PaperlayerView exact revision on arXiv

Source-linked AI summary

Beyond Empirical Support: Structured Outlier Generation via Sinkhorn Optimal Transport

Haixiang Sun, Andrew L. Liu

arXiv:2609.31470v1stat.MLcs.LGmath.OC

TL;DR

Rare outliers are difficult to study because finite training data cover them poorly, while existing synthesis approaches often rely on low-support criteria. SBOG uses Sinkhorn transport support with semantic constraints to generate controlled outliers, and experiments report informative outputs across time-series and image tasks that improve downstream robustness evaluation.

  • Problem

    Outlier generation remains relatively scarce, and existing approaches often use sparse neighborhoods, low-likelihood regions, or classifier-boundary crossings as support criteria.

    Source: §1 · p. 1

  • Method

    SBOG uses Sinkhorn Outlier Energy to select weakly supported latent boundary candidates, while semantic feasibility and decoding constraints produce controlled, class-consistent outliers.

    Source: §1 · p. 2, §3.1 · p. 5

  • Results

    Experiments on time-series anomaly generation and image outlier synthesis demonstrate informative synthetic outliers across modalities and improved downstream robustness evaluation.

    Source: §5 · p. 10

  • Takeaways & Limitations

    The Sinkhorn-DRO connection justifies SBOG's support-boundary acquisition rule, not a claim that the algorithm solves a downstream task-specific DRO problem.

    Source: §3.1 · p. 5

    SBOG depends on representation quality, transport-cost design, semantic anchoring, and generator expressiveness; weak embeddings or limited decoders may reduce outlier fidelity and utility.

    Source: §5 · p. 10

Abstract

from arXiv · show

Outliers are essential for evaluating and improving the robustness of machine learning systems, especially when future distributions may differ significantly from historical training data. In high-stakes applications, robustness often depends on rare cases that finite datasets fail to capture, making simple resampling or perturbation insufficient for stress scenario generation. Existing outlier synthesis methods typically rely on sparse neighborhoods, low support latent regions, or classifier boundary crossings, which can be heuristic, unstable, and tied to specific modalities or architectures. We therefore propose Sinkhorn Boundary Outlier Generation (SBOG), a structured framework for latent-space outlier generation that couples Sinkhorn optimal transport geometry with distributionally robust boundary modeling. The resulting Sinkhorn-induced support cost guides the sampler toward weakly supported boundary regions, while semantic constraints prevent uncontrolled drift from the intended context, yielding controlled deviations from the in-distribution reference measure rather than arbitrary sparse-region samples. Experiments on time series anomaly generation and image outlier synthesis show that our framework produces informative, semantically controlled outliers and improves downstream robustness evaluation across modalities, providing a foundation for stress scenario generation beyond empirical support.

1 Introduction

Outliers matter for robustness because rare cases are poorly covered by finite datasets, but generating useful outliers remains less studied than detecting them. The paper proposes SBOG, which uses Sinkhorn transport support and semantic constraints to generate controlled boundary cases across modalities.

1 Introduction

  • Rare, atypical cases are poorly represented in finite training datasets, yet can determine whether models behave robustly in deployment.The paper frames outlier generation as a complement to detection, monitoring, and mitigation.Source: §1 · p. 1
  • Existing synthesis methods use sparse neighborhoods, low-likelihood regions, or classifier-boundary crossings, but low support alone can produce semantically drifting or invalid samples.Such pipelines are also often tailored to time series or image synthesis.Source: §1 · pp. 1–2
  • SBOG combines Sinkhorn transport-based support scoring, boundary-anchor selection, and semantic feasibility to generate controlled latent-space outliers.Its Sinkhorn Outlier Energy is used as a boundary score rather than as a standalone density estimator.Source: §1 · p. 2
  • Across time-series and image benchmarks, SBOG generates informative, semantically controlled outliers and improves downstream robustness evaluation.The Sinkhorn-DRO dual connects its energy-based boundary decisions to a robust optimization interpretation.Source: §1 · p. 2

2 Preliminaries

The preliminaries introduce normalized entropic Sinkhorn transport and its distributionally robust optimization formulation. They then describe latent representations and a sampler that generates outliers relative to the in-distribution embedding set and decodes them to inputs.

