unifies geometry-aware scoring, density sensitivity, and adaptive local structure the way ADERH does.

Q2 - Limited Grid Search

To ensure fair comparisons, we equalized total compute (~9–12 evals/method), not per-axis grid granularity, accounting for varying hyperparameter dimensions and runtimes. Parameter ranges centered on author-recommended defaults from original papers or widely used libraries.

Since unsupervised anomaly detection lacks labels, traditional tuning (e.g., CV) is infeasible and risks bias. As highlighted in [1] and [2], defaults are standard for consistency and scale. We followed this precedent to maintain parity.

Crucially, we went further, conducting systematic grid searches for both isolation-based and deep learning baselines. Across 20+ benchmarks, our method ranked #1 on average under both realistic (defaults) and idealized (tuned) conditions.

To further solidify the evaluation, we extended the grid for isolation-based baselines from 9 to 25 configurations:

• ADERH (proposed method): Varying the ensemble size
  $n_{\mathrm{esti}} \in \{100, 150, 200, 300, 500\}$
  and the random-subset size
  $\omega \in \{8, 18, 24, 30, 36\}$

• INNE: Varying
  $n_{\mathrm{esti}} \in \{100, 150, 200, 300, 500\}$
  and max subsample size
  $\{8, 18, 24, 30, 36\}$

• IForest: Varying
  $n_{\mathrm{esti}} \in \{100, 150, 200, 300, 500\}$
  and max samples per tree
  $\{128, 200, 256, 300, 500\}$

• EIF: Same grid as IForest.

We evaluate a representative subset of datasets, sufficient to capture the main trends. The results validate the original findings: ADERH consistently outperforms isolation-based baselines, showing robustness across all parameter settings. Even with 25 optimized configurations, no baseline surpassed ADERH in multiple settings.

---

Table: Performance comparison of four methods across various datasets

Bold = best score (rank 1)

Q3 - Parameter Choices

With fixed compute $N_{\text{hyp}} = n\omega$ (ADERH scales linearly), risk is minimized by choosing $\omega$ just large enough to cover all normal modes (typically low-teens) and allocating the rest to ensemble size $n$. This strategy balances contamination control with the variance bound established in the paper.

Variance

From Equation 14, the ensemble variance satisfies:

$$\operatorname{Var}[I(x)] \le \frac{1}{4n}$$

Substituting $n = \frac{N_{\text{hyp}}}{\omega}$ gives:

$$\frac{\omega}{4N_{\text{hyp}}}$$

This reveals that larger $\omega$ increases variance by reducing the number of subsets available for averaging.

---

Contamination Bias

For anomaly rate $\rho \ll 1$, the chance that a subset of size $\omega$ includes at least one anomaly is:

$$p_{\mathrm{cont}}(\omega) = 1 - (1 - \rho)^\omega$$

Each contaminated subset perturbs the score by at most $c \le 1$ (since $\text{WPitch} \in [0,1]$), so the squared bias is bounded by:

$$c^2 \cdot p_{\mathrm{cont}}(\omega)^2$$

For small $\rho\omega$, this grows like $(\rho\omega)^2$ and eventually saturates—worsening with larger $\omega$.

---

Coverage

A subset must sample at least one representative from each of the $k$ normal clusters or no hypersphere can be centered in that region. Coupon-collector tails give:

$$\Pr[\text{miss a cluster}] \le k e^{-\frac{\omega}{k}}$$

To guarantee failure probability below $\delta$, it suffices that:

$$\omega \ge k \ln(k/\delta)$$

---

Mean-Squared Error (MSE) Proxy

Combining the three pieces yields a proxy for mean-squared error:

$$\mathrm{MSE}(\omega) \le \frac{\omega}{4N_{\text{hyp}}} + c^2\left[1 - (1 - \rho)^{\omega}\right]^2 + \mathbf{1}_{\{\omega < k \ln(k/\delta)\}}$$

• The indicator drops to zero once $\omega$ hits the coupon-collector threshold (i.e., enough to sample each cluster once).
• Beyond this, both error terms grow with $\omega$, so the optimal $\omega$ is the smallest feasible.

Feasibility isn't purely statistical: random pairing requires ~10–20 points per subset to ensure full cluster coverage and pairing diversity. This reflects a contamination–variance tradeoff:

• First, using 256 subsets isn’t inherently optimal—it reflects a safe, low-contamination $\omega$ under computational limits.
• Second, $n\omega$ isn’t decisive:
  • $n$ controls variance
  • $\omega$ controls bias
  • Their balance ensures reliable isolation.

