pith. sign in

arxiv: 2511.15083 · v2 · submitted 2025-11-19 · 💻 cs.LG · eess.SP

Fourier-KAN-Mamba: A Novel State-Space Equation Approach for Time-Series Anomaly Detection

Xiancheng Wang , Lin Wang , Rui Wang , Zhibo Zhang , Minghang Zhao This is my paper

Pith reviewed 2026-05-17 20:11 UTC · model grok-4.3

classification 💻 cs.LG eess.SP
keywords time-series anomaly detectionstate-space modelMambaFourier transformKolmogorov-Arnold Networkhybrid architecturetemporal gating
0
0 comments X

The pith

A hybrid architecture fuses Fourier frequency extraction, Kolmogorov-Arnold networks, and Mamba state-space modeling to detect anomalies in time-series data more effectively than prior approaches.

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

The paper proposes Fourier-KAN-Mamba as a new model for spotting anomalies in time-series data from industrial and sensor sources. It combines a Fourier layer to pull out multi-scale frequency patterns, Kolmogorov-Arnold Networks to represent nonlinear relationships, and the Mamba selective state-space model for efficient long-sequence processing. An added temporal gating step helps separate normal behavior from anomalies. Experiments on the MSL, SMAP, and SWaT datasets show the model beats existing state-of-the-art methods.

Core claim

The Fourier-KAN-Mamba model integrates a Fourier layer for multi-scale frequency features, Kolmogorov-Arnold Networks for stronger nonlinear representation, the Mamba selective state-space model for long-sequence efficiency, and a temporal gating control mechanism to better distinguish normal from anomalous patterns in time-series data.

What carries the argument

The Fourier-KAN-Mamba hybrid architecture, which stacks a Fourier layer to extract frequency features, a KAN module for nonlinear mapping, a Mamba state-space block for sequence modeling, and temporal gating to highlight anomalies.

If this is right

  • Better performance on industrial monitoring and fault diagnosis tasks that rely on sensor time series.
  • More efficient handling of long sequences compared with transformer-based alternatives for anomaly detection.
  • Improved separation of subtle anomalous patterns through the added gating mechanism.
  • Potential reduction in false alarms when deployed in real-time monitoring systems.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same hybrid structure could be tested on related tasks such as time-series forecasting or classification without major redesign.
  • Combining frequency and state-space components this way might extend to other sequential domains like audio or financial data.
  • Further scaling experiments on streaming or very high-dimensional series would clarify practical limits.

Load-bearing premise

The specific combination of Fourier, KAN, Mamba, and gating layers will capture complex temporal patterns and nonlinear dynamics more reliably than existing models across different datasets without needing heavy per-dataset tuning.

What would settle it

Running the model on a new time-series anomaly dataset or under added noise conditions and finding no statistically significant gain over strong baselines would challenge the central claim.

Figures

Figures reproduced from arXiv: 2511.15083 by Lin Wang, Minghang Zhao, Rui Wang, Xiancheng Wang, Zhibo Zhang.

Figure 1
Figure 1. Figure 1: KMA-AD Architecture: (a) Description of the main structure of the KMA-AD model; (b) Sources of the [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Model Hyperparameter Sensitivity Experiments: [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
read the original abstract

Time-series anomaly detection plays a critical role in numerous real-world applications, including industrial monitoring and fault diagnosis. Recently, Mamba-based state-space models have shown remarkable efficiency in long-sequence modeling. However, directly applying Mamba to anomaly detection tasks still faces challenges in capturing complex temporal patterns and nonlinear dynamics. In this paper, we propose Fourier-KAN-Mamba, a novel hybrid architecture that integrates Fourier layer, Kolmogorov-Arnold Networks (KAN), and Mamba selective state-space model. The Fourier layer extracts multi-scale