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

1 INTRODUCTION

• A range approaching the join can cause false negatives, while a range approaching the meet can cause false positives on nominal points in U \ W.These distance predictions coincide with reconstruction errors only under P2's nearest-point projection; arbitrary learned ranges need not lie between the meet and join.Source: §1 · p. 2
• Excess learned range produces join blindness: anomalies in V \ U can receive zero reconstruction error, while anomalies outside V remain detectable.A reconstruction objective constrains the model on nominal samples but does not bound its range outside them.Source: §1 · p. 2
• Limited capacity can favor the meet W, discarding component-specific directions and assigning positive errors to nominal points in U \ W.The paper gives sufficient conditions for this preference in rank-constrained MSE-optimal linear autoencoders and extends the mechanism through local tangent approximations.Source: §1 · p. 2
• The compact nominal union U is optimal among nominal-faithful ranges, but any linear reconstruction map preserving U must also contain its join V.The paper therefore identifies nonlinearity as necessary to preserve U without expanding to V.Source: §1 · pp. 2–3
• Dynamic Push and Pull trains on controlled perturbations, pulling reconstructions toward clean nominal samples and pushing them from perturbed inputs without anomaly examples or class labels.Nested manifold carving applies this compaction recursively in latent space while using sufficient capacity to reduce premature contraction toward shared directions.Source: §1 · p. 3
• The paper tests predicted representation dynamics and evaluates anomaly detection, unseen image degradations from nominal-only data, and unlabelled ECG pretraining for downstream classification.The remedies target unsupervised representation learning more generally; anomaly detection is one application of the learned geometry.Source: §1 · p. 3

2 RELATED WORK

• Autoencoders score anomalies by reconstruction error, but unseen anomalies may still be reconstructed with small error, motivating regularization and architectural constraints.Prior methods include latent compactness, sparse, denoising, and masked reconstruction objectives, as well as prototype memories, latent density models, and adversarial or one-class components.Source: §2 · p. 3
• The mechanisms and conditions that make autoencoder-based anomaly detection fail or reliable remain insufficiently understood.Bouman & Heskes (2025) report accurate anomaly reconstruction through interpolation between nominal groups and extrapolation beyond observed nominal data.Source: §2 · pp. 3–4
• The paper interprets broad-range failures as join blindness and identifies meet preference as the opposing loss of component-specific nominal directions.Unlike denoising methods that reconstruct clean samples from perturbations, Dynamic Push and Pull adds an explicit separation margin; Nested Manifold Carving recursively targets off-union latent directions.Source: §2 · p. 4

3.1 DYNAMIC PUSH-PULL LEARNING

• Push–pull training pulls a perturbed nominal sample's reconstruction toward the clean sample and pushes it from the perturbed input by a margin proportional to perturbation severity.The experiments retain an ordinary reconstruction objective on unperturbed nominal samples, which anchors those reconstructions alongside the push–pull term.Source: §3.1 · p. 4
• Under the stated small-loss conditions, Lemma 1 bounds nominal range distance and requires perturbed-sample reconstruction error to meet the push margin, subject to nominal proximity to U.For exact nominal membership and normal perturbations, Push is inactive at the exact Pull optimum when α ≤ 1; off-union distance can permit α > 1.Source: §3.1 · pp. 4–5, Figure 2 · p. 4
• The guarantee is local to sampled perturbations; suppressing join blindness throughout V \ U requires the sampler to probe that region sufficiently.Without sufficient probing, only sampled neighborhoods are certified as carved.Source: §3.1 · p. 5

3.2 NESTED MANIFOLD CARVING

• Nested Manifold Carving freezes earlier encoder–decoder pairs and trains a new pair on their latent representation, then decodes through preceding levels.Each level preserves nominal information while rejecting perturbed latent inputs; sufficient latent width helps retain component-specific directions and avoid premature meet preference.Source: §3.2 · p. 5
• Under the stated compatibility, capacity, and exact-optimization conditions, successive levels remove probed off-union regions without sacrificing nominal coverage.The construction also depends on nonlinearity and perturbation coverage.Source: §3.2 · p. 5

3.3 POS-PREDICTED REPRESENTATION DYNAMICS

• As representations become more PoS-compact, nominal components are predicted to separate relatively more, while off-union anomalies move toward nominal components.Consequently, nominal structure can become more compact even as nominal and anomalous representations become less distinguishable, potentially yielding worse features for a classifier tasked with distinguishing them.Source: §3.3 · p. 5