Q4 - Parameter Stability

ADERH demonstrates strong robustness and stability across diverse datasets, showing low standard deviations regardless of dimensionality or sample size (e.g., Pendigits: 0.004, Skin: 0.013). This aligns with its theoretical foundation, which is agnostic to data characteristics. Ensemble configurations consistently outperform single models, confirming the theoretical benefits of aggregation. Moreover, default parameters yield performance close to grid-searched optima, supporting the bias–variance prediction that ADERH operates in a flat region. Despite the unsupervised nature of anomaly detection, ADERH achieves state-of-the-art performance with minimal tuning.

[1] Fang, Zeyu, et al. "Towards a Unified Framework of Clustering-based Anomaly Detection." Forty-second International Conference on Machine Learning (ICML 2025).

[2] Han, Songqiao, et al. "Adbench: Anomaly detection benchmark." Advances in neural information processing systems 35 (2022).

We thank the reviewer for the valuable feedback.

W1 - General Suggestions for Improvement

We thank the reviewer for the valuable suggestions for improvement and will reduce repetition—especially around concepts like Rarity and Deviation—by consolidating overlapping explanations. Introductory sections will be tightened to eliminate redundancy, and essential content from the Appendix will be integrated into the main text where it supports key results like Theorem 3.15.

Q1 - Choice of Norm

Due to its convenience and common usage, the Euclidean ($\ell_2$) distance was selected as the default norm in ADERH. However, all core steps rely only on the axioms of a metric, not on Euclidean geometry. Consequently, the algorithm and theory extend unchanged to any alternative metric. Complex data (e.g., images) is typically embedded into vector spaces with well-defined metrics. In our experiments, we used datasets such as Leather, Bottle, Toothbrush, and MNIST-Stripe, whose features were originally obtained using a ResNet18 feature extractor.

Q2 - Non-spherical Data Patterns

Unlike OCSVM’s global boundary, ADERH uses local hyperspheres with Pitch and NDensity that may adapt to complex manifolds and avoid flagging anomalies as normals. On one synthetic two-moons dataset (with anomalies, density variation, and complex shape), ADERH achieves top results vs. multiple SOTA baselines. Due to space constraints, we kindly refer the reviewer to our detailed response under Reviewer gbWb for a full discussion of this point.

Q3 - Differences to Studies Mentioned in the Review

We highlight key differences from ADERH to [1,2,3] in the revised Related Work and present new comparative experiments using the original evaluation protocol across multiple benchmark datasets.

MV‑ERM and MV‑SRM [1]

Main Ideas in MV‑ERM and MV‑SRM.
The framework described in [1] recasts minimum‑volume‑set estimation as an empirical‑risk‑minimisation task: MV‑ERM chooses, within a prescribed class of candidate regions, the set of smallest reference‑measure volume whose penalised empirical mass still reaches the target level $\alpha$, which yields distribution‑free finite‑sample bounds and strong universal consistency; MV‑SRM embeds the same complexity penalty directly in the objective, so the optimiser simultaneously tunes structural complexity—e.g., histogram bandwidth or decision‑tree depth—to the data.

Difference to ADERH and MV‑ERM / MV‑SRM
ADERH replaces the axis-aligned partitions of MV-ERM and MV-SRM with randomly centered, multi-scale hyperspheres, avoiding VC-dimension overhead and rigid geometry assumptions. Its anomaly score—based on boundary proximity and local sparsity—is continuous, variance-bounded by $1/(4n)$, and scales linearly with data size and dimension. This yields a parameter-light, geometry-adaptive alternative aligned with the minimum-volume principle, but free from explicit volume estimation.

[1] C. Scott and R. Nowak, “Learning minimum volume sets,” Journal of Machine Learning Research, vol. 7, pp. 665–704, April 2006.

---

GEM [2]

Main Ideas in Geometric Entropy Minimisation (GEM).
GEM [2] casts anomaly detection as a minimal-wiring problem: it selects the $K = \lfloor \rho n \rfloor$ points whose $k$-NN graph (or MST) has minimal total edge length, yielding the tightest subset covering mass $\rho$ and converging to the distribution’s minimum-entropy set. At test time, a candidate is anomalous if inserting it increases total wiring more than any inlier, with the edge-length jump providing a calibrated score. GEM is scale-free and geometrically adaptive, but computationally heavy due to leave-one-out graph updates and sensitive to edge-length estimates.

