Summary
In the context of anomaly detection, the authors introduce the hypothesis that the underlying density function tends to have lower variance over normal samples than over abnormal samples. Under this hypothesis, they proposed variance-stabilized density estimation: it learns a density estimator using normal samples only with regularization for the sample variance of density estimates on normal samples. In this paper, the authors adopt probabilistic normalized network with feature permutation ensemble as a density estimator. Through experiments, the authors claim that the proposed anomaly detector, which uses a score based on the estimated density estimates, achieves higher accuracy than existing detectors. Also, they performed other experiments to verify their hypothesis and for ablation study.
Weaknesses
4. In several parts, the mathematical description is unclear. For example, the authors do not write the definitions of $X\_N$ and $X\_A$ in the first paragraph of Section 3, $x^{(<i)}$ in (2), $\nabla\_x^{(i)}$ in (4), $h\_A$ and $h\_b$ in (5), and skewed Gaussians of Section 4.1.
5. The justification for the introduction of the hypothesis $\sigma\_n^2<\sigma\_a^2$ is not sufficiently clear. This hypothesis implies that the underlying densities over normal samples are stable globally, not locally. I cannot agree this formulation on the basis of their description alone. Seeing the formulation of the score function, I understood that the authors defined normal samples as samples on $x$ with large $p\_X(x)$ and abnormal samples as samples on $x$ with small $p\_X(x)$. If this is true, I think that $\sigma\_a^2$ tends to be small and that the authors' hypothesis requires the uniformity (stability) of $p\_X(x)$ for $x\in X\_N$ strongly. I don't think that this point, which sounds somewhat odd, is adequately explained (this comment is related to the next comment as well).
6. Figure 2: Here, the authors have attempted to verify the hypothesis, $\sigma\_n^2<\sigma\_a^2$. The authors applied an estimated density instead of an inaccessible underlying density. In a domain $X\_{low}$ with a low density $p\_X(x)$, a density estimate $\hat{p}\_{\theta}(x)$ often have high variance ($Var[\hat{p}\_{\theta}(x)]$ gets large at each $x\in X\_{low}$, where Var is taken over all the draws of sample set). I suspect that this issue may just have greatly affected the experimental results, but have you done enough experiments to rule this suspicion out? Please write how to estimate the density. I consider this experiment to be quite important in motivating the study, considering also the 5th comment.
7. The lead up to the formulation of the proposed method (1) is not clear. The authors should descript the reason why (1) uses only $x\in X\_N$. It is not enough to write “because previous studies have done so”. Make your paper as self-contained as possible. Also, why did (1) cut out only part of the hypothesis $\sigma\_n^2<\sigma\_a^2$ and adopt the regularization term $\lambda\hat{\sigma}\_n^2$? Clarify the reason why the regularization term $\lambda\hat{\sigma}\_n^2/\hat{\sigma}\_a^2$ or $-\lambda\hat{\sigma}\_a^2$ is bad? I worry that the regularization term $\lambda\hat{\sigma}\_n^2$ may have a different effect on the estimation than what the authors expected.
8. I see that the formulation of the proposed method in Section 3 consists of the former part "Regularized density estimation Following ... (low variance) density estimate" and latter part "In recent years, ... (Jaffe et al., 2015) to anomaly detection". Although these are two independent proposals, the discussion throughout the paper is biased toward the former half. The author can use another architecture for a density estimator. If the authors want to claim that the latter part is also an important proposal, they should more clearly state the arguments that support it. For example, experimental results in section 4.2 for OURS without regularization (i.e., $\lambda=0$) should also be reported at least. If otherwise, the readers cannot understand whether the goodness of OURS is based on the former half or the latter half, or both (ablation study in Section 4.3 is insufficient; see the 13rd comment).
9. The authors write "we leverage this property to robustify our estimate" in the part "Feature permutation ensemble". Robustness against what?
10. Did you compute $\sigma\_n^2$ and $\sigma\_a^2$ (not $\hat{\sigma}\_n^2$ and $\hat{\sigma}\_a^2$) for the synthetic data used in Section 4.1? Please tell me values. Do they satisfy $\sigma\_n^2<\sigma\_a^2$?
11. In experiments in Sections 4.1 and 4.2, how to select the regularization parameter $\lambda$ is not descripted.
What are candidates for $\lambda$ (the description "in the range [1, 10]" is unclear)? Although I saw the program code, I could not find any indication that the authors selected $\lambda$ via a proper validation process. If the authors reported results for $\lambda$, which had the best test performance, then the experimental procedure is problematic and Table 1 is inacceptable. Also, if so, increasing candidates for $\lambda$ easily decrease the reported test performance. Furthermore, the range [1, 10] seems to be narrow generally; is that enough exploration?
12. The choice of hyper-parameters for the baseline methods is poor. For example, the authors fixed $k$ as 5 for $k$-NN. This is not a fair comparison.
13. Why did the authors use only 25 datasets in experiments in Section 4.3? Why does Figure 5 write results for 24 datasets? The authors used 52 datasets in Section 4.2. Absence of results for a part of datasets in Sections 4.3 and 4.4 could raise unnecessary doubts and degrade the reliability of the entire experiment. Since this would be undesirable for the authors, it would be better to take measures such as modifying or putting the rest of the results in the Appendix.
14. Most of existing studies use F1-score, not AUC. This may be due to the incompatibility of AUC and imbalance data (which often appear in the context of anomaly detection). Although there exist existing studies using AUC as well, but why do the authors dare to use AUC and not report F1 score?
15. $y$-axis label $\sigma\_a^2/\sigma\_n^2$ of Figure 2 is misleading (it is an estimate); ”extreme” points in the second paragraph of Section 1 is a latex error of ``extreme'' points; right side of Figure 4 (b) is out of text-length; "proposed in (Jaffe et al., 2015)," in p. 4 is error of citation manner (use \citet{}); NTL 2022 in Table 1 is an error of NTL 2021. There may be other minor writing errors. Please check again.