4.1 A CONTROLLED STUDY OF THE PREDICTED FAILURE MODES

• Stage 0 detects anomalies removed by its projection, while nested carving detects off-union structure retained in the frozen Stage-0 representation.Digit 8 has large image error but small Stage-1 latent error; digit 1 has small image error but large latent error and becomes predominantly 7-like after carving.Source: §4.1 · p. 6, Figure 3 · p. 6
• Push and Pull improves single-stage and nested models, while a reconstruction-only nested module provides no further gain because it admits an identity solution.The controlled ablations attribute improvement to explicitly carving the learned range, rather than additional capacity, depth, or perturbation alone; denoising alone does not explain AE-PP's further Stage-1 improvement.Source: §4.1 · p. 6, Table 1 · p. 7

4.2 TESTING THE PREDICTED REPRESENTATION DYNAMICS

• Across all 16 tested Push and Pull configurations, nominal relative separation increased and all four anomaly-proximity indicators decreased.Ordinary AE increased relative separation in 7/16 configurations and satisfied all four proximity predictions in 14/16.Source: Table 2 · p. 7, §4.2 · p. 7

4.3 STANDARD ONE-CLASS ANOMALY DETECTION

• 70.77% F1 on Arrhythmia versus 60.00% for the strongest baseline, and 71.73% AUC-ROC on CIFAR-10 versus 69.6%.AE-PoS improves AE-PP from 67.69% to 70.77% F1 on Arrhythmia and from 71.30% to 71.73% AUC-ROC on CIFAR-10.Source: Table 3 · p. 7, Table 4 · p. 8, §4.3 · p. 8
• The one-class benchmarks use nominal-only training, while checkpoints are selected on validation data containing both nominal and anomalous samples.AE-PP adds Dynamic Push and Pull to the reconstruction-only AE baseline; AE-PoS additionally applies nested carving to the frozen Stage-0 representation.Source: §4.3 · p. 7

4.4 DETECTING UNSEEN IMAGE DEGRADATIONS

• On high-quality images, the study evaluates detection of unseen noise, blur, rain, and underwater degradations, including realistic motion blur and degradations absent from training.Training perturbations are limited to parametric blur kernels, sensor noise, resampling, and JPEG compression; the supplied passages state no comparative detection values.Source: §4.4 · pp. 8–9, Table 5 · p. 8

4.5 TRANSFERABLE ECG REPRESENTATIONS

• Nested carving improves performance across all five ECG classification tasks using a frozen representation and a linear diagnostic classifier.The ECG recordings are unlabeled during representation learning; only the classifier is trained on labels.Source: §4.5 · p. 9, Table 7 · p. 9

5 CONCLUSION AND LIMITATIONS

• Dynamic Push and Pull and Nested Manifold Carving target join blindness and meet preference by better approximating the compact nominal union prescribed by PoS.The paper reports improvements in anomaly detection and transferable representation learning.Source: §5 · p. 9
• Across all 16 Push and Pull configurations, nominal components disentangle and off-union samples move toward the nominal union.The paper also notes that increased compactness may weaken downstream discrimination between off-union samples and nominal data.Source: §5 · p. 9
• In the controlled MNIST example, digit 8 has large reconstruction error but little latent change, whereas digit 1 is detected mainly by Stage 1 and becomes predominantly 7-like.The paper presents these outcomes as matching the two-stage geometric prediction.Source: §5 · p. 9
• The reported limitations include metric dependence, manual depth selection, non-unique projections of structured anomalies, and dynamics validated only in a controlled multi-component setting.These limitations are discussed in Appendix G.Source: §5 · p. 9

STATEMENTS

• The ECG experiments use de-identified datasets and evaluate representation learning, not clinical diagnostic validity; deployment requires independent validation and appropriate regulatory and ethical review.The work uses existing benchmark datasets and involves no new data collection or human-participant interaction.Source: STATEMENTS · p. 10

A MEET–UNION–JOIN GEOMETRY: DEFINITIONS AND PROOFS

• The appendix applies meet, union, and join relations fiberwise in local tangent coordinates, without assuming globally linear manifolds or networks.When geometric distance is interpreted as reconstruction score, the analysis assumes the ideal nearest-point projection prescribed by P2; generic autoencoder outputs need not be nearest points.Source: A MEET–UNION–JOIN GEOMETRY:… · p. 16

A.1 AXIOMATIC DEPENDENCE OF THE RESULTS

• Table 8 distinguishes postulates used directly in constructions or results from those needed only to transfer geometric statements to neural-network interpretations.The stated results apply to input or latent representation spaces, with level-specific symbols understood at each level.Source: Table 8 · p. 16, §A.1 · p. 16

