Fairness-aware Anomaly Detection via Fair Projection

Feng Xiao; Jicong Fan; Xiaoying Tang

REVIEW 4 major objections 6 minor 23 references

Fairness-aware Anomaly Detection via Fair Projection

T0 review · 4 major / 6 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read Mapping each demographic group to one shared compact distribution enforces group fairness in unsupervised anomaly detection.

desk verdict A genuinely useful fairness-through-target-distribution idea for unsupervised AD, with a sensible threshold-free metric, but the printed objective and the transfer assumptions need fixing before the fairness guarantees can be taken at face value. read the letter →

arxiv 2505.11132 v1 pith:GKKFMJ2S submitted 2025-05-16 cs.LG

Feng Xiao , Xiaoying Tang , Jicong Fan This is my paper · ORCID

classification cs.LG

keywords fairness-awareanomalydetectionunsupervisedgroupfairnessfairprojectionSinkhorndistancethreshold-freemetricdemographicparitypredictiveequality

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper's thesis is that group fairness in unsupervised anomaly detection can be achieved by construction, not by adding fairness penalties. It proposes learning one projection that maps the normal training data of every demographic group onto the same simple, compact target distribution; because the anomaly score is the norm of the projected point, identical projected distributions across groups imply identical flagging rates at every threshold. The paper proves this implication (Propositions 1 and 2) and states two assumptions under which the guarantee extends from normal training data to unseen anomalies, yielding equal opportunity or demographic parity. A threshold-free fairness metric, ADPD, is introduced so fairness can be evaluated across all thresholds without choosing a cutoff. On six real-world benchmarks, including balanced and skewed group splits, the method reports a better accuracy-fairness trade-off than existing fair anomaly detectors.

What carries the argument

The central object is a fair projection $h_\phi$, an encoder trained to minimize, for every protected group $s$, the Sinkhorn distance between $h_\phi(X_{S=s})$ and a common target distribution $D_z$, together with a reconstruction term that preserves the information needed for detection. The target distribution is a truncated isotropic Gaussian, chosen because its compactness gives a reliable density-based decision boundary and its identity across groups makes the induced score function group-fair. The argument runs through the chain Proposition 2 to Proposition 1: zero total distribution distance implies $h(x)$ is independent of the protected attribute, which implies that any threshold-based decision function on top of $h$ yields equal flagging probabilities across groups.

What would settle it

Train Im-FairAD on two groups using only normal data, then evaluate on a test set where anomalies for one group are deliberately placed much farther from the normal manifold than anomalies for the other group; if the between-group difference in flagging rates on these anomalies is large while the training normal ADPD is near zero, the transferability assumption fails.

Watch

Extended reading notes

Core claim

The central claim is that a single learned projection $h$ can simultaneously serve as the anomaly score and as the fairness mechanism. The paper shows that if the normal data of every demographic group are mapped to the same simple, compact target distribution (a truncated isotropic Gaussian), then the projected features are independent of the protected attribute, so for any threshold the flagging probability is equal across groups. More precisely, Proposition 2 states that when the summed Sinkhorn distance between each group's projected distribution and the common target is zero, equation (9) holds and $h(x)$ is independent of the sensitive attribute; Proposition 1 then guarantees that any score function built on $h$ satisfies predictive equality on the training data. Under the stated transferability assumptions, the same reasoning extends to equal opportunity or demographic parity on unseen anomalies. The paper also introduces ADPD, a threshold-free fairness metric, and reports experiments on six benchmarks showing that the proposed Im-FairAD and Ex-FairAD achieve a better trade-off between detection accuracy and fairness than existing fair anomaly detection baselines.

Load-bearing premise

The method never sees anomalies during training, so the guarantee that fair flagging rates on normal data carry over to anomalies rests on the assumption that the fairness violation on unseen anomalies is at most a small constant times the violation on training normal data.