2.1 Sinkhorn Distributionally Robust Optimization

  • Sinkhorn OT compares candidate distributions through couplings and an entropic transport cost, normalized so the minimum discrepancy over candidate marginals is zero.The discrepancy defines a Sinkhorn ambiguity set around the nominal distribution.Source: §2.1 · p. 3
  • The Sinkhorn-DRO objective has a one-dimensional dual whose log-moment-generating term exponentially tilts toward high-loss values, yielding softmax-type robustification.The robustification is governed by the ambiguity radius, regularization, and dual temperature.Source: §2.1 · p. 3

2.2 Outlier Generation

  • The generation setup learns normalized latent embeddings aligned with class conditioning vectors and treats the empirical embedding set as a latent reference measure.Outlier generation is framed as producing new latent points relative to the geometry induced by that set.Source: §2.2 · p. 3
  • An outlier sampler places most of its latent mass outside the in-distribution region, then maps sampled latent candidates back to inputs through a conditional generator.Repeating the procedure yields an auxiliary out-of-distribution dataset.Source: §2.2 · p. 3

3 SBOG: Sinkhorn Boundary Outlier Generation

SBOG scores latent candidates by their transport support cost, selects weakly supported but semantically feasible boundary points, and decodes them into outliers. The method connects this acquisition rule to Sinkhorn-DRO and uses Sinkhorn neighborhoods in representation alignment; its theoretical guarantees are conditional on stated stability and acceptance conditions.

3.1 Outlier Sampling under Latent Space

  • SOE measures the transport cost of supporting a latent candidate from the in-distribution reference measure, with larger energy indicating more expensive support.In the special case of uniform empirical support with squared Euclidean cost, it is a monotone transform of an unnormalized RBF-KDE score; beyond that setting it is not equivalent to KDE.Source: §3.1 · pp. 4–5
  • SBOG selects top-energy embeddings as anchors, perturbs them locally, and accepts proposals above the SOE threshold that also satisfy semantic feasibility before decoding the highest-energy feasible candidate.This couples a smooth transport-support contour with semantic anchoring and generative validity rather than relying on low support alone.Source: §3.1 · p. 5
  • The Sinkhorn-DRO dual identifies SOE as a support-normalization cost, justifying threshold-based boundary acquisition but not claiming that SBOG solves a downstream task-specific DRO problem.A downstream loss or uncertainty score can be added for a task-aware variant.Source: §3.1 · p. 5
  • For a fixed candidate pool, the SOE threshold decisions can match their population counterparts with sufficient samples, although the bound can loosen when the normalizer is small.The population-SOE sampler converges in total variation under the stated energy-error, anchor-set, threshold, and acceptance conditions; this is not alignment with an independently defined target boundary.Source: §3.1 · p. 6

3.2 Sinkhorn-DRO Alignment for Boundary-Aware Representations

  • Sinkhorn-DRO alignment robustifies pointwise class alignment over local transport neighborhoods, making representation geometry more compatible with SOE boundary generation.When the radius is zero and the reference measure is a point mass, the robust alignment loss reduces to standard alignment.Source: §3.2 · pp. 6–7

4 Experiments

The experiments evaluate SBOG for time-series anomaly generation and image outlier synthesis. They test downstream performance, sampler components, threshold sensitivity, and semantic anchoring.

4.1 Time Series Generation

  • CARLA-SBOG achieves the best average AU-PR and F1 among CARLA variants for downstream time-series anomaly detection.The evaluation uses five multivariate benchmarks and point-wise scoring without Point Adjustment.Source: §4.1 · pp. 7–8, Table 1 · p. 8
  • Sampler ablations show SBOG maintains high anchor fidelity while inducing stronger distribution shift, with a better margin–diversity trade-off than ablated and support-score variants.The comparison holds the trained encoder fixed; SBOG moves negatives farther along the boundary while retaining high intra-negative diversity.Source: §4.1 · p. 8
  • Performance is stable across a broad middle range of energy thresholds, while overly high thresholds reduce both energy margin and feasible-anchor coverage.The reported diagnostics are Gap and Fill.Source: §4.1 · p. 8, Table 2 · p. 8