Difference to ADERH and GEM
GEM constructs global graphs and uses fixed neighborhoods, making it prone to dense-boundary false positives. ADERH, in contrast, avoids a global view and builds many twin hyperspheres from random point pairs and scores points via the product of Pitch (boundary ratio) and NDensity (inverse occupancy). Anomalies emerge only when both signals align, enabling robust separation under high dimensionality and heterogeneous cluster shapes, with no graph construction or tuning beyond $n$ and $\omega$.

[2] Hero, Alfred. "Geometric entropy minimization (GEM) for anomaly detection and localization." Advances in Neural Information Processing Systems 19 (2006).

---

DTM [3]

Main Ideas in Distance-to-a-Measure (DTM).
In DTM [3], a small probability window $m$ (or its discrete proxy $k = \lceil m n \rceil$) and a power parameter $q$ define, for every point $x$, the average radius $r_p(x)$ needed to enclose each mass level $p \le m$, yielding the score:

$$d_{P,m,q}(x) = \left( \frac{1}{m} \int_0^m r_p(x)^q\, dp \right)^{1/q}$$

Small values indicate dense interior points, while large values suggest potential anomalies. The original paper analyses an unbagged estimator, yet the authors’ GitHub reproduces a baseline named aNNE that simply wraps DTM inside scikit-learn's BaggingRegressor, training multiple DTM models on random subsamples and averaging their outputs—an ensemble trick that smooths variance but leaves the underlying DTM logic untouched.

Difference to ADERH and DTM
While DTM and its bagged forms rely on global distance graphs, ADERH uses a purely local, randomized ensemble of hyperspheres to score points by edge proximity and sparsity. This avoids the instability of nearest-neighbor distances near dense boundaries, offering stable anomaly detection in high dimensions with fixed hyperparameters and no reliance on global structure.

[3] Gu, Xiaoyi, Leman Akoglu, and Alessandro Rinaldo. "Statistical analysis of nearest neighbor methods for anomaly detection." Advances in Neural Information Processing Systems 32 (2019).

---

Experimental Results

We evaluate ADERH, GEM, and DTM under both default and optimized settings to assess detection performance and robustness.

• ADERH:
  $n_{\text{esti}} \in \{100, 200, 300\}$
  $\omega \in \{8, 18, 24\}$

• GEM (default $k = 5$):
  $k \in \{5, 10, 20, 30, 40, 50\}$

• DTM (default $n_{\text{esti}} = 100$, subsample size = 256):
  Uses scikit-learn's BaggingRegressor over DTM, per the protocol in [3]
  $n_{\text{esti}} \in \{100, 200, 300\}$
  Subsample size $\in \{128, 256, 300\}$

We report AUC-ROC scores under both default and optimized settings. The tables below present a representative subset of evaluated datasets due to space constraints. However, all datasets from the main paper were evaluated. Across the full benchmark, ADERH achieves the best average rank under both default (1.30 vs. 2.85 for GEM, 1.60 for DTM) and grid search settings (1.29 vs. 2.43 for GEM, 1.86 for DTM), confirming its consistent superiority. ADERH consistently ranks top on most datasets and never performs the worst. Its robustness across configurations stems from its random pairing and multi-scale design, which adapts to varying data without relying on global heuristics like GEM. ADERH shows a statistically significant improvement over GEM and DTM under both default and grid search settings (paired Wilcoxon signed-rank test with Bonferroni-Holm correction; $p < 0.05$, $\alpha = 0.05$). Its pair-and-halve strategy with density-weighted scoring enables scalable, multi-scale isolation with provable variance control.

We will release our updated codebase after the review process concludes, in accordance with the NeurIPS policy, which prohibits updates during the review phase.

Table: AUC-ROC performance under default setup

Comparison between ADERH, GEM [2], and DTM [3]
*Bold = best score (rank 1)

---

Table: AUC-ROC performance with hyperparameter optimization; best result per method is reported.

Comparison between ADERH, GEM [2], and DTM [3]
*Bold = best score (rank 1)

We thank the reviewer for the helpful feedback.

