PaperlayerView exact revision on arXiv

Source-linked AI summary

Generalist Representation, Specialist Detection: TS-Router for Time-Series Anomaly Detection

Tian Lan, Yifei Gao, Yimeng Lu, Xuming An, Meng Wang, Yue Pan, Wenjun He, Chenghao Liu, Chen Zhang

arXiv:2610.00978v1cs.LGcs.AI

TL;DR

TSAD detector suitability varies across series, and transferable representations do not by themselves settle which deviations are anomalous. TS-Router uses pretrained representations and simulated-task competence supervision to select specialists for unlabeled targets. It achieves the best overall average rank among the evaluated baselines.

  • Problem

    Detector suitability varies across target series, while foundation-model approaches commonly pair transferable representations with fixed anomaly criteria.

    Source: §1 · p. 2

  • Method

    TS-Router predicts heterogeneous specialists’ relative competence from pretrained temporal representations, using labeled simulated tasks for soft routing supervision and fitting only selected specialists without target anomaly labels.

    Source: §1 · pp. 2–3, §4.1 · p. 5

  • Results

    TS-Router achieves the best overall average rank among evaluated baselines, with controlled ablations supporting adaptive specialist selection and multiple frozen TSFM encoders.

    Source: §1 · p. 3

  • Takeaways & Limitations

    Foundation models can coordinate specialized detection criteria through competence routing rather than directly imposing a universal anomaly criterion.

    Source: §1 · p. 2

    The regret analysis is stated for a fixed mean-pooled representation, bounded specialist competences, and the specified simulated and real task distributions.

    Source: §4.4 · p. 6

Abstract

from arXiv · show

Time-series anomaly detection (TSAD) is difficult to generalize across datasets because heterogeneous temporal dynamics imply different notions of normality and favor different detection criteria. While time-series foundation models provide transferable representations, coupling them with a fixed anomaly-scoring mechanism can overlook this variation. This motivates a different perspective on foundation-model-based TSAD: using foundation models to coordinate specialized anomaly criteria rather than directly imposing a universal one. Based on this view, we propose \textbf{TS-Router}, a generalist-representation, specialist-detection framework that estimates the relative competence of heterogeneous anomaly detectors from pretrained temporal representations and selects suitable specialists for each target series. To avoid relying on specialist-performance labels from real tasks, we derive soft competence supervision from specialists' relative performance on labeled simulated tasks. At deployment, routing requires no target anomaly labels, and only the selected specialists are fitted unsupervisedly on the target series. We bound Top-\(k\) set-competence regret under representation coverage and conditional competence stability. Across 16 real-world benchmarks and four complementary evaluation metrics, TS-Router achieves the best overall average rank. Controlled ablations with multiple frozen TSFM encoders further support the use of pretrained representations for competence estimation and adaptive specialist selection. The code is available at https://anonymous.4open.science/r/TS-Router-D8FF.

1 INTRODUCTION

TSAD detector suitability varies across series, while transferable foundation-model representations do not by themselves resolve which deviations are anomalous. TS-Router uses those representations to estimate specialist competence and route target series to selected detectors.

1 INTRODUCTION

  • Detector suitability varies across series, while fixed forecasting or reconstruction errors can fail to address variation in anomaly criteria across targets.The introduction frames the open question as how foundation models should contribute beyond directly defining an anomaly criterion.Source: §1 · p. 2
  • TS-Router separates transferable temporal representation from target-dependent anomaly decisions, using representations to estimate competence and select suitable specialists.Only selected detectors are fitted unsupervisedly on the target series, without target anomaly labels.Source: §1 · p. 2
  • Soft routing supervision comes from specialists’ relative performance on labeled simulated series, and the analysis bounds Top-k set-competence regret under explicit transfer assumptions.The framework is motivated by scarce real anomaly annotations.Source: §1 · pp. 2–3
  • TS-Router achieves the best overall average rank among evaluated baselines, with controlled ablations supporting adaptive specialist selection and multiple frozen TSFM encoders.The reported experiments cover diverse TSAD benchmarks.Source: §1 · p. 3

2 RELATED WORK

Prior work includes target-specific detectors, time-series foundation models adapted to anomaly detection, and automated detector-selection methods. The section identifies transferable competence estimation for unseen series as an open challenge.

