When Does Unsupervised Learning Succeed or Fail? A PoS Perspective on Reconstruction-Based Anomaly Detection

Mehmet Yamaç; Yagmur Mustu; Muhammad Numan Yousaf; Lei Xu; Marcel van Gerven

cs.LGSep 23, 2026

When Does Unsupervised Learning Succeed or Fail? A PoS Perspective on Reconstruction-Based Anomaly Detection

Authors: Mehmet Yamaç, Yagmur Mustu, Muhammad Numan Yousaf, Lei Xu, Marcel van Gerven

Organizations: Tampere University, Tampere, Finland · Radboud University, Nijmegen, The Netherlands

Abstract

Reconstruction-based unsupervised learning can fail in two opposing ways: a model may reconstruct anomalies too accurately or discard valid nominal variation. Using the Pursuit of Subspaces hypothesis, we characterize these failures through the meet, union, and join geometries induced by the nominal components. Excess learned range produces join blindness, while insufficient capacity produces meet preference and loss of nominal fidelity. We show that the compact nominal union is optimal among nominal faithful ranges and generally requires a nonlinear reconstruction map. Based on this geometry, we introduce Dynamic Push and Pull, which learns from controlled perturbations without anomaly labels, and nested manifold carving, which applies the same principle recursively in latent space. Experiments confirm the predicted changes in latent geometry across every tested Push and Pull configuration. The proposed methods improve reconstruction-based anomaly detection across standard benchmarks and unseen image degradations, while also improving pretrained ECG representations for downstream classification. These results connect reconstruction failures to identifiable geometric conditions and provide practical mechanisms for learning compact representations.

Figures & tables

Panel of Figure 1 — Figure 1: Toy geometry of successful and failed anomaly detection. The shaded plane denotes the join $\mathcal{V}$ of the two nominal subspaces; dark blue and ochre denote nominal and anomalous inputs, light blue and yellow their reconstructions, and dashed segments reconstruction errors. (a) A linear network with range $\mathcal{V}$ detects anomalies outside the join through their nonzero reconstruction errors. (b) The same range is blind to anomalies in $\mathcal{V}\setminus\mathcal{U}$ , which can be reconstructed as accurately as nominal data. (c) A nonlinear network learns the compact union $\mathcal{U}$ , preserving the nominal components while producing nonzero errors for anomalies both inside and outside $\mathcal{V}$ . (d) Rank deficient learning produces a low dimensional compromise range $T_{\mathrm{under}}\subset\mathcal{V}$ that poorly represents both nominal and anomalous data, inflating both scores and reducing their fidelity to nominal union membership.

Figure 2: Ideal and realizable regimes of Lemma 1 . (a) If

s_{n}\in\mathcal{U}

and

e

is normal to

\mathcal{U}

, exact Pull gives

\widehat{s}_{n}=s_{n}

. For

\alpha\leq 1

, Push is inactive at this optimum but supplies a gradient whenever its margin is violated. (b) If

\operatorname{dist}_{d}(s_{n},\mathcal{U})=\eta(s_{n})>0

, margins

\alpha>1

become admissible and impose a constraint unavailable to Pull alone. The triangle inequality gives

\alpha\delta_{e}\leq\delta_{e}+\eta(s_{n})

, recovering the zero-push-error case of Eq. ( 6 ).

Figure 3: Geometric interpretation of the two-stage MNIST experiment. At Stage 0, the nominal digits

4

5

, and

7

define a compact union

\mathcal{U}

within a coarser learned range

\mathcal{V}

. An anomaly such as

8

, lying outside

\mathcal{V}

, produces a large Stage-0 reconstruction error. By contrast, digit

1

can lie inside

\mathcal{V}\setminus\mathcal{U}

and therefore remain undetected at Stage 0. Stage 1 operates in the frozen Stage-0 latent space and further carves the retained range, projecting such off-union representations toward a nominal component. Anomalous information already removed by Stage 0 is unavailable to Stage 1.

Configuration	F1	ROC-AUC	PR-AUC
Bouman-Heskes baseline	0.693	0.866	0.936
Higher-capacity AE	0.780	0.893	0.962
AE + mask	0.766	0.881	0.957
AE + PP	0.792	0.897	0.963
AE + nested + PP	0.798	0.909	0.967
AE + nested + PP + sparsity	0.808	0.928	0.972

Table 1: Controlled MNIST study with nominal digits

4,5,7

. The baseline reproduces Bouman and Heskes (2025) with a two-dimensional bottleneck. The remaining models use the same backbone with Stage-0 latent dimension

9

. Nested configurations report the complete Stage-1 reconstruction score. PP denotes push–pull. Complete results are provided in Tables 10 and 11 .