Editorial extensions

If this is right

If the projection is learned well, the resulting detector satisfies predictive equality on training normal data for any threshold, without any explicit fairness regularization term.
Under the transferable fairness assumption, the same score distribution implies equal opportunity on unseen anomalies, so fairness is not limited to the training distribution.
The threshold-free ADPD metric gives a single number summarizing fairness over all thresholds, resolving the sensitivity of fairness-ratio evaluations.
Skewed group sizes do not by themselves inflate unfairness, because each group is matched to the same target distribution; the reported ADPD stays nearly flat from balanced to skewed splits.
The method is applicable beyond tabular data, with experiments on image data showing the same shared-projection mechanism works.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The common-target mechanism could be transferred to other one-class problems, such as fraud or medical screening, where the cost of a false alarm differs by group, by choosing the target distribution to reflect those costs.
A direct extension would estimate the ratio of fairness violation on unseen anomalies to that on training normal data, turning the paper's transferability assumption into an empirically checkable quantity rather than a postulate.
Because the fairness guarantee is stated at the level of the learned representation, the same projection could be paired with any downstream thresholding rule, inviting a compositional view of fair anomaly detection.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Two things to know. First, the central idea is good: learn a projection that pushes each demographic group's normal data onto the same compact target distribution, then use the norm of the projection as an anomaly score. That is a clean, end-to-end way to enforce group-level score parity without adversarial training, and it is not in the cited prior work. Second, as printed, the main objective in Eq. (11) does not implement that idea: it sums Sinkhorn(h_phi(X), Z) over groups, with the same full sample X each time, so the sum has no group-specific effect. That is almost certainly a typo (Eq. (10) has the correct group-wise term), but it has to be fixed.

What the paper does well: the assumption analysis in Section 3.2 (Claim 1) is a useful statement of why unlabeled anomalies make demographic parity and equal opportunity hard in unsupervised AD. The experiments are unusually broad: balanced and skewed splits, tabular and image data, and a reasonable set of fair-AD baselines. On COMPAS, Adult, and Credit, the AUC/ADPD trade-off genuinely favors Im-FairAD and Ex-FairAD. The ADPD metric is a sensible threshold-free summary of flagging-rate disparity across thresholds, and it improves on the threshold-dependent fairness ratio for comparison purposes.

The soft spots, in order of seriousness. (1) Proposition 2's proof relies on M >= 0; the Sinkhorn objective as defined in Eq. (6) includes an unnormalized entropy term + alpha sum P log P, which can be negative, so the nonnegativity step is false as written. Using a proper Sinkhorn divergence or KL-regularized OT fixes it, but the proof needs to be redone. (2) Assumptions 2 and 3 are vacuous in the form stated: any nonzero violation on training normals gives some finite mu, and "small" is never defined. The rigorous claim should be limited to predictive equality on the training normal data; equal opportunity and demographic parity for unseen anomalies are empirical properties here, not consequences of the theory. The authors should say that plainly. (3) Some empirical claims outrun the tables: on CelebA both proposed methods have lower AUC than Deep SVDD, and on Titanic the locked-in F1 of 66.66 with 1.00 fairness ratio suggests the detector is flagging everything, which makes the fairness comparison trivial there.

Bottom line: this is a solid, fixable paper with a useful idea and mostly honest experiments. I would send it to review. The authors should be asked to correct Eq. (11), repair the Sinkhorn proof, either bound or explicitly de-emphasize Assumptions 2 and 3, and soften the CelebA and Titanic claims.

Referee Report

4 major / 6 minor

Summary. The paper studies group fairness in unsupervised anomaly detection (UAD). It argues that without additional assumptions, demographic parity and equal opportunity cannot be guaranteed in the unsupervised setting, and it introduces two assumptions under which fairness on training normal data transfers to unseen anomalies. The proposed method, FairAD, learns a projection that maps normal data of each demographic group to a common compact target distribution, so that anomaly scores (distances to the origin in the projected space) are comparable across groups; two variants are given: Im-FairAD, which uses a shared target distribution without an explicit fairness term, and Ex-FairAD, which adds a Sinkhorn-based fairness regularizer on the score distributions. The paper also proposes ADPD, a threshold-free fairness metric, and evaluates the methods on six datasets under balanced and skewed group splits, reporting improved accuracy-fairness trade-offs relative to several baselines.

Significance. If the central mechanism is correct, this is a valuable contribution: fairness in unsupervised anomaly detection is understudied, and the idea of achieving group fairness by mapping all groups to a common target distribution is simple, avoids adversarial training, and is potentially task-compatible. The empirical evaluation is reasonably extensive (multiple tabular and image datasets, balanced/skewed splits, threshold-free and threshold-dependent metrics, ablations, contamination experiments), and the proposed ADPD metric is a useful addition to the evaluation toolbox. However, as detailed below, two load-bearing theoretical pieces—the printed Im-FairAD objective and the proof of Proposition 2—are not currently sound as written, and the transferability assumptions are vacuous as formalized. The empirical results may still support the method's practical utility, but the theoretical claims need substantial revision.

major comments (4)