4.2 Image Generation

  • SOE anchor selection outperforms k-NN sparsity and KDE-RBF scoring on token alignment, diversity, and feasible yield with the remaining pipeline fixed.This ablation tests image outlier generation.Source: Table 5 · p. 9, §4.2 · p. 9
  • Moderate smoothing yields larger empirical margins and lower flip rates, whereas very small ε makes the energy landscape sharp and unstable.The threshold sweep uses the same trained encoder.Source: Figure 3 · p. 9, §4.2 · p. 9
  • Semantic anchoring improves intended-class purity, token alignment, cross-class margin, and contamination control compared with an unanchored candidate pool.The comparison evaluates the top 250 candidates using the same downstream ranking rule; the accompanying t-SNE shows anchored candidates near intended class boundaries.Source: Table 6 · p. 9, §4.2 · p. 10

5 Conclusion

SBOG uses Sinkhorn Outlier Energy with semantic feasibility and decoding constraints to generate controlled outliers, and experiments report informative samples across time-series and image tasks. Its fidelity and utility depend on representation, transport-cost, semantic-anchoring, and generator quality.

5 Conclusion

  • Experiments on time-series anomaly generation and image outlier synthesis show informative synthetic outliers and improved downstream robustness evaluation.The conclusion reports these outcomes across modalities.Source: §5 · p. 10
  • SBOG uses transport-support energy to identify weakly supported boundary regions, with semantic feasibility and decoding constraints for controlled, class-consistent outliers.The paper also connects SOE to kernel support and the Sinkhorn-DRO dual.Source: §5 · p. 10
  • SBOG depends on representation quality, transport-cost design, semantic anchoring, and generator expressiveness; weak embeddings or limited decoders may reduce outlier fidelity and utility.The conclusion identifies adaptive Sinkhorn-temperature and SOE-threshold calibration as future work.Source: §5 · p. 10

A Related works

  • DRO representation work mainly reweights losses or stabilizes features, whereas this framework uses Sinkhorn-DRO geometry to define latent-space outlyingness and synthesis.The related-work discussion contrasts the framework with approaches focused on robust representations and loss refinement.Source: A Related works · p. 16
  • Time-series outlier research is largely detection-oriented, while dedicated synthesis remains scarce and is often auxiliary to detector training.The paper positions its generated samples for downstream evaluation and analysis.Source: A Related works · p. 16
  • Image synthesis methods use empirical density, sparse-neighborhood, or boundary-crossing heuristics to localize candidates; the paper contrasts these with its Sinkhorn geometric definition.The related-work discussion says image-level methods have advanced visual quality but retain heuristic candidate localization.Source: A Related works · p. 16

B Discussion of Sinkhorn Boundary Outlier Energy

The discussion interprets Sinkhorn Outlier Energy as an entropic support cost: larger values correspond to candidates less supported by the reference geometry. The energy also appears in the Sinkhorn-DRO dual objective.

B Discussion of Sinkhorn Boundary Outlier Energy

  • Larger SOE values correspond to smaller kernel mass under the reference measure, indicating weaker support and greater likelihood of being an outlier under the proposed criterion.The text describes the energy as the entropic variational cost associated with a candidate and the reference measure.Source: B Discussion of Sinkhorn Boundary… · pp. 16–17
  • SOE is introduced as an entropic support cost; for an empirical reference measure, the discussion relates it to kernel support.The supplied lemma passage names the entropic support cost but does not provide its full result.Source: B Discussion of Sinkhorn Boundary… · p. 17
  • The Sinkhorn-DRO dual decomposition contains an SOE term, so minimizing that objective necessarily optimizes an objective containing outlier energy.The decomposition is stated for the robust log-moment objective.Source: B Discussion of Sinkhorn Boundary… · p. 18

C Comparison of Sample Complexity

The comparison contrasts hard neighborhood selection with SOE thresholding: k-NN incurs a local-scale sample burden in weakly supported regions, while SOE has a finite-candidate stability guarantee under stated conditions.