Predicted change	AE	AE + PP
Increased nominal relative separation, $\Delta S>0$	$7/16$	$\mathbf{16/16}$
All four anomaly proximity indicators decrease	$14/16$	$\mathbf{16/16}$

Table 2: Observed PoS representation dynamics. Each objective includes 16 configurations obtained from four normal digit sets, two masking choices, and two sparsity choices.

Baselines

Ours

Dataset

OC-SVM

DSEBM-e

AnoGAN

DAGMM

ALAD

RCGAN

DSVDD

GOAD

MEMAE

MEMGAN

TOLL

AE-PP

AE-PoS

Arrhythmia

53.03

45.18

46.01

42.42

49.83

51.52

54.14

34.79

52.00

51.13

55.72

60.00

67.69{\pm}5.75

\mathbf{70.77{\pm}5.75}

Table 3: One-class anomaly detection on Arrhythmia ( Guvenir et al., 1997 ) , measured by F1 score (%). AE-PP and AE-PoS results are reported as mean

\pm

standard deviation over 10 seeds. Bold indicates the highest mean.

	Baselines											Ours
	OCGAN	VAE	DSVDD	DSEBM	DAGMM	IF	AnoGAN	ALAD	MEMAE	MEMGAN	TOLL	AE-PP	AE-PoS
Average	65.6	58.3	64.8	60.4	54.4	55.5	61.8	59.3	58.5	65.3	69.6	71.30	71.73

Table 4: One-class anomaly detection on CIFAR-10 ( Krizhevsky, 2009 ) , measured by AUC-ROC (%). Values are averaged over the ten nominal-class tasks. Full class-wise results, including uncertainty over 10 seeds for AE-PP and AE-PoS, are reported in Table 23 .

Figure 4: Effect of the margin scale

\alpha

on Stage-0 AE-PP. Arrhythmia uses F1; CIFAR-10 and CIFAR-100 use AUC-ROC.

	AE (0.83M)		VAE (83.7M)		AE-PP (0.83M)		AE-PoS (1.67M)
Family	ROC	PR	ROC	PR	ROC	PR	ROC	PR
Additive noise (1)	1.000	1.000	0.998	0.998	1.000	1.000	1.000	1.000
Blur (7)	0.568	0.536	0.133	0.334	0.889	0.881	0.903	0.896
Rain (5)	0.440	0.462	0.791	0.687	0.946	0.931	0.958	0.938
Underwater (3)	0.295	0.378	0.449	0.439	0.967	0.962	0.985	0.981
Overall (16)	0.504	0.512	0.452	0.506	0.929	0.919	0.941	0.932

Table 5: Detection of unseen degradations in high-quality images. Entries are unweighted averages within each degradation family, reported as AUC-ROC and PR-AUC. Parentheses give the number of evaluation settings; the overall row averages all 16 settings. Complete results are provided in Table 26 . Bold indicates the best result in each metric.

	AE	VAE	AE-PP	AE-PoS
Clean	63.04	27.59	43.42	43.07
Blur 1.0	62.72	37.01	30.51	31.79
$\Delta$	$-0.32$	$\mathbf{+9.42}$	$-12.91$	$-11.28$

Table 6: Reconstruction PSNR (dB) under the strongest REDS blur. Higher PSNR indicates lower reconstruction error. VAE is the only model whose reconstruction quality improves substantially under blur, reversing the expected anomaly-score ordering.

	PTB-XL				ICBEB
Encoder	Super	Sub	Form	Rhythm	CPSC2018
ECG-CPC	0.8630	0.8860	0.8730	0.9380	0.9020
ECG-FM	0.8730	0.8660	0.7820	0.8650	0.9060
MERL	0.8870	0.8950	0.8640	0.8770	0.9140
ST-MEM	0.8954	0.8935	0.8548	0.9463	0.9559
Stage 0	0.8969	0.8917	0.8523	0.9449	0.9530

Table 7: Frozen-encoder AUROC. Stage 0 and Stage 1 denote our depth-controlled models. Bold indicates the best result in each column.

Appendix figures & tables24 assets

Supplementary material from the paper’s appendix.

Appendix

Result or construction	P1	P2	P3	P4
Score ordering, union optimality, and linear obstruction		$\circ$
Join blindness and meet preference as PoS failures	$\checkmark$	$\circ$
Dynamic Push and Pull construction	$\checkmark$	$\checkmark$	$\checkmark$
Push and Pull geometric guarantees		$\checkmark$
Nested Manifold Carving and the nested-range relation	$\checkmark$	$\checkmark$	$\checkmark$	$\checkmark$
Increasing relative separation of nominal components	$\checkmark$			$\checkmark$

Appendix

Table 8: Dependence of the main results and constructions on the PoS postulates. A checkmark denotes direct use, while an open circle denotes that the postulate is required only to transfer a geometric statement to its neural-network interpretation. Blank entries rely only on the stated hypotheses and standard geometry.