[Section 3.3.2, Eq. (11)] The stated objective for Im-FairAD is min_{φ,ψ} Σ_{s∈S} Sinkhorn(h_φ(X), Z) + (β/n) Σ_i ∥x_i − g_ψ(h_φ(x_i))∥². The first term does not depend on s, so the sum over groups equals |S|·Sinkhorn(h_φ(X), Z), which is exactly the fairness-unaware objective in Eq. (7). As printed, Eq. (11) therefore does not implement the group-wise distribution matching described in Eq. (10) and in the surrounding text, and the claim that Im-FairAD achieves fairness through a shared target distribution is not supported by the printed objective. The authors should correct Eq. (11) to use Sinkhorn(h_φ(X_{S=s}), Z) inside the sum and verify that the reported Im-FairAD results were obtained with that corrected objective.
[Appendix A.3, proof of Proposition 2] The proof of Proposition 2 relies on the assertion that M(·,·) is a distance metric with M(·,·)≥0 and that M(P,Q)=0 implies P=Q. However, the Sinkhorn objective defined in Eq. (6) includes the term α Σ_{i,j} P_{ij} log P_{ij}, which is non-positive and can make the objective negative; moreover, with entropy regularization the optimal value is not a metric and zero value does not imply equality of distributions. Therefore the chain of implications in Eq. (17) is not justified for the Sinkhorn distance as used in Eq. (11). The proposition should either be stated for a genuine distributional distance (e.g., Sinkhorn divergence, Wasserstein distance, or MMD with a characteristic kernel) with a proof that zero distance implies distributional equality, or the entropy regularization should be defined so that the resulting quantity is nonnegative and has the identity-of-indiscernibles property.
[Section 3.2, Assumptions 2 and 3] Assumptions 2 and 3 are vacuous as formalized. For any detector with E>0, the inequality Ẽ ≤ μE holds for μ = Ẽ/E, and the condition that μ is 'small' is never defined; thus the assumption does not constrain the model. If E=0, the assumption requires Ẽ=0, which is an unverified strong condition that the paper does not establish. Consequently, the statement that solving (10) together with Assumption 2 or 3 yields equal opportunity or demographic parity on unseen anomalies is not a formal derivation; the rigorous content reduces to predictive equality on the training normal set. The authors should either (i) provide a concrete quantitative bound on μ and τ derived from the generative model or from measurable quantities, or (ii) explicitly reframe the claim as an empirical regularity rather than a theoretical guarantee. The current supporting arguments—that existing methods 'implicitly verified' the assumptions and that test-set ADPD values are small—do not establish a uniform bound on μ.
[Section 3.4 and Appendix G] The ADPD metric is presented as a threshold-free measure of demographic parity. With finite samples, ADPD=0 holds if and only if the empirical score distributions are equal across groups, which is stronger than demographic parity at any single threshold; the paper should state this explicitly so that the metric is not overclaimed as an exact proxy for demographic parity. This is a conceptual clarification rather than a technical error, but it affects how the experimental comparisons are interpreted.

minor comments (6)

[Section 4.1] There are corrupted placeholder strings in the manuscript text after Figure 2 and Figure 3 (e.g., '/uni00000013/...'), which appear to be artifacts of the source file and should be removed before publication.
[Eq. (6)] The quantity in Eq. (6) is called a 'Sinkhorn distance,' but with the added entropy term it is not a distance in the mathematical sense; consider renaming it to 'entropic optimal transport cost' or 'Sinkhorn loss' throughout the paper.
[Section 3.3.2, Eq. (12)] In the Ex-FairAD objective, the notation Sinkhorn(ζ(h_φ(X_{S=si})), ζ(h_φ(X_{S=sj}))) should be clarified: ζ(h_φ(X)) denotes the set of anomaly scores for samples in that group, and the Sinkhorn cost is computed on these one-dimensional score sets. Using 1-D scores is a special case that could be replaced by a simpler closed-form distance; a remark would help reproducibility.
[Table 14] The column heading 'APDP' in Table 14 should be 'ADPD' for consistency with the rest of the paper.
[Appendix D.2] The ablation study for Ex-FairAD reports results without the fairness term, but the hyperparameter λ is varied only on three datasets; a sensitivity analysis for λ on the remaining datasets (Titanic, SP, Credit) would strengthen the claim that the fairness term is necessary and that performance is not overly sensitive to λ.
[Appendix B.2] The Sinkhorn entropy coefficient α is fixed to 0.1 in all experiments, but no sensitivity analysis is reported; given that Proposition 2's proof depends on the distance-like properties of M, a brief study of α's effect on both fairness and detection would be useful.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the fairness derivation is constructive and the transfer to unseen anomalies is explicitly assumed, not derived.