A.2 MEET, UNION, AND JOIN

• The reference sets satisfy W ⊆ U ⊆ V, but this chain does not impose an ordering on an arbitrary learned range T.U is generally not a subspace, whereas W and V are subspaces.Source: §A.2 · p. 16

A.2.1 LINEAR AUTOENCODERS ARE NON-COMPACT LEARNERS OF SUBSPACE UNIONS

• When U differs from V, a linear autoencoder whose range contains U must also contain V, so it cannot have learned range exactly U.Under the ideal nearest-point projection, off-union directions V \ U become zero-score blind regions; exact union carving therefore requires a nonlinear complete reconstruction map, unless U is already a subspace.Source: §A.2.1 · p. 17

A.3 OPTIMALITY OF THE NOMINAL UNION AMONG GEOMETRICALLY FAITHFUL RANGES

• The expected geometric score gap equals expected anomaly distance for faithful ranges, but does not by itself determine AUC.Its transfer to reconstruction error requires the network to realize the nearest-point projection prescribed by P2.Source: §A.3 · p. 18
• Under nominal support on U, U is an optimal geometrically faithful range because every larger faithful range can only reduce anomaly distance.The nominal distance terms vanish for all such ranges, so the score gap is maximized at U.Source: §A.3 · p. 18
• A larger range can tie U for a fixed anomaly distribution when its distance matches U's for almost every anomaly.Thus, optimality need not be unique for a fixed distribution.Source: §A.3 · p. 18

A.4 JOIN BLINDNESS

• If the learned range is the join V and anomalies lie in V, nominal and anomalous samples receive identical zero scores, yielding AUC 1/2 under the half-credit convention for ties.This is the limiting case of join blindness under the ideal geometric score.Source: §A.4 · p. 18

A.5 MEET PREFERENCE AND FIDELITY LOSS

• A globally MSE-optimal linear autoencoder retains shared directions first when the component-specific covariance eigenvalues are lower; at rank dim W, its range is exactly W.The sufficient result assumes balanced sampling, mutually orthogonal component-specific spaces, and the stated eigenvalue condition.Source: §A.5 · p. 19
• The shared-direction result does not imply universal convergence to W: exact collapse occurs at rank dim W, while other ranks retain less or additional component-specific directions.Mutual orthogonality of component-specific spaces is a simplifying sufficient assumption.Source: §A.5 · p. 19
• Meet collapse to W gives positive geometric scores to nominal samples with nonzero component-specific content outside W, causing loss of nominal fidelity under P2.Limited capacity may favor shared directions while discarding component-specific ones.Source: §A.5 · pp. 18–19
• Meet collapse predicts fidelity loss, not a universal chance-AUC outcome; chance AUC requires additional conditions such as identical nominal and anomaly score distributions.The toy comparison likewise reports that the rank-deficient model has large errors for both groups, while the compact-union model preserves low nominal error and larger anomaly scores.Source: §A.5 · pp. 19–20, Table 9 · p. 20

A.6 PROOF OF THE SEPARATE PULL AND PUSH GUARANTEES

• The separate pull-and-push proof combines distance bounds to establish the stated guarantees in Eqs. (5) and (6).It bounds nominal distance to the learned range using both its lower push-related bound and an upper bound via U.Source: §A.6 · p. 20

B DETAILED CONSTRUCTION OF NESTED MANIFOLD CARVING

• At each latent level, the method trains a new encoder–decoder pair on the current nominal representation, freezes it, and repeats push–pull at the next level.The pull term preserves the nominal latent representation, while the push term discourages reproducing its perturbed counterpart.Source: B DETAILED CONSTRUCTION OF NESTED… · pp. 20–21
• Below the required latent width, some distinct nominal points become indistinguishable, making exact reconstruction of the union impossible.The construction chooses the new latent dimension before training to preserve the current nominal union.Source: B DETAILED CONSTRUCTION OF NESTED… · p. 20
• Preserving nominal distinctions does not ensure compact-union carving: a linear faithful stage on a non-subspace union must contain its join, so exact carving also requires nonlinearity and informative perturbations.A sufficient local capacity choice is latent width at least the preceding width; nominal fidelity must still be learned by the pull objective.Source: B DETAILED CONSTRUCTION OF NESTED… · p. 21