Target-Specific Time-Series Anomaly Detection

  • Target-specific detectors use varied inductive biases, so their suitability can vary across series and deployment still requires selection and adaptation.Methods include reconstruction, discriminative, and classical statistical or geometric approaches.Source: §2 · p. 3

Time-Series Foundation Models for Anomaly Detection

  • TSFMs transfer temporal representations, but anomaly-detection adaptations generally pair them with a unified detector or fixed scoring mechanism across heterogeneous targets.Direct forecasting or reconstruction errors may perform comparably to simple local-statistics baselines.Source: §2 · p. 3

Automated Detector Selection

  • Automated selection methods use handcrafted descriptors, historical tasks, surrogate evaluations, or candidate execution, leaving transferable competence estimation on unseen series open.Examples include MetaOD, Unsupervised Model Selection, Choose Wisely, and AutoTSAD.Source: §2 · p. 3

3 PROBLEM FORMULATION

The formulation treats detector choice as selecting a fixed-budget set of heterogeneous specialists based on a target series’ temporal representation. It distinguishes set competence from the utility of fused anomaly scores.

TSAD with Heterogeneous Specialists

  • Each specialist produces anomaly scores under a fixed label-free protocol, and its competence is the evaluation utility of those scores against task labels.The formulation considers univariate series; multivariate targets use variable-wise routing followed by score aggregation.Source: §3 · pp. 3–4

Specialist Selection with a Fixed Budget

  • For a fixed budget k, selection quality is the mean individual competence of a k-specialist set, with TopK identifying the maximizing set.These set utilities compare equal-sized selections before score fusion and are not the detection utility of an averaged score sequence.Source: §3 · p. 4

Representation-Conditioned Competence

  • A representation-conditioned choice can improve selection over a globally fixed set, while the gap to hindsight selection measures information unavailable from the representation.Transferable representations can inform which specialist criteria to deploy without directly defining a universal anomaly criterion.Source: §3 · p. 4

Competence Routing

  • The router predicts specialist competence from the encoded series and selects its Top-k specialists before fitting them without target labels and combining their normalized scores.The representation-conditioned selection precedes detector fitting.Source: §3 · p. 4

4 METHOD

TS-Router trains a competence router using simulated tasks and frozen pretrained temporal representations, then routes unlabeled target series before fitting selected specialists. Its theoretical analysis bounds Top-k routing regret under stated assumptions.

4 METHOD

  • The framework freezes a foundation encoder, trains a lightweight router, and at deployment routes before fitting only selected specialists for unlabeled detection.The stages separate temporal representation learning from target-adaptive anomaly decisions.Source: §4 · pp. 4–5

4.1 SYNTHETIC COMPETENCE SUPERVISION

  • Simulated competence tasks vary normal temporal dynamics and anomaly mechanisms, probing specialist suitability without attempting to reproduce the real-data distribution.The specialist pool uses lightweight detectors with heterogeneous inductive biases, and known simulated labels evaluate competence rather than fit the specialists.Source: §4.1 · p. 5
  • Specialists’ relative performance on simulated tasks becomes soft competence targets that preserve ordering and pairwise gaps, including comparable specialists.The resulting routing dataset supervises suitability as a function of observable temporal context.Source: §4.1 · p. 5

4.2 FOUNDATION-GUIDED COMPETENCE ROUTING

  • A frozen pretrained encoder supplies fixed temporal representations, and a lightweight router predicts the full specialist-competence distribution from pooled token embeddings.The encoder is trained under its native objective independently of the competence supervision; only router parameters are optimized to match the soft targets.Source: §4.2 · pp. 5–6