Learned geometry	$d_{\mathrm{nom,train}}$	$d_{\mathrm{nom,test}}$	$d_{\mathrm{anom}}$	$\frac{d_{\mathrm{anom}}}{d_{\mathrm{nom,test}}}$	AUROC
Join (linear AE, $2\mathrm{D}$ )	$0.134\pm 0.11$	$0.143\pm 0.10$	$0.136\pm 0.12$	$0.95$	$0.456$
Compact union (ReLU AE, $2\mathrm{D}$ )	$0.226\pm 0.12$	$0.244\pm 0.11$	$1.624\pm 0.56$	$6.64$	$1.000$
Rank deficient underlearning (linear AE, $1\mathrm{D}$ )	$1.364\pm 0.51$	$1.386\pm 0.44$	$1.435\pm 0.90$	$1.04$	$0.422$

Appendix

Table 9: Toy verification of the learned geometries and their score consequences. The linear two dimensional model learns the join, the ReLU model approximates the compact union, and the one dimensional linear model exhibits rank deficient underlearning. The meet is trivial in this toy, so the final case illustrates nominal fidelity loss rather than exact meet collapse. Distances report the mean and standard deviation of reconstruction scores.

Normal digits	Aug.	Objective	Sparse	Stage 0		Stage 1
				ROC-AUC	F1	ROC-AUC	F1
0,2,4,5,6,8,9	mask	AE	–	$0.700\pm 0.015$	$0.401\pm 0.038$	$0.702\pm 0.017$	$0.402\pm 0.040$
0,2,4,5,6,8,9	mask	AE	yes	$0.685\pm 0.008$	$0.373\pm 0.045$	$0.688\pm 0.012$	$0.382\pm 0.038$
0,2,4,5,6,8,9	mask	AE+PP	–	$0.711\pm 0.011$	$0.427\pm 0.029$	$\mathbf{0.822\pm 0.016}$	$0.493\pm 0.028$
0,2,4,5,6,8,9	mask	AE+PP	yes	$0.708\pm 0.020$	$0.400\pm 0.030$	$0.782\pm 0.024$	$0.472\pm 0.048$
0,2,4,5,6,8,9	none	AE	–	$0.704\pm 0.016$	$0.410\pm 0.032$	$0.710\pm 0.017$	$0.411\pm 0.035$

Appendix

Table 10: Per-stage MNIST anomaly-detection performance. At each stage

\ell

, the anomaly score is the input-space reconstruction error

d(s,\widehat{s}^{(\ell)})

, where

\widehat{s}^{(\ell)}

is the complete reconstruction through that stage. No latent-space or multistage score fusion is used. Each entry reports the mean

\pm

sample standard deviation over seeds

42

–

46

. AE denotes the nominal reconstruction objective

\mathcal{L}_{\mathrm{AE}}

, while AE+PP additionally includes

\lambda_{\mathrm{PP}}\mathcal{L}_{\mathrm{PP}}

. The best result within each nominal-digit set and stage is shown in bold .

Normal digits	Hierarchy	Push	Sparsity	F1	ROC-AUC	PR-AUC
	level	pull
Baseline (4,5,7)				0.693	0.866	0.936
4,5,7				0.780	0.893	0.962
4,5,7	✓			0.780	0.893	0.962
4,5,7			✓	0.772	0.890	0.960
4,5,7	✓		✓	0.772	0.889	0.960

Appendix

Table 11: Full factorial ablation on MNIST with nominal digits

4,5,7

. The first row reproduces the two-dimensional autoencoder baseline of Bouman and Heskes (2025) . Rows without a hierarchy tick use the Stage-0 reconstruction score; ticked rows use the complete Stage-1 reconstruction decoded to the input space. Values are means over different seeds. Best results are shown in bold and second-best results are underlined .

Objective	Stage	F1	ROC-AUC	PR-AUC
AE	0	0.780	0.893	0.962
AE	1	0.780	0.893	0.962
DAE	0	0.783	0.895	0.962
DAE	1	0.783	0.895	0.962
AE with Push and Pull	0	0.792	0.897	0.963
AE with Push and Pull	1	0.798	0.909	0.967

Appendix

Table 12: Matched comparison of reconstruction, denoising, and Push and Pull on MNIST with nominal digits

4,5,7

. Masking and sparsity are disabled to isolate the training objective. Stage 1 uses the complete nested reconstruction decoded to input space, without score fusion.