C DETAILED MNIST ABLATION STUDY

• 0.816 ROC-AUC and 0.489 F1 were reached at Stage 1, up from 0.722 ROC-AUC and 0.436 F1 at Stage 0, when digit 1 was anomalous and no masking or sparsity was used.Nested carving contributes most when Stage 0 leaves anomaly directions within its coarse range; when digit 1 was nominal, the same configuration changed only from 0.921 to 0.922 ROC-AUC and from 0.748 to 0.749 F1.Source: C DETAILED MNIST ABLATION STUDY · p. 22
• 0.929 ROC-AUC and 0.815 F1 were reached by masked AE+PP with sparsity at Stage 1 for nominal digits {4, 5, 7}.Masking can complement Push and Pull in a nested objective, although masking alone has no consistent carving advantage.Source: C DETAILED MNIST ABLATION STUDY · p. 22
• A reconstruction-only nested module leaves performance essentially unchanged, while Push and Pull supplies the carving signal; sparsity and masking are optional, non-uniformly additive regularizers.Sparsity strengthens nested Push and Pull in the {4, 5, 7} experiment but can reduce performance for other nominal sets.Source: C DETAILED MNIST ABLATION STUDY · p. 22
• 0.792 F1 and 0.897 ROC-AUC at Stage 0, rising to 0.798 F1 and 0.909 ROC-AUC at Stage 1, for Push and Pull on nominal digits {4, 5, 7}.Matched denoising results were 0.783 F1 and 0.895 ROC-AUC at Stage 0, with Stage 1 unchanged; the AE baseline reached 0.780 F1 and 0.893 ROC-AUC at Stage 0.Source: C DETAILED MNIST ABLATION STUDY · p. 24, Table 12 · p. 24

C.1 GEOMETRIC CONTRIBUTION OF NESTED LATENT CARVING

• For digit 8, the image channel fires on 87.6% of samples, while only 0.6% of detections depend exclusively on the latent score, consistent with anomalous directions removed before Stage 1.Its retained representation becomes close to nominal digit 5, leaving little for deeper latent processing.Source: §C.1 · pp. 24–25
• For digit 1, the image channel fires on 12.3% of samples, the latent channel on 57.6%, and 48.1% are detected exclusively in latent space.Without the Stage 1 latent score, 80.8% of detected digit 1 samples would be missed; nested carving frequently makes its reconstruction 7-like, consistent with movement toward a nearby nominal component.Source: §C.1 · pp. 25–26, Figure 5 · p. 26

D.1 SEPARATION RATIO

• The Separation Ratio divides mean inter-normal-cluster distance by mean cluster radius, so it can rise through class separation or contraction in place.The table reports the two terms separately; under PushPull, the ratio rises as radii fall roughly 8–10% while inter-cluster distance also falls more slowly.Source: §D.1 · pp. 26–27, Table 14 · p. 27

D.2 PROXIMITY RATIO

• The Proximity Ratio measures median anomaly distance to its nearest normal-cluster center relative to that cluster’s radius, while IC normalizes distance by mean inter-normal-centroid distance.IC is unchanged by uniform rescaling and asks whether anomalies approach normal classes faster than the space contracts.Source: §D.2 · pp. 27–29, Table 15 · p. 28
• The two proximity measures have interpretive limits: cluster shrinkage can distort the radius-normalized ratio, and the pooled median can represent one anomaly class when the class count is small and odd.The section therefore also reports unnormalized anomaly-distance changes, both sample-weighted and pooled by class.Source: §D.2 · pp. 28–29

E.1 BENCHMARKS AND PROTOCOL

• The one-class benchmarks include MNIST, Fashion-MNIST, CIFAR-10, CIFAR-100 coarse superclasses, and Arrhythmia; image training uses nominal-class data only.Image tasks use AUC-ROC, while Arrhythmia labels the top 15% of test anomaly scores as anomalous before F1 calculation.Source: §E.1 · p. 29

E.2 COMPETING METHODS

• AE, AE-PP, and AE-PoS share the TOLL-family architecture and optimization settings, but not its compactness objective; AE-PP adds Push and Pull, and AE-PoS further adds Stage 1.The controlled ablation holds backbone, data splits, and training budget fixed across AE, AE-PP, and AE-PoS.Source: §E.2 · p. 30, §E.5 · p. 31

E.4 PUSH-PULL NEGATIVE VIEWS

• AE-PP’s negative views are generated from nominal training samples for the Push–Pull objective, not labeled anomaly data.At Stage 1, CIFAR perturbations are applied to the frozen Stage-0 latent; MNIST, Fashion-MNIST, and Arrhythmia reuse the Stage-0 transformation.Source: §E.4 · p. 30, §E.3 · p. 30