full rationale

The central derivation is self-contained rather than circular: Proposition 2 (Appendix A.3, Eqs. 17-20) shows that zero Sinkhorn distance from each group's projected normal data to a common target distribution implies equality of the projected distributions across groups, and Proposition 1 (Appendix A.2, Eqs. 14-16) shows that any score function on equal representations preserves predictive equality. This is a constructive sufficient condition, not a redefinition of the conclusion. Detection accuracy is measured against held-out anomalies (AUC and F1 in Tables 5-8), and the ADPD fairness metric is applied uniformly to all methods, including on test anomalies that are never used in training; low ADPD for Im-FairAD is therefore not a fitted value renamed as a prediction. The only formally questionable step is the transfer from training normal data to unseen anomalies: Assumptions 2 and 3 (Section 3.2) assert bounds E-tilde <= mu*E and E-hat <= tau*E with 'small' mu and tau, but no concrete bounds are proved, and if E = 0 the first assumption reduces to E-tilde = 0, which is exactly equal opportunity on anomalies. That makes the advertised equal-opportunity or demographic-parity guarantee conditional on an unverified and potentially vacuous premise; however, the paper explicitly labels these as assumptions rather than theorems, so this is a correctness or evidential gap, not a circular reduction. Self-citations (Cai and Fan 2022, Fu et al. 2024, Xiao et al. 2025) appear only as background for standard anomaly-detection objectives and are not load-bearing for the fairness argument; no uniqueness theorem or ansatz is imported from the authors' prior work. Overall, no equation in the paper is identical to its input by construction, and no fitted parameter is relabeled as a prediction.

Assumptions & free parameters 6 free parameters · 5 assumptions · 0 invented entities

The claimed guarantee (fairness by construction) holds only at zero Sinkhorn loss; in practice the loss is positive and no finite-sample bound is given. The transfer of fairness to anomalies depends on Assumptions 2/3, whose constants are unbounded, and on an assumed map Q from anomalies to normals. Several hyperparameters that materially affect the result (β, λ, latent dimension, truncation range, Sinkhorn iterations) are not reported. No new physical or conceptual entities are introduced.

free parameters (6)

beta (reconstruction weight) = not reported per dataset; ablation over {0.001..1000}
Weights the reconstruction term against the Sinkhorn matching; the final value used for reported results is not given.
lambda (Ex-FairAD fairness weight) = not reported per dataset; ablation over {0.001..1000}
Weights the cross-group score-matching term in Ex-FairAD; main results do not report the chosen value.
alpha (Sinkhorn entropy coefficient) = 0.1
Set to 0.1 in all experiments; controls the entropy regularization of the transport plan.
latent dimension m = not specified
Dimension of the projection/target distribution; not reported per dataset.
target distribution truncation range = not specified
The truncated Gaussian's boundaries are never stated, yet they set the compactness of the decision region.
Sinkhorn iteration count = not specified
Referenced as t in Appendix F; actual value used in experiments is not reported.

assumptions (5)

domain assumption Assumption 1 (Learnable abnormality): an unknown score function T* ranks normal training samples and separates normal from anomalous data
Standard compactness assumption for unsupervised AD; stated in Section 3.2, Assumption 1.
ad hoc to paper Assumption 2 (Transferable fairness): there exists small μ≥1 such that Ẽ ≤ μE
Bridges fairness on training normal data to unseen anomalies; the constant μ is unbounded, making the assumption vacuous as stated. Section 3.2.
ad hoc to paper Assumption 3 (Generalizable parity): there exists small τ≥1 such that Ê ≤ τE
Extends fairness to all unseen normal and anomalous data; same unbounded-constant issue. Section 3.2.
ad hoc to paper Existence of a low-complexity map Q with D̃_x = Q(D_x)
Invoked to explain transfer of fairness to test anomalies via P∘Q. Section 4.1, paragraph after Figure 3.
ad hoc to paper The Sinkhorn objective in Eq. (6) is a distance metric with M≥0
Used in Proposition 2 proof; with +αΣP log P the objective can be negative and is not a metric. Appendix A.3 and Eq. (6).

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Cite this review

Pith. "Pith review of Fairness-aware Anomaly Detection via Fair Projection." pith.science (2026). https://pith.science/paper/GKKFMJ2S

@misc{pith2026250511132,
  author       = {Pith},
  title        = {Pith review of: Fairness-aware Anomaly Detection via Fair Projection},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GKKFMJ2S}},
  note         = {Machine review of arXiv:2505.11132}
}

read the original abstract

Unsupervised anomaly detection is a critical task in many high-social-impact applications such as finance, healthcare, social media, and cybersecurity, where demographics involving age, gender, race, disease, etc, are used frequently. In these scenarios, possible bias from anomaly detection systems can lead to unfair treatment for different groups and even exacerbate social bias. In this work, first, we thoroughly analyze the feasibility and necessary assumptions for ensuring group fairness in unsupervised anomaly detection. Second, we propose a novel fairness-aware anomaly detection method FairAD. From the normal training data, FairAD learns a projection to map data of different demographic groups to a common target distribution that is simple and compact, and hence provides a reliable base to estimate the density of the data. The density can be directly used to identify anomalies while the common target distribution ensures fairness between different groups. Furthermore, we propose a threshold-free fairness metric that provides a global view for model's fairness, eliminating dependence on manual threshold selection. Experiments on real-world benchmarks demonstrate that our method achieves an improved trade-off between detection accuracy and fairness under both balanced and skewed data across different groups.