4.3 OFFLINE TARGET ROUTING AND DETECTION

  • At offline deployment, the router uses the complete unlabeled observed series; selected specialists fit the designated prefix, score the suffix, and contribute standardized scores to a fused sequence.Neither target labels nor encoder or router parameter updates are used.Source: §4.3 · p. 6

4.4 THEORETICAL ANALYSIS

  • Theorem 1 bounds simulated-to-real Top-k routing regret under the stated assumptions, separating representation information, soft-target prediction, and competence transfer at a fixed budget.The accompanying KL lemma bounds Top-k competence regret from prediction divergence; the analysis uses a fixed mean-pooled representation and specified simulated and real task distributions.Source: §4.4 · pp. 6–7

5 EXPERIMENTS

The experiments assess TS-Router’s overall detection performance, adaptive specialist routing, and representation choices. Across 16 benchmarks and four detection metrics, the reported comparisons favor the complete routing-and-adaptation pipeline and frozen TSFM representations.

5.1 EXPERIMENTAL SETUP

  • The evaluation covers 16 benchmarks and four complementary detection metrics, with routing assessed by NDCG@k and Hit@k against VUS-PR-based competence rankings.Threshold-dependent metrics use oracle thresholds selected from evaluation labels after score generation; those labels are not used for routing, fitting, or score generation.Source: §5 · p. 7, §5.1 · pp. 7–8

5.2 EXPERIMENTAL RESULTS

  • TS-Router has the highest mean detection scores and best average ranks under all four metrics across 16 benchmarks.Table 1 summarizes performance across the benchmarks; the comparison includes direct zero-shot models and target-fitted unsupervised detectors with differing deployment protocols.Source: §5.2 · pp. 8–9, Table 1 · p. 8
  • TS-Router Top-3 outperforms the hindsight-best fixed specialist and full-pool ensemble, while also outperforming the globally fixed Best Top-3 set.Top-3 improves over Router Top-1 under all four detection metrics; the comparisons distinguish adaptive selection and selective fusion from globally fixed or indiscriminate alternatives.Source: §5.2 · p. 9, Table 2 · p. 9
  • All three frozen TSFM encoders achieve higher VUS-PR and NDCG than conventional feature representations, with metric-specific leaders across encoders.TimeRCD has the highest NDCG and Hit scores and the best VUS-PR and Standard-F1, while MOMENT has the highest Affiliation-F1 and F1T.Source: §5.2 · pp. 9–10, Table 2 · p. 9

6 CONCLUSION

The conclusion presents TS-Router as a framework that uses transferable temporal representations to estimate specialist competence and select detectors, with selective score fusion improving detection in the reported experiments. Its transfer depends on simulated-task coverage, and its multivariate setting uses variable-wise routing.

6 CONCLUSION

  • TS-Router estimates specialist competence from transferable temporal representations and selects suitable detectors, using labeled simulations for soft routing supervision.The conclusion reports a Top-k set-competence regret bound under explicit transfer assumptions.Source: §6 · p. 10
  • The experiments evaluate a representation-to-competence-to-detection pathway in which foundation representations inform specialist suitability and selective score fusion improves detection.Results with multiple frozen TSFM encoders show that this role is not confined to a single pretrained model.Source: §6 · p. 10
  • Competence transfer is shaped by how well the simulated task distribution covers variations relevant to unseen targets.For multivariate series, the study uses variable-wise routing to isolate the pathway; incorporating cross-variable interactions is identified as a natural extension.Source: §6 · p. 10

A.1.1 PROOF OF LEMMA 1

The proof controls Top-k selection regret by relating mismatched selected sets to softmax probabilities, including a lower bound on probability mass at the selection boundary.