E.6 PUSH–PULL SENSITIVITY

• The matched margin ablation compares Pull alone at α = 0 with explicit Push separation for α > 0, and reports consistent improvement beyond denoising.The authors report most gains by α = 2 or 3, with performance broadly stable thereafter; validation selects α = 3 for Arrhythmia, MNIST, and Fashion-MNIST, and α = 2 for CIFAR-10 and CIFAR-100.Source: §E.6 · p. 31
• Performance plateaus at α = 2 or 3, consistent with the stated prediction that the Push gradient no longer depends on α once the zero-Push-error bound is exceeded for most training pairs.On Arrhythmia, α ∈ {3, 4, 5, 6} produces identical results.Source: §E.6 · p. 31

E.7.1 DATASET CONSTRUCTION

• The anomaly evaluation covers 16 settings across noise, REDS blur, GoPro motion blur, rain, and underwater degradation.REDS uses five blur levels; underwater groups are divided into high, medium, and low severity.Source: §E.7.1 · p. 34
• For Kodak24, REDS, and paired GoPro, clean counterparts define nominal data; rain, underwater, and the second GoPro protocol use clean DIV8K crops.The DIV8K comparisons are corpus-level rather than scene-paired and may reflect differences in image content and native resolution.Source: §E.7.1 · pp. 34–35

E.7.2 COMPARED MODELS

• The compared AE architecture uses a 9-channel Stage-0 latent, and Stage 1 applies a 9 → 27 → 9 mapping to the frozen latent without changing spatial resolution.The stated latent-to-input dimensionality ratio for Stage 0 is 0.75.Source: §E.7.2 · p. 35

E.8 RESULTS

• AE-PP and AE-PoS combine reliable detection of unseen degradations with high nominal reconstruction fidelity.Plain AE reconstructs degraded inputs as well as nominal images, while VAE loses nominal detail and increasingly represents blur; AE-PoS preserves nominal content while excluding unseen degradations from its learned target set.Source: Table 26 · p. 34, Table 27 · p. 35, §E.8 · p. 36
• A lightweight degradation-detection stage could route only images flagged as degraded to expensive restoration networks, reducing expected latency and energy consumption when most captured images are already sufficiently good.This is presented as a potential use in mobile imaging pipelines.Source: §E.8 · p. 36

F TRANSFERABLE ECG REPRESENTATIONS

• The capacity-controlled two-stage design splits ST-MEM-style learning into two successive stages, each with a six-block encoder and four-block decoder, matching the original total encoder depth.Stage 1 freezes Stage 0 and applies masked latent reconstruction to its representation.Source: F TRANSFERABLE ECG REPRESENTATIONS · pp. 36–37
• Both stages use the same unlabelled pretraining pool of 164,972 ECG recordings, with no additional ECG data introduced after Stage 0.The pool contains 34,808 Ningbo and 130,164 CODE-15 recordings.Source: F TRANSFERABLE ECG REPRESENTATIONS · p. 37
• Evaluation uses frozen-encoder linear probes on four PTB-XL tasks and ICBEB/CPSC-2018, reporting test AUROC from the validation-selected checkpoint.The reported Table 7 results use the complete labelled training split and average over five seeds.Source: F TRANSFERABLE ECG REPRESENTATIONS · pp. 37–38
• Stage 1 improves upon Stage 0 on all five ECG downstream tasks and achieves the highest AUROC in every comparison column.Because the two-stage encoder matches the original ST-MEM backbone’s total depth, the comparison indicates gains from successive refinement rather than increased total encoder depth alone.Source: F TRANSFERABLE ECG REPRESENTATIONS · p. 39

• Pixel ℓ2 distance can make anomalies close to the nominal union difficult to distinguish; 87% of digit 9 samples in the {4, 5, 7} experiment evade both scores.The authors point to task-appropriate metrics or geometry-based anomaly scores as future directions, subject to Lemma 1's metric conditions.Source: G EXPANDED LIMITATIONS AND FUTURE… · p. 39
• The number of nested carving levels is selected manually; a proposed label-free stopping criterion would track held-out nominal perturbations that still violate the Push margin.Whether that fraction can reliably determine when further carving is needed remains open.Source: G EXPANDED LIMITATIONS AND FUTURE… · p. 39
• For anomalies outside M's reach, nearest nominal points may be nonunique; the authors conjecture structured anomalies may lie on isometric images Φγ(M), where inverse mapping enables projection onto those manifolds.They leave conditions for identifying such inverse transformations to future work.Source: G EXPANDED LIMITATIONS AND FUTURE… · p. 39