		Channel fires		Disjoint outcomes
Digit	Role	Recon.	Subsp.	Recon. only	Subsp. only	Neither	Subsp.-only share of det.
0	anomaly	$0.952{\pm}0.044$	$0.325{\pm}0.113$	$0.632{\pm}0.127$	$0.006{\pm}0.008$	$0.042{\pm}0.038$	$0.6\%$
1	anomaly	$\mathbf{0.123{\pm}0.094}$	$\mathbf{0.576{\pm}0.137}$	$0.028{\pm}0.015$	$\mathbf{0.481{\pm}0.101}$	$0.396{\pm}0.126$	$\mathbf{80.8\%}$
2	anomaly	$0.965{\pm}0.008$	$0.444{\pm}0.098$	$0.531{\pm}0.099$	$0.010{\pm}0.005$	$0.024{\pm}0.005$	$1.1\%$
3	anomaly	$0.827{\pm}0.033$	$0.129{\pm}0.077$	$0.705{\pm}0.093$	$0.008{\pm}0.004$	$0.165{\pm}0.030$	$0.9\%$
6	anomaly	$0.581{\pm}0.062$	$0.388{\pm}0.080$	$0.249{\pm}0.081$	$0.056{\pm}0.031$	$0.364{\pm}0.050$	$8.8\%$

Appendix

Table 13: Per-image flag decomposition of the two-channel or-rule. Entries are fractions of test images, with thresholds calibrated on held-out nominal images to a joint

5\%

false-positive rate (

q\approx 97

). Results are mean

\pm

standard deviation over 12 runs. Nominal digits are the negative class; hence, their detection share is undefined.

Figure 5: Representative reconstructions at successive stages. Anomalous directions removed at Stage 0 cannot be recovered from the Stage 0 latent representation. Digit $1$ , which remains inside the coarse Stage 0 range, is modified by nested carving and frequently becomes $7$ -like, consistent with projection toward a nearby nominal component.

normal set	mask	loss	sparsity	$\Delta S\ \uparrow$	$\Delta\bar{D}_{\text{inter}}$	$\Delta\bar{S}_{\text{intra}}$
				(%)	(%)	(%)
$\{4,5,7\}$	none	MSE	–	+0.1	+0.0	-0.0
	none	MSE	yes	-0.2	-0.1	+0.1
	mask	MSE	–	-0.1	+0.0	+0.1
	mask	MSE	yes	-0.8	-0.9	-0.1
	none	PP	–	+7.6	-3.1	-10.0

Appendix

Table 14: Do the normal classes separate from one another? Every column is a relative change from

x

\hat{x}

\bar{D}_{\text{inter}}

is how far apart the classes sit;

\bar{S}_{\text{intra}}

is how wide the classes are;

S=\bar{D}_{\text{inter}}/\bar{S}_{\text{intra}}

combines the two into a single separation score, so it can rise either because the classes move apart or because they contract in place. Under PushPull the rise in

S

is driven by

\bar{S}_{\text{intra}}

falling by roughly

8

–

10\%

while

\bar{D}_{\text{inter}}

also falls, only more slowly. MSE leaves all three unchanged. The arrow marks the direction of improvement, the best value is bold and the second best is underlined .

normal set	mask	loss	sparsity	$\Delta d_{\text{samp}}\ \downarrow$	$\Delta d_{\text{pool}}\ \downarrow$	$\Delta\mathrm{IC}\ \downarrow$	$\Delta\rho\ \downarrow$
				(%)	(%)	(%)	(%)
$\{4,5,7\}$	none	MSE	–	-0.4	-0.1	-0.5	-0.1
	none	MSE	yes	-0.5	-0.3	-0.4	-0.6
	mask	MSE	–	-1.0	-0.8	-1.0	-0.6
	mask	MSE	yes	-1.1	-1.7	-0.2	-1.0
	none	PP	–	-12.4	-15.3	-9.6	-2.2

Appendix

Table 15: Do the anomalies move into the normal clusters? All columns are relative changes from

x

\hat{x}

, and all are built from the same quantity

d(a)

, the distance from an anomalous point to its nearest normal centroid.

\Delta d_{\text{samp}}

is the median of

d(a)

over all anomalous points, while

\Delta d_{\text{pool}}

takes the median within each anomaly class and then the median across classes.

\Delta\mathrm{IC}

reports whether the anomalies approached faster than the space itself contracted.

\Delta\rho

divides instead by the radius of each point’s own nearest cluster. The arrows mark the direction of improvement, and the best value in each column is set in bold and the second best is underlined .

Dataset	Training pool	Validation	Test	Input shape	Problems
MNIST	50 000	10 000	10 000	$1{\times}28{\times}28$	10
Fashion-MNIST	50 000	10 000	10 000	$1{\times}28{\times}28$	10
CIFAR-10	40 000	10 000	10 000	$3{\times}32{\times}32$	10
CIFAR-100	40 000	10 000	10 000	$3{\times}32{\times}32$	20
Arrhythmia	452 records (fixed stratified split)			274 features	1

Appendix

Table 16: One-class benchmark statistics. For the image datasets, the training count denotes the full pool before restriction to the nominal class. All our variants and seeds use the same fixed splits.