A.1.1 PROOF OF LEMMA 1

  • For competence vectors with range at most one, every selection regret is bounded by αk; when k=K, the regret and the second term in Cτ,K,k vanish.The proof then restricts subsequent arguments to k<K.Source: §A.1.1 · p. 15
  • The k-th largest softmax probability is at least 1/((k − 1)e^(1/τ) + K − k + 1) when osc(v) ≤ 1.The proof obtains the bound by shifting scores by v(k) and bounding the terms above and below the selection boundary.Source: §A.1.1 · p. 15
  • For differing Top-k sets, the proof pairs exchanged indices and uses their score gaps and softmax probability ratios to bound selection regret.The paired score gaps lie in [0,1], and the resulting probability differences are bounded using the stated chord inequality.Source: §A.1.1 · p. 15
  • The proof combines the established bounds to conclude Lemma 1.It takes the smaller of the bounds from Eqs. (21) and (24), together with αk from Eq. (17).Source: §A.1.1 · p. 16

A.1.2 PROOF OF THEOREM 1

The proof derives Top-k set-competence guarantees from conditional competence prediction, accounting for representation coverage and competence stability. It then relates set recovery to fused detection utility through the competence gap at the selection boundary.

A.1.2 PROOF OF THEOREM 1

  • The proof decomposes conditional KL loss and shows its prediction-dependent term is minimized by the conditional mean soft competence.Combining this argument with the theorem's bounds establishes the stated regret inequalities, including the k = K case.Source: §A.1.2 · pp. 17–18
  • The regret bound accounts for representation coverage and changes in relative competence, with the latter controlled by conditional competence stability.The proof bounds changes in competence gaps using δ and uses a change-of-measure argument for representation coverage.Source: §A.1.2 · p. 18

A.1.3 CONDITIONAL SOFTMAX DISTORTION

  • Conditional softmax distortion measures the mismatch between competence gaps and log-ratios of conditional mean soft targets, and remains explicit when nonzero.A conditional concentration condition yields a target-averaged bound involving r(Z)^2/(8τ).Source: §A.1.3 · pp. 18–19
  • Distortion is zero when the representation determines pairwise competence differences, even if overall competence varies through a shared task-dependent offset.In that case the soft target is conditionally deterministic.Source: §A.1.3 · p. 19

A.1.4 UTILITY HIERARCHY AND REPRESENTATION INFORMATION

  • More informative representations cannot reduce the best attainable conditional competence at a fixed selection budget, but this does not guarantee better performance for every trained router.The conditional optimum selects the k specialists with the largest conditional mean competence.Source: §A.1.4 · p. 19

A.1.5 FROM SET RECOVERY TO SCORE FUSION

  • The selected-set mismatch bound depends on the competence gap between the kth and (k+1)th specialists, linking set recovery to fused-utility agreement.As routing regret vanishes, the expected fused-utility difference converges to zero under the stated boundary condition.Source: §A.1.5 · pp. 20–21

A.1.6 SCOPE OF THE GUARANTEE AND EXPERIMENTAL INTERPRETATION

The guarantee concerns specialist selection at a fixed budget, while experiments also examine the detection pipeline and budget trade-offs. TS-Router uses simulated competence supervision and label-free routing and fitting, evaluated on eleven genuinely univariate and five multivariate benchmarks.

A.1.6 SCOPE OF THE GUARANTEE AND EXPERIMENTAL INTERPRETATION

  • The guarantee compares selectors through mean individual competence at a fixed budget; fused detection utility and performance comparisons across budgets are separate questions.Routing metrics assess predicted specialist rankings, whereas detection metrics assess the complete fitting-and-fusion pipeline.Source: §A.1.6 · p. 21

A.2 OVERALL TRAINING AND DEPLOYMENT PROCEDURE

  • TS-Router freezes a pretrained temporal encoder, learns routing targets from labeled simulated tasks, then routes before fitting only selected specialists on an unlabeled target.The benchmark protocol applies routing and detection independently to each variable for multivariate targets, then aggregates variable-wise scores.Source: §A.2 · p. 22

B.1.1 BENCHMARK DATASETS

  • Evaluation uses eleven genuinely univariate collections and five multivariate benchmarks, excluding decomposed multivariate series from the univariate track.The univariate collections also support controlled routing and representation analyses that avoid multivariate variable-wise aggregation.Source: §B.1.1 · p. 24

B.1.2 BASELINE METHODS AND DEPLOYMENT PROTOCOLS

The experiments compare zero-shot and target-fitted detectors under a shared no-target-label constraint, then describe TS-Router’s specialist pool, evaluation protocol, and implementation. Across datasets and metrics, detector strengths vary, and no competing detector consistently dominates.