C Comparison of Sample Complexity

  • Under a local mass upper bound, hard k-NN cannot reliably resolve neighborhood scale Δr unless the sample size is on the order of kΔr^-m.This is presented as the cost of estimating a local counting or radius event in weakly supported regions.Source: C Comparison of Sample Complexity · p. 19
  • SOE thresholding has a finite-candidate stability guarantee under explicit normalizer and margin conditions.The comparison holds the candidate pool fixed and describes the guarantee for threshold decisions.Source: C Comparison of Sample Complexity · pp. 18–19

D Population-SOE consistency of the sampler

The consistency analysis compares the empirical sampler with its population-SOE counterpart and gives conditions for termination and convergence. It also distinguishes this consistency from alignment with an independently specified target boundary.

D Population-SOE consistency of the sampler

  • When n ≥ p0 > 0 and a_n + b_n(e_n) < p0, the empirical sampler terminates almost surely and its output law is bounded in total variation from the population-SOE law.The bound also holds after applying a shared conditional decoder; convergence in probability follows when the energy error and sampler-disagreement terms vanish and threshold and acceptance conditions hold with probability tending to one.Source: D Population-SOE consistency of the… · pp. 21–22
  • The sampler comparison couples anchor draws and shared proposals; disagreement is controlled by anchor changes, threshold errors, and instability of the feasible proposal maximizer.The threshold event accounts for errors in both candidate energy and calibrated threshold; the same semantic test on shared proposals means no semantic-margin assumption is needed.Source: D Population-SOE consistency of the… · pp. 21–22
  • A sufficient uniform energy-estimation condition gives e_n = O(log n/n) for fixed latent dimension and stated constants, but consistency also requires vanishing selection-margin terms.The concentration argument assumes an i.i.d. global reference bank and does not itself cover data-dependent trimming, local-reference restrictions, or an encoder fitted on that same bank.Source: D Population-SOE consistency of the… · p. 23
  • Conditional boundary precision and coverage follow when target regions have sufficient population-SOE mass, and the same statements hold after a shared decoder.This result is conditional on the fitted bank; it does not establish that SOE-selected regions match a task-defined boundary or that outputs sharing the bank are unconditionally independent.Source: D Population-SOE consistency of the… · pp. 23–24

E Controlled Numerical Validation

The controlled validation tests boundary generation against a method-independent population oracle using three stable VAR processes and several kinds of perturbation. SBOG places more samples in the defined boundary band and achieves greater average coverage than competing generators, while worst-region coverage remains low.

E Controlled Numerical Validation

  • The oracle uses three stable VAR processes with perturbations to local means, innovation covariances, and transition dynamics, defining its target band by an exact population log-likelihood ratio.The population oracle is distinct from the population-SOE sampler in Theorem 3.5.Source: E Controlled Numerical Validation · p. 24
  • Evaluation compares SBOG with RandomNeg, kNNNeg, and KDENeg using boundary precision, semantic validity, coverage, SW2, and held-out detection AUPR.The measurements assess boundary alignment, coverage, and downstream utility.Source: E Controlled Numerical Validation · p. 24
  • SBOG places more samples in the independently defined boundary band and achieves greater average boundary coverage than competing generators, while maintaining high semantic validity.Generated samples also support higher AUPR on held-out oracle data, but worst-region coverage remains low.Source: E Controlled Numerical Validation · p. 24, Table 7 · p. 24

F Time-Series Experimental Details

The time-series experiments keep the CARLA pipeline and model schedule fixed across five datasets, changing only the input channel count by benchmark. The sampler uses shared SOE settings, and the downstream detector uses a common training configuration.

F.1 Pipeline parameters

  • Across SMD, MSL, SMAP, SWaT, and WADI, the CARLA pipeline shares its encoder architecture and optimization schedule; only input channel count varies by dataset.The channel count equals the number of benchmark variates.Source: §F.1 · p. 24
  • The shared time-series SOE sampler uses ε = 0.02, a threshold at the 95th-percentile refined ID energy plus 0.04, Gaussian proposal noise σ = 0.02, and 128 proposals per anchor.Its anchor pool is the top ⌈0.02n⌉ high-energy ID points, capped at 256 and lower-bounded at 8; the reference measure trims the top 8% highest-energy ID points.Source: §F.1 · pp. 24–25
  • The downstream detector uses the same ResNet backbone and 5-NN softassignment across datasets, with 50 Adam epochs and fixed anomaly contamination of 0.99.Detection scores are point-wise, following [27].Source: §F.1 · p. 25

F.2 Quality and stability metrics

The appendix defines measures of generated-negative quality and stability, then gives the image synthesis configuration. These cover latent fidelity and diversity, distributional differences, and the setup for decoding accepted latents into images.