Dataset	AE	AE-PP	AE-PoS
MNIST	94.74 $\pm$ 4.65	99.38 $\pm$ 0.42	99.35 $\pm$ 0.51
Fashion-MNIST	91.60 $\pm$ 5.34	94.83 $\pm$ 3.95	94.87 $\pm$ 4.04
CIFAR-10	68.96 $\pm$ 7.29	71.30 $\pm$ 6.70	71.73 $\pm$ 6.60
CIFAR-100	65.42 $\pm$ 9.09	68.56 $\pm$ 8.89	68.87 $\pm$ 8.97
Arrhythmia (F1)	60.77 $\pm$ 7.26	67.69 $\pm$ 5.75	70.77 $\pm$ 5.75

Appendix

Table 17: Plain AE, AE-PP and AE-PoS on the one-class benchmarks.

				Push-pull
Dataset	Latent	LR	Batch	Scale	Weight
Arrhythmia	16	$10^{-3}$	100	3.00	1.00
MNIST	128	$10^{-4}$	1000	3.00	1.00
Fashion-MNIST	256	$10^{-4}$	500	3.00	1.00
CIFAR-10	64	$10^{-2}$	50	2.00	1.00
CIFAR-100	64	$10^{-3}$	100	2.00	1.00

Appendix

Table 18: Stage-0 AE-PP settings. Latent width, learning rate, and batch size follow Ghafourian et al. (2025) ; scale and weight specify Dynamic Push and Pull. All runs use Adam without weight decay for 2,000 iterations and validate every 20 iterations.

				Push-pull
Dataset	Encoder path	LR	Batch	Space	Scale	Weight
Arrhythmia	$16\!\to\!8\!\to\!2$	$10^{-3}$	100	latent	1.00	1.00
MNIST	$128\!\to\!256\!\to\!256$	$10^{-4}$	1000	pixel	2.00	1.00
Fashion-MNIST	$256\!\to\!384\!\to\!384$	$10^{-4}$	500	pixel	2.00	4.00
CIFAR-10	$64\!\to\!32\!\to\!32$	$10^{-2}$	50	latent	2.00	1.00
CIFAR-100	$64\!\to\!96\!\to\!96$	$10^{-3}$	100	latent	2.00	1.00

Appendix

Table 19: Stage-1 AE-PoS settings. Stage 1 is trained within the frozen Stage 0 model for each nominal class and seed. The decoder mirrors the listed MLP encoder path.

Dataset	Stage 0	Stage 1
MNIST	CutPaste with three rotated patches	Same CutPaste configuration
Fashion-MNIST	OneOf : CutPaste (area $0.15$ – $0.4$ , two patches), patch shuffle (grid 2), $90^{\circ}$ rotation, and phase scramble ( $0.3$ )	Same OneOf configuration
CIFAR-10	OneOf : phase scramble ( $0.3$ ), patch shuffle (grid 2), CutPaste, and channel shuffle	Gaussian noise on the Stage-0 latent, with relative $\sigma=0.03$ and $p=0.8$
CIFAR-100	OneOf : phase scramble ( $0.3$ ), patch shuffle (grid 2), CutPaste, and channel shuffle	Gaussian noise on the Stage-0 latent, with relative $\sigma=0.03$ and $p=0.8$
Arrhythmia	Feature shuffle with fraction $0.2$	Same feature-shuffle configuration

Appendix

Table 20: Negative views used by Dynamic Push and Pull. OneOf samples one transformation per input with weights

3{:}1{:}1{:}1

in the listed order.