B.1.2 BASELINE METHODS AND DEPLOYMENT PROTOCOLS

  • Zero-shot baselines apply pretrained models without target-specific detector fitting, while target-fitted baselines learn from each unlabeled target series.TS-Router uses a fixed pretrained representation and routing policy before fitting only selected specialists; comparisons share the constraint of no target anomaly labels.Source: §B.1.2 · pp. 24–25

B.1.3 SPECIALIST POOL

  • TS-Router uses a fixed pool of eleven lightweight specialists spanning distinct inductive biases, and fits only selected specialists on each unlabeled target.The same pool supplies simulated competence supervision and real-target deployment; deployment selects Top-3 candidates.Source: §B.1.3 · pp. 25–26

B.1.4 EVALUATION METRICS

  • Four complementary metrics—VUS-PR, Affiliation-F1, F1T, and Standard-F1—cover threshold-independent range-aware, event-level, and point-wise detection views.VUS-PR also provides the competence utility used to construct routing supervision.Source: §B.1.4 · pp. 26–27
  • F1 scores use evaluation-label oracle thresholds selected separately for each method, target series, and F1 metric, limiting operational threshold-selection claims.Evaluation labels are not used for model or router training, specialist fitting, or routing; VUS-PR is threshold-independent.Source: §B.1.4 · p. 27

B.1.5 IMPLEMENTATION DETAILS

  • The frozen TimeRCD encoder supplies representations for a router trained on 850,000 labeled simulated tasks; at deployment, only selected specialists fit the target prefix.The router and encoder remain frozen at deployment, and selected specialists score the test suffix.Source: §B.1.5 · pp. 27–28

B.2.1 PER-DATASET PERFORMANCE

  • Relative detector strengths vary substantially across datasets and metrics, and no competing detector consistently dominates across heterogeneous targets.This pattern includes both pretrained zero-shot and target-fitted unsupervised methods.Source: §B.2.1 · p. 28

B.2.2 DETAILED REPRESENTATION ABLATION

The representation ablation compares router inputs while holding the router, specialist pool, supervision, and selection pipeline fixed. Results show variation among representations and specialists, while the Top-k and pool-size analyses examine selection-budget and pool-size effects.

B.2.2 DETAILED REPRESENTATION ABLATION

  • With the router and selection pipeline fixed, the representation ablation varies only the input representation, isolating its information for predicting specialist suitability.All variants use the same MLP, eleven-specialist pool, competence supervision, Top-3 selection, and normalized mean fusion.Source: §B.2.2 · p. 28
  • TimeRCD has the best average VUS-PR and Standard-F1, while MOMENT leads Affiliation-F1 and F1T and remains close to TimeRCD on F1T.Handcrafted representations can excel on particular datasets but vary more across temporal domains; Chronos and MOMENT are weaker on VUS-PR and Standard-F1.Source: §B.2.2 · pp. 28–29

B.2.3 SPECIALIST COMPLEMENTARITY

  • Specialist strengths vary substantially by dataset, and no individual specialist consistently dominates across targets.Examples include subspace methods on NEK, neighborhood methods on MGAB and SED, KMeansAD on Power, and SR on Stock and YAHOO.Source: §B.2.3 · p. 30
  • TS-Router scores 56.36, 88.11, 49.62, and 47.73 versus 39.68, 84.34, 41.01, and 36.43 for the best fixed specialist under VUS-PR, Affiliation-F1, F1T, and Standard-F1, respectively.The comparison indicates that the aggregate advantage is not explained by repeatedly selecting one globally strong detector.Source: §B.2.3 · p. 30