F.2 Quality and stability metrics

  • Energy gap measures accepted negatives' excess over the acceptance threshold, while feasible fill ratio measures the fraction of anchors with at least one accepted proposal.A larger energy gap means negatives sit deeper in the high-energy region, although the gap can shrink as the threshold rises into the ID energy tail.Source: §F.2 · p. 25
  • Distributional metrics include RBF MMD², sliced Wasserstein-2, and intra-negative pairwise cosine diversity.Higher MMD² indicates negatives form a distribution detectable as different from ID; sliced Wasserstein-2 averages over 1000 random unit-norm one-dimensional projections.Source: §F.2 · p. 25
  • Quality and stability metrics include anchor fidelity to the proposing anchor's pseudo-label and cross-class margin relative to pseudo-class means.The negatives are evaluated against in-distribution embeddings and pseudo-labels in the encoder's 4-D unit-sphere latent space using cosine ground cost.Source: §F.2 · pp. 25–26

G.1 Configuration

  • Image experiments use ResNet-34 features from CIFAR-100 and ImageNet-100, class text anchors, and per-class banks of 1000 ID features; accepted latents are decoded with Stable Diffusion v1.5 conditioned on the source class token.The per-class threshold is the 0.95-quantile plus a default-zero margin; proposals use σ = 0.012 and M = 256, with a local reference of 128 nearest neighbors and seed-cosine floor 0.7.Source: §G.1 · p. 26

G.2 Definition of Diagnostic Metrics

The diagnostic metrics define how the paper measures outlier energy, semantic and class alignment, diversity, calibration stability, and generation feasibility. They are reported per class as mean ± standard deviation across 100 ImageNet-100 classes.

G.2 Definition of Diagnostic Metrics

  • Empirical decision margin is the distance of accepted candidates’ energy from the threshold; larger values indicate more confident decisions and serve as a proxy for ΔE.Theorem 3.4 relates sample complexity to 1/ΔE².Source: §G.2 · p. 27
  • Flip Rate@256 measures how often decisions change when thresholds are recalibrated from 256 uniformly sampled ID examples; lower rates indicate closer agreement with the full bank.The reported rate averages 30 random calibration subsets; Theorem 3.4 predicts this regime when ΔE is well-conditioned.Source: §G.2 · p. 27
  • Token cosine measures similarity to the class text token, while diversity is one minus mean off-diagonal cosine and reflects spread across boundary directions.The time-series generated-negative diversity uses the same formula; higher values indicate coverage of more boundary directions rather than clustering.Source: §G.2 · pp. 27–28
  • Yield multiplies the semantic-filter pass rate by the mean Sinkhorn outlier energy of passing proposals, capturing both on-class proposal frequency and outlier strength.Each anchor uses 32 Gaussian perturbations at σ = 0.012; proposals pass when cos(v, tc)≥τsem.Source: §G.2 · p. 27
  • Intended-class purity measures the fraction of candidates whose nearest ID neighbour across classes belongs to the intended class; higher values indicate class-manifold fidelity.The nearest-neighbour bank uses 150 samples per class.Source: §G.2 · p. 27
  • Fidelity is prototype cosine similarity, contamination is the fraction assigned to another class, and positive margin means a negative is closer to its pseudo-class than any other.In the matched-prototype regime, Fidelity + Contam ≈1.Source: §G.2 · p. 28
  • MMD2 measures squared maximum-mean-discrepancy between generated negatives and empirical positives; larger values indicate stronger distributional shift desirable for contrastive negative mining.The Gaussian RBF kernel uses the median heuristic bandwidth.Source: §G.2 · p. 28
  • A moderate ϵ with a small margin gives the most reliable generation setting, preserving outlier strength and stable, feasible synthesis while semantic alignment changes little.Stricter acceptance reduces energy advantage and feasible outliers, especially at very small ϵ.Source: Figure 5 · p. 28, §G.2 · p. 28
Loading 2609.31470v1…