Normal

OC-SVM

OCGAN

VAE

DCAE

AnoGAN

KDE

DSVDD

MEMAE

MEMGAN

Toll

AE-PP

AE-PoS

98.6

\pm

0.0

99.8

99.7

97.6

\pm

0.7

98.0

\pm

0.3

96.6

\pm

1.3

97.1

\pm

0.0

98.0

\pm

0.7

99.3

\pm

0.1

99.3

\pm

0.1

99.8

\pm

0.1

99.91

\pm

0.02

99.90

\pm

0.02

99.5

\pm

0.0

99.9

98.3

\pm

0.6

97.3

\pm

0.4

99.2

\pm

0.6

98.9

\pm

0.0

99.7

\pm

0.1

99.8

\pm

0.0

99.9

\pm

0.0

99.9

\pm

0.0

99.94

\pm

0.01

99.95

\pm

0.01

82.5

\pm

0.1

94.2

93.6

85.4

\pm

2.4

88.6

\pm

0.5

85.0

\pm

2.9

79.0

\pm

0.0

91.7

\pm

0.8

90.6

\pm

0.8

94.5

\pm

0.1

96.2

\pm

1.6

99.24

\pm

0.05

99.28

\pm

0.05

88.1

\pm

0.0

96.3

95.9

86.7

\pm

0.9

89.9

\pm

0.4

88.7

\pm

2.1

86.2

\pm

0.0

91.9

\pm

1.5

94.7

\pm

0.6

95.7

\pm

0.4

97.9

\pm

0.6

99.14

\pm

0.11

99.15

\pm

0.12

94.9

\pm

0.0

97.5

97.3

86.5

\pm

2.0

92.7

\pm

0.6

89.4

\pm

1.3

87.9

\pm

0.0

94.9

\pm

0.8

94.5

\pm

0.4

96.1

\pm

0.4

97.8

\pm

0.4

99.06

\pm

0.03

99.03

\pm

0.05

77.1

\pm

0.0

98.0

96.4

78.2

\pm

2.7

85.5

\pm

0.8

88.3

\pm

2.9

73.8

\pm

0.0

88.5

\pm

0.9

95.1

\pm

0.1

93.6

\pm

0.3

98.1

\pm

0.6

99.33

\pm

0.03

99.34

\pm

0.04

Appendix

Table 21: Mean and standard deviation of AUC-ROC in % on MNIST ( Deng, 2012 ) classes over 10 seeds. The highest value in each row is in bold.

	Baselines							Ours
Nominal	DAGMM	DSEBM	ADGAN	GANomaly	GeoTrans	ARNet	TOLL	AE-PP	AE-PoS
T-shirt/top	42.1	91.6	89.9	80.3	99.4	92.7	$91.9{\pm}0.2$	$95.12{\pm}0.09$	$94.92{\pm}0.10$
Trouser	55.1	71.8	81.9	83.0	97.6	99.3	$98.7{\pm}0.0$	$98.91{\pm}0.06$	$99.24{\pm}0.08$
Pullover	50.4	88.3	87.6	75.9	91.1	89.1	$91.3{\pm}0.2$	$\mathbf{92.47{\pm}0.20}$	$92.14{\pm}0.12$
Dress	57.0	87.3	91.2	87.2	89.9	93.6	$92.4{\pm}0.2$	$95.24{\pm}0.19$	$\mathbf{95.40{\pm}0.11}$
Coat	26.9	85.2	86.5	71.4	92.1	90.8	$91.5{\pm}0.1$	$92.92{\pm}0.22$	$\mathbf{93.25{\pm}0.27}$

Appendix

Table 22: One-class anomaly detection on Fashion-MNIST ( Xiao et al., 2017 ) , measured by AUC-ROC (%). Values with uncertainty are reported as mean

\pm

standard deviation over 10 seeds. Bold indicates the highest mean in each row.

Baselines

Ours

Nominal

OCGAN

VAE

DSVDD

DSEBM

DAGMM

AnoGAN

ALAD

MEMAE

MEMGAN

TOLL

AE-PP

AE-PoS

Airplane

75.7

70.0

61.7{\pm}4.1

41.4{\pm}2.3

56.0{\pm}6.9

60.1{\pm}0.7

67.1{\pm}2.5

64.7{\pm}2.6

66.5{\pm}0.9

73.0{\pm}0.8

75.0{\pm}4.3

75.53{\pm}2.18

\mathbf{75.99{\pm}1.94}

Automobile

53.1

38.6

65.9{\pm}2.1

57.1{\pm}2.0

48.3{\pm}1.8

50.8{\pm}0.6

54.7{\pm}3.4

38.7{\pm}0.8

46.4{\pm}0.1

52.5{\pm}0.7

67.6{\pm}3.2

68.65{\pm}1.46

\mathbf{69.10{\pm}1.32}

Bird

64.0

67.9

50.8{\pm}0.8

61.9{\pm}0.1

53.8{\pm}4.0

49.2{\pm}0.4

52.9{\pm}3.0

67.0{\pm}0.7

66.0{\pm}0.1

67.2{\pm}0.1

56.2{\pm}2.8

59.47{\pm}1.95

59.99{\pm}1.94

Cat

62.0

53.5

59.1{\pm}1.4

50.1{\pm}0.4

51.2{\pm}0.8

55.1{\pm}0.4

54.5{\pm}1.9

59.2{\pm}0.3

52.9{\pm}0.1

57.3{\pm}0.2

\mathbf{64.6{\pm}1.6}

63.41{\pm}0.47

63.91{\pm}0.58

Deer

72.3

74.8

60.9{\pm}1.1

73.3{\pm}0.2

52.2{\pm}7.3

49.8{\pm}0.4

65.1{\pm}3.2

72.7{\pm}0.6

72.8{\pm}0.1

73.9{\pm}0.9

67.7{\pm}1.4

69.47{\pm}0.81

70.01{\pm}0.99

Appendix

Table 23: One-class anomaly detection on CIFAR-10 ( Krizhevsky, 2009 ) , measured by AUC-ROC (%). AE-PP and AE-PoS results are reported as mean

\pm

standard deviation over 10 seeds. Bold indicates the highest mean in each row.