Figures

Figures reproduced from arXiv: 2505.11132 by the authors.

**Figure 2.** Accuracy-fairness trade-off on balanced splitting. Note that the baselines, FairOD and CFAD are [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

**Figure 3.** Fairness of all baselines on abnormal data [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: Accuracy-fairness trade-off on skewed splitting. [PITH_FULL_IMAGE:figures/full_fig_p018_4.png]

**Figure 5.** Figure 5: Fairness of all baselines on abnormal data from test set. [PITH_FULL_IMAGE:figures/full_fig_p019_5.png]

**Figure 6.** Figure 6: Average AUC and ADPD (all test set) with [PITH_FULL_IMAGE:figures/full_fig_p019_6.png]

**Figure 7.** Figure 7: Average AUC and ADPD (all test set) with [PITH_FULL_IMAGE:figures/full_fig_p022_7.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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The details of each dataset are as follows: Table 2: Statistics of datasets. Dataset Type Dimension Sensitive Variable Normal Set Abnormal Set Adult tabular 14 gender income≤50K income>50K COMPAStabular 8 race no recidivism within 2 yearsrecidivism within 2 years Credit tabular 23 age no default payment next monthdefault payment next month Titanic tabular...

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [2]

The details of each dataset are as follows: Table 2: Statistics of datasets. Dataset Type Dimension Sensitive Variable Normal Set Abnormal Set Adult tabular 14 gender income≤50K income>50K COMPAStabular 8 race no recidivism within 2 yearsrecidivism within 2 years Credit tabular 23 age no default payment next monthdefault payment next month Titanic tabular...

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[2] [4]

Table 3: Balanced data splitting on Adult, COMPAS, CelebA and Credit

Note that there is no balanced splitting for Titanic and SP due to the limitations of extremely uneven samples size across different demographic groups for Titanic and insufficient sample size for SP. Table 3: Balanced data splitting on Adult, COMPAS, CelebA and Credit. PV denotes the protected variable. ‘Nor.’ and ‘Abnor.’ denote normal sample and abnorm...

work page 2000

[3] [5]

A framework for determining the fairness of outlier detection

Ian Davidson and Selvan Suntiha Ravi. A framework for determining the fairness of outlier detection. In ECAI 2020, pages 2465–2472. IOS Press,

work page 2020

[4] [7]

Fairness through awareness

Cynthia Dwork, Moritz Hardt, Toniann Pitassi, Omer Reingold, and Richard Zemel. Fairness through awareness. In Proceedings of the 3rd Innovations in Theoretical Computer Science Conference, Jan 2012a. doi: 10.1145/2090236.2090255. URL http://dx.doi.org/10.1145/2090236. 2090255. Cynthia Dwork, Moritz Hardt, Toniann Pitassi, Omer Reingold, and Richard Zemel...

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How we analyzed the compas recidivism algorithm

Jeff Larson, Surya Mattu, Lauren Kirchner, and Julia Angwin. How we analyzed the compas recidivism algorithm. ProPublica (5 2016), 9(1):3–3,

work page 2016

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Fairod: Fairness-aware outlier detection

Shubhranshu Shekhar, Neil Shah, and Leman Akoglu. Fairod: Fairness-aware outlier detection. In Proceedings of the 2021 AAAI/ACM Conference on AI, Ethics, and Society , Jul

work page 2021

[7] [13]

This indicates that a skewed split poses a more intractable problem for fairness than a balanced split

From the Table 12 and Table 13, we have the following observations: • In the transition from a balanced split to a skewed split, it can be observed that equal opportunity, as measured by ADPD and EO, exhibits significantly larger values in most cases which means a skewed split tends to introduce more unfairness compared to a balanced split, which is consi...

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Deep domain confusion: Maximizing for domain invariance

Eric Tzeng, Judy Hoffman, Ning Zhang, Kate Saenko, and Trevor Darrell. Deep domain confusion: Maximizing for domain invariance. arXiv preprint arXiv:1412.3474,

arXiv