Q1 - Curse of Dimensionality for Assumption 3.1

Assumption 3.1 relies on the notion that normal points lie within radius $\sigma$ of some local cluster center $\mu_j$, while anomalies fall outside a margin $\sigma + \delta$. In low-dimensional settings, such a margin $\delta$ offers a clean geometric separation between dense inlier regions and sparser outlier zones. However, this separation becomes fragile in high-dimensional regimes due to the well-known curse of dimensionality.

Specifically, for data vectors $x \in \mathbb{R}^d$ with independent, isotropic components, the Euclidean norm satisfies $\|x\|_2 \approx \sqrt{d}$ in expectation, with relatively small variance. As a consequence, both inter-point distances and intra-cluster radii naturally scale with $\sqrt{d}$. Thus, in unscaled coordinates, the fixed gap $\delta$ postulated in Assumption 3.1 effectively vanishes unless it, too, grows with $\sqrt{d}$.

Nonetheless, the conceptual intent of Assumption 3.1 survives under a reinterpretation. Rather than demanding a fixed absolute margin $\delta$, it is more appropriate to require a relative margin $\tau = \delta / \sigma$—a dimensionless ratio reflecting how far anomalies lie from their nearest clusters, scaled by the local intra-cluster spread. This formulation is standard in robust statistics and aligns with Mahalanobis-distance-based interpretations, where each feature is normalized to unit variance. In this view, anomalies are required to lie outside a radius $(1 + \tau)\cdot \sigma$, a condition that remains meaningful even in high-dimensional spaces.

The ADERH algorithm implicitly respects this scaling. Each hypersphere is defined by a randomly chosen pair of points, with radius $R = \frac{1}{2}\|x - y\|$. Since distances themselves scale with $\sqrt{d}$, the hypersphere radii adjust naturally to the ambient geometry. The Pitch term—computed as $\mathrm{dist}(x, c)/R$—is invariant under uniform scaling of the space, and the NDensity term depends purely on counts rather than metric magnitudes. Therefore, the anomaly score produced by ADERH is unaffected by global distance inflation. Moreover, ensemble averaging across $n$ such hyperspheres reduces the variance of these scores by a factor of $1/(4n)$, preserving their discriminative power even when some hyperspheres are degraded by high-dimensional effects.

In conclusion, although the curse of dimensionality erodes the efficacy of fixed Euclidean margins, Assumption 3.1 remains valid when interpreted in relative terms. ADERH maintains its robustness under this reformulation, particularly when features are normalized such that $\sigma \approx 1$, allowing the parameters $\delta$ and $\tau$ to be treated interchangeably. Empirically, ADERH shows strong performance on high-dimensional datasets like Leather, Toothbrush, or Census.

For a quick sanity check, we generated a high-dimensional dataset with 100,000 features, consisting of clustered normal points and injected anomalies. On this data, ADERH achieves a perfect AUC-ROC of $1.00$.

The code for this is shown below:

# High-dimensional anomaly dataset generation
from sklearn.datasets import make_blobs
import numpy as np

n_samples = 500
n_features = 100000
n_clusters = 3
anomaly_fraction = 0.05
n_anomalies = int(n_samples * anomaly_fraction)
n_normals = n_samples - n_anomalies

# Generate normal data (3 blobs)
X_normal, _ = make_blobs(n_samples=n_normals,
                         n_features=n_features,
                         centers=n_clusters,
                         cluster_std=5.0,
                         random_state=42)

min_val = X_normal.min()
max_val = X_normal.max()
X_anomaly = np.random.uniform(min_val, max_val, size=(n_anomalies, n_features))

# Combine data
X = np.vstack([X_normal, X_anomaly])

# Labels: 0 = normal, 1 = anomaly
Y = np.hstack([np.zeros(n_normals), np.ones(n_anomalies)])

Practical remark. If the adverse effects of high dimensionality remain pronounced in a given dataset, one can first apply a dimension-aware feature transform—such as UMAP. These transforms mitigate distance concentration while approximately conserving relative cluster geometry. After this preprocessing step, ADERH is run in the transformed space: both its Pitch and (normalized) NDensity terms still operate on relative scales, so the theoretical guarantees and the $\tau$-margin interpretation continue to hold.

Q2 - Capturing Heterogeneous Distributions

By capturing heterogeneous distributions, we mean that our approach is able to identify anomalies situated near structures with different characteristics, such as varying densities.

We can demonstrate the beneficial properties of ADERH empirically, for example. While we are not permitted to add new results to the anonymous GitHub repository during the rebuttal, we conducted additional evaluations on the two moons dataset from sklearn, modifying it to exhibit varying densities and injecting both local and global outliers.