B.2.4 EFFECT OF TOP-k SELECTION

  • VUS-PR rises from 49.00 at k = 1, then scores decline as k grows toward 10 after a small selection budget.This sweep evaluates the full selection-and-fusion pipeline; the fixed-k set-competence bound does not predict monotonic fused detection improvement.Source: §B.2.4 · pp. 30–31, Figure 3 · p. 32

B.2.5 EFFECT OF SPECIALIST-POOL SIZE

  • Enlarging the specialist pool from 6 to 11 improves detection with Top-3 selection, retraining the router for each pool size.Figure 4 reports unweighted means across eleven univariate benchmarks for VUS-PR, F1T, Standard-F1, and Affiliation-F1.Source: §B.2.5 · p. 32, Figure 4 · p. 32

B.2.6 ROUTER-HEAD ABLATION

The router-head ablation changes the competence-target metric while holding the representation, specialist pool, selection, and test benchmarks fixed. The F1T head yields the strongest mean scores across the reported detection metrics.

B.2.6 ROUTER-HEAD ABLATION

  • The comparison changes only the competence-target metric; the specialist pool, frozen TimeRCD representation, Top-3 selection, and test set remain fixed.The four heads use VUS-PR, Affiliation-F1, F1T, or Standard-F1 targets.Source: §B.2.6 · p. 32
  • F1T-head routing gives the highest mean score under all five reported detection metrics, including VUS-PR (58.24) and Affiliation-F1 (89.17).The VUS-PR head is close on VUS-PR at 56.36, while the Standard-F1 head has the lowest VUS-PR, VUS-ROC, and Affiliation-F1 values in this comparison.Source: §B.2.6 · p. 32

B.2.7 ADDITIONAL ALGORITHM-SELECTION TRANSFER SETTINGS

The transfer experiments compare TS-Router with classical selectors using synthetic histories, limited labeled in-distribution support, or real histories from other datasets. TS-Router leads the leave-one-dataset-out averages, while support access narrows or reverses some metric gaps.

Synthetic-to-Real Selection

  • With synthetic competence histories and shared Top-3 fusion, TS-Router has the highest average performance under all four detection metrics.Classical selectors remain competitive on particular datasets, but their relative performance varies substantially across targets.Source: §B.2.7 · p. 33, Table 11 · p. 33

In-Distribution Selection with Limited Support

  • With labeled support from the Tuning distribution, MetaOD reaches VUS-PR 55.91 versus TS-Router’s 57.40, while MSAD leads Affiliation-F1, F1T, and Standard-F1.MSAD scores 90.92, 49.76, and 48.86 on those metrics versus TS-Router’s 88.89, 48.79, and 48.01; classical selectors receive roughly 10% support and TS-Router does not adapt.Source: §B.2.7 · p. 34, Table 12 · p. 34

Leave-One-Dataset-Out Transfer

  • The leave-one-dataset-out comparison holds each of eleven benchmarks out in turn; classical selectors train on the other ten datasets, while TS-Router uses its fixed Syn-RCD policy.Classical selectors deploy their top-ranked specialist, whereas TS-Router uses default Top-3 normalized mean fusion without real-data retraining or target-specific router adaptation.Source: §B.2.7 · pp. 34–35, Table 13 · p. 35

B.2.7 ADDITIONAL ALGORITHM-SELECTION TRANSFER SETTINGS

  • TS-Router averages 56.36, 88.11, 49.62, and 47.73 versus the strongest classical-selector averages of 38.43, 80.99, 33.17, and 31.43 under VUS-PR, Affiliation-F1, F1T, and Standard-F1, respectively.In leave-one-dataset-out transfer, classical selectors train on the other ten real collections and deploy Top-1; TS-Router retains its Syn-RCD policy and Top-3 fusion.Source: §B.2.7 · p. 35, Table 13 · p. 35
  • TS-Router’s Router Top-1 variant also exceeds the strongest classical-selector average under each of the four metrics, extending the comparison to single-specialist selection.The reported claim compares against classical-selector averages across metrics.Source: §B.2.7 · p. 35
Loading 2610.00978v1…