	Baselines							Ours
Nominal	DAGMM	DSEBM	ADGAN	GANomaly	GeoTrans	ARNet	TOLL	AE-PP	AE-PoS
Aquatic mammals	43.4	64.0	63.1	57.9	74.7	77.5	$70.3{\pm}3.4$	$69.30{\pm}1.55$	$69.62{\pm}1.07$
Fish	49.5	47.9	54.9	51.9	68.5	70.0	$66.2{\pm}1.1$	$69.99{\pm}1.49$	$\mathbf{70.38{\pm}1.41}$
Flowers	66.1	53.7	41.3	36.0	74.0	62.4	$80.3{\pm}0.7$	$80.13{\pm}1.06$	$\mathbf{80.85{\pm}1.06}$
Food containers	52.6	48.4	50.0	46.5	81.0	76.2	$58.2{\pm}1.5$	$66.80{\pm}3.91$	$67.27{\pm}3.95$
Fruit and vegetables	56.9	59.7	40.6	46.6	78.4	77.7	$80.1{\pm}1.3$	$79.78{\pm}0.72$	$\mathbf{80.36{\pm}0.62}$

Appendix

Table 24: One-class anomaly detection on the CIFAR-100 coarse superclasses ( Krizhevsky, 2009 ) , measured by AUC-ROC (%). Values with uncertainty are reported as mean

\pm

standard deviation over 10 seeds. Bold indicates the highest mean in each row.

	Baseline	AE-PP ( $w=1$ )
Dataset	AE ( $\beta=0$ )	$\alpha=0$	$\alpha=1$	$\alpha=2$	$\alpha=3$	$\alpha=4$	$\alpha=5$	$\alpha=6$
Arrhythmia (F1)	$60.77{\pm}7.26$	$63.08{\pm}3.24$	$63.08{\pm}3.24$	$65.38{\pm}4.05$	$\mathbf{67.69{\pm}6.07}^{\dagger}$	$\mathbf{67.69{\pm}6.07}$	$\mathbf{67.69{\pm}6.07}$	$\mathbf{67.69{\pm}6.07}$
MNIST	$94.74{\pm}4.65$	$98.54{\pm}1.38$	$99.04{\pm}0.77$	$99.37{\pm}0.44$	$\mathbf{99.38{\pm}0.42}^{\dagger}$	$99.32{\pm}0.45$	$99.28{\pm}0.48$	$99.26{\pm}0.50$
Fashion-MNIST	$91.60{\pm}5.34$	$93.75{\pm}4.60$	$94.12{\pm}4.43$	$94.74{\pm}4.02$	$\mathbf{94.83{\pm}3.97}^{\dagger}$	$94.74{\pm}3.99$	$94.67{\pm}3.94$	$94.69{\pm}3.95$
CIFAR-10	$68.96{\pm}7.29$	$69.32{\pm}7.36$	$69.48{\pm}7.13$	$\mathbf{71.30{\pm}6.73}^{\dagger}$	$71.16{\pm}6.81$	$71.05{\pm}6.78$	$71.23{\pm}6.97$	$71.29{\pm}7.06$
CIFAR-100	$65.42{\pm}9.09$	$66.78{\pm}9.26$	$67.18{\pm}9.17$	$68.56{\pm}8.91^{\dagger}$	$68.45{\pm}8.98$	$68.59{\pm}8.69$	$\mathbf{68.69{\pm}8.53}$	$68.57{\pm}8.72$

Appendix

Table 25: Effect of the Push–Pull margin scale

\alpha

. AE is the reconstruction-only baseline obtained using the common TOLL backbone with its compactness coefficient set to

\beta=0

. All AE-PP configurations use

w=1

;

\alpha=0

retains Pull while disabling Push. Results are reported as mean

\pm

standard deviation. Bold indicates the highest mean in each row, and the dagger marks the value of

\alpha

selected on the validation set.

Dataset	Clean reference	# Images	AE		VAE		AE-PP		AE-PoS
Dataset	Clean reference	# Images	ROC	PR	ROC	PR	ROC	PR	ROC	PR
Additive noise
Kodak24 ( $\sigma=25$ )	Own GT	24	1.0000	1.0000	0.9983	0.9983	1.0000	1.0000	1.0000	1.0000
Blur
REDS blur 0.2	Own GT	200	0.7268	0.6589	0.3552	0.4024	0.6083	0.5725	0.6002	0.5766
REDS blur 0.4	Own GT	200	0.6581	0.5683	0.2168	0.3484	0.7597	0.7239	0.7808	0.7574

Appendix

Table 26: Detection of unseen image degradations. The clean reference defines the nominal class: paired clean images (Own GT) are used when available; otherwise, a fixed set of clean DIV8K crops is used. AUC-ROC and PR-AUC are computed using reconstruction error as the anomaly score. Bold indicates the best result in each row and metric.