Normal points lie along a curved, non-convex manifold, intentionally violating the single-cluster (or single-blob) assumption underpinning classical boundary-based models such as the one-class SVM. This makes the dataset particularly well-suited for evaluating algorithms that must infer non-linear decision boundaries in complex, topologically non-trivial structures.

We evaluated ADERH, INNE, IForest, DIF, LODA, and ECOD on this dataset. On this challenging benchmark, ADERH achieves the best overall ranking. Competing methods, including INNE and Isolation Forest, rank lower, while even deep learning-based approaches such as DIF perform worse. These results highlight ADERH's ability to model intricate data geometry and reject anomalies effectively.

Table: Performance on the Two Moons dataset with injected anomalies

Ranked by ROC-AUC

The code used to create the dataset is shown below:

# Two Moons dataset with injected outliers
from sklearn.datasets import make_moons
import numpy as np

X, Y = make_moons(
    n_samples=(1000, 300),
    noise=0.05,
    random_state=42
)

outliers = np.array([
    [-0.5, -0.5],
    [ 0.0,  0.7],
    [-1.5,  0.8],
    [ 1.5,  0.25],
    [ 2.0,  1.0],
    [ 1.0, -0.15],
    [-1.0, -0.12]
])

X1 = np.vstack([X, outliers])
y_full = np.hstack([
    np.zeros(len(X)),
    np.ones(len(outliers))
])

Q3 - Empty Set in Eq. 12

Yes, that is correct. Since the hyperspheres considered in Eq. 12 are created using a subset $S_i \subset \mathcal{D}$ and Eq. 12 is calculated for all points in the dataset $\mathcal{D}$, there is no guarantee that all points $x \in \mathcal{D}$ will be covered by at least one hypersphere (this is in contrast to points $x \in S_i$, which are guaranteed to be within at least one hypersphere). In practice, however, this situation does not occur very often, as the random pairing of points usually results in some hyperspheres with large radii and, therefore, almost complete coverage of the data space.

Improvement Suggestions and Limitations

We thank the reviewer for the many suggestions to improve our manuscript, and we incorporated those in the revised version:

1. Related work & refs.

   • L32 – Added Deep iForest (DIF).
   • L34 – Replaced Hawkins with Barnett & Lewis 1974; Aggarwal 2017.

2. Text, notation, flow

   • L41 – Split iForest description vs. weakness.
   • §2 – Condensed; shortened LOF details.
   • L130 – Dropped rarity parenthetical.
   • Assump 3.1 – Defined $J$; removed “typical”.
   • L176 – Added forward-ref to radius variability.
   • Def 3.7 – Defined $X_{\mathcal{H}}$ in Eq. 4, and we will make this clearer.
   • Eq 9/10 – \max fixed; $c_{\mathcal{H}}$ typo should be $C(\mathcal{H})$ as defined in L191.
   • Fig 2 – We will improve this for clarity.

3. Theory & proofs

   • A.3 – adjusted.
   • Lemma 3.10 – Stated a.s. convergence; removed “$\ll$”.
   • L220–222 – Ensembling mitigates impact, but individual subsets remain vulnerable. We'll address this.
   • Lemma 3.13 – adjusted.

Also, we will add more information regarding the limitations of ADERH.

We thank the reviewer for the valuable feedback.

Q1 - Proper Citations of Baseline Methods

We thank the reviewer for this indication. The section 'Experimental Setup' will be expanded to include full bibliographic details for every comparator.

Q2 - Recent state-of-the-art Baselines

Following the reviewers’ request, we will incorporate UniCAD [1], one of the most recent publicly available anomaly detection methods (ICML 2025), into our comparison experiments. Since this experiment is extensive, we consider only a representative subset of datasets. Nevertheless, this selection is sufficient to capture and reflect the key trends and main findings. The results of those experiments, using the implementation and hyperparameters as referenced in the original paper, are shown in the table below. We see that ADERH achieves superior results in the majority of cases.
This improvement is statistically significant under a paired Wilcoxon signed-rank test with Bonferroni–Holm correction at α = 0.05 (p-value < 0.05).
We will release our updated codebase after the review process concludes, in accordance with the NeurIPS policy, which prohibits updates during the review phase.

Table: Extended AUC-ROC performance comparison between ADERH and UniCAD [1] across additional datasets.

---

[1] Fang, Zeyu, et al. "Towards a Unified Framework of Clustering-based Anomaly Detection." Forty-second International Conference on Machine Learning (ICML 2025).

Q3 - Reporting of Statistical Results

In our submission, Table 1 reports for each dataset the mean result of 15 runs for non-deterministic algorithms like ADERH (three splits with five seeds each) and three runs for deterministic algorithms. Also, the other Tables report the mean of all tested configurations. In the final version, we will mention this directly in the corresponding captions of the tables. We agree that including standard deviations (std) provides additional insight into the stability and robustness of each method. We will add all std values in the revised manuscript and believe these additions address the reviewer’s concern and strengthen the empirical rigor of our evaluation. For the main competing baselines — ADERH, INNE, and IForest — we report the average standard deviation across all datasets below:

• ADERH — mean ROC-AUC std ≈ 0.0133, mean AP std ≈ 0.0317
• INNE — mean ROC-AUC std ≈ 0.0241, mean AP std ≈ 0.0515
• IForest — mean ROC-AUC std ≈ 0.0235, mean AP std ≈ 0.0417

These results show that ADERH exhibits the lowest standard deviation, suggesting it is the most stable and consistent method across various datasets.

Q4 - Missing Visualizations

Effect of Pitch and NDensity

Appendix O presents a numerical ablation study (Table 7), where various design choices are individually analyzed, highlighting the overall advantage of the final ADERH configuration. This study demonstrates that each component contributes significantly to ADERH’s strong performance. To further support these findings, we will include additional figures in the revised version, which illustrate the score distributions before and after combining Pitch and NDensity.

Varying Radii through Random Pairing

Indeed, a central claim of our approach is that random pairing of points naturally produces hyperspheres of varying radii, enabling adaptive coverage across multiple scales. To empirically substantiate this claim, we analyzed the hypersphere radii generated by ADERH using real-world data. This analysis appears in Appendix C, where Figure 3 presents histograms showing the distribution of hypersphere radii across six benchmark datasets. The histograms in Figure 3 reveal diverse and data-dependent radius distributions. In some cases (e.g., Satimage-2, Waveform), the radii are concentrated around smaller values, indicating large dense neighborhoods. In other cases (e.g., Skin, Musk), the distributions are more spread out or even multi-modal, reflecting the presence of both compact and sparse regions. These results offer direct empirical confirmation of our theoretical analysis in Section 3 and Appendix B, which shows that random pairing leads to both intra- and inter-cluster combinations. The net effect is that hyperspheres span a wide range of radii, without the need for manual scale selection.

Additive vs. Multiplicative Combination of Pitch and NDensity

In our submission, we address this question empirically in Appendix Q through a comprehensive ablation study. Here, we compare the performance of the multiplicative combination $(1 - \text{NDensity}) \cdot \text{Pitch}$ with the additive variant $(1 - \text{NDensity}) + \text{Pitch}$ using 21 datasets and AUC-PR as the primary metric.

The results in Table 9 show that the multiplicative strategy consistently achieves better or comparable precision-recall performance, with a significantly lower average rank (1.29 versus 1.71). This supports our theoretical motivation: the multiplicative form yields high anomaly scores only when both boundary proximity and low local density are simultaneously present—i.e., anomalies that are both spatially isolated and lie near hypersphere boundaries. In contrast, the additive form can assign high scores when only one of these two conditions is met, leading to inflated anomaly scores in some dense regions. Nonetheless, we agree that complementary visualizations would make this distinction more tangible and interpretable. We will add visual examples showcasing the strengths of a multiplicative combination to Appendix Q in the revised version.

Ensemble Scoring Mechanism

We fully agree that demonstrating the robustness and variance reduction of the ensemble beyond theoretical bounds enhances the transparency and interpretability of our method. To address this, we will include a direct empirical comparison between the ensemble score and the top-performing single subset (Subset #1) across a diverse collection of datasets in the manuscript. The ensemble consistently yields higher AUC-ROC values than any individual subset, demonstrating its robustness and generalization benefits. The results confirm our theoretical claims and align with the intuition that averaging over multiple diverse scoring functions reduces sensitivity to idiosyncratic failure modes of individual subsets, thereby increasing robustness.

Suggestions

We appreciate the helpful suggestions of the reviewer that have already led to additional numerical results. In addition, new figures will be added to the final manuscript to illustrate specific components of our methodology.