Dataset	Clean reference	# Images	AE		VAE		AE-PP		AE-PoS
Dataset	Clean reference	# Images	Clean	Degr.	Clean	Degr.	Clean	Degr.	Clean	Degr.
Additive noise
Kodak24 ( $\sigma=25$ )	Own GT	24	57.15	26.49	26.49	18.86	44.58	23.64	43.65	23.55
Synthetic blur
REDS blur 0.2	Own GT	200	63.04	62.06	27.59	29.11	43.42	42.55	43.07	42.38
REDS blur 0.4	Own GT	200	63.04	62.41	27.59	30.84	43.42	41.00	43.07	40.88

Appendix

Table 27: Average reconstruction PSNR (dB), computed between each input

s

and its reconstruction

\hat{s}

, for clean and degraded images. Higher PSNR indicates lower reconstruction error. The # Images column reports the number of images in each evaluation setting. “Own GT” denotes the paired clean ground truth associated with each degraded image, whereas “DIV8K” denotes the fixed clean DIV8K evaluation subset.

Panel of Figure 6 — Figure 6: Underwater degradation at three severities (columns). Top row: the input $s$ ; below it each network’s reconstruction $\hat{s}$ with its PSNR and the absolute difference $|s-\hat{s}|$ . All difference panels share one intensity gain.

Panel of Figure 7 — Figure 7: REDS under increasing blur. Columns show the same scene at five blur strengths; rows show the input $s$ , each model’s reconstruction $\hat{s}$ with its PSNR, and the absolute residual $\lvert s-\hat{s}\rvert$ . The VAE increasingly represents the blurred component: its reconstruction PSNR rises from $29.2$ to $37.3$ dB as the blur strengthens, causing the residual to decrease. In contrast, AE-PoS excludes the blur from its learned target set, producing increasingly visible residuals and a PSNR decrease from $43.5$ to $31.1$ dB. All residual panels use the same intensity gain. On the corresponding clean image, the reconstruction PSNRs are $62.4$ , $27.6$ , and $43.9$ dB for AE, VAE, and AE-PoS, respectively.

Panel of Figure 8 — Figure 8: Nominal fidelity and degradation rejection. For each input $s$ , the figure shows the reconstruction $\hat{s}$ , its PSNR, and the absolute residual $\lvert s-\hat{s}\rvert$ for AE, VAE, and AE-PoS. From top to bottom, the examples are a nominal Kodak24 image, a nominal GoPro frame, its motion-blurred counterpart, and a rainy Test2800 image. AE reconstructs both nominal and degraded inputs almost identically, providing high fidelity but weak anomaly separation. VAE loses substantial detail on nominal images and represents the motion-blurred input more accurately. AE-PoS preserves nominal image content while excluding motion blur and rain from its learned target set, thereby exposing these degradations in the residual. All residual panels use the same intensity gain.

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Jul 14, 2026, cs.CVJ/K move · Enter open · S save

Statistical Non-linear Reconstruction Loss for Image Anomaly Detection

Nguyen Minh Tri, Hoang Khuong Duy, Huynh Cong Viet Ngu

AIC-Lab, FPT University

Outliers Suppression

Open paper AskPDF arXiv

Reconstruction-based methods are a cornerstone of unsupervised image anomaly detection, but they remain vulnerable to \emph{outlier leakage}, where standard mean squared error (MSE) loss drives the model to faithfully reconstruct anomalous patterns. We propose a Non-linear Reconstruction Loss that applies a sigmoid-based squashing function to suppress high-magnitude features, preventing outliers from dominating optimization while preserving sensitivity to normal patterns. In addition, we introduce a statistical calibration scheme that selects the scaling factor

k

from the confidence interval (CI) of the normal feature distribution, enabling data-driven control of the suppression strength. Our approach achieves competitive or superior anomaly detection performance compared to state-of-the-art methods, reaching 99.0% Image-AUROC and 97.3% Pixel-AUROC on MVTec-AD, and 95.3% Image-AUROC and 99.0% Pixel-AUROC on VisA. These results indicate that non-linear gradient suppression is an effective mechanism for mitigating outlier leakage and improving anomaly localization in unified industrial inspection settings. The implementation is available at https://github.com/mintii13/Statistical-Non-linear-Reconstruction-Loss.git.

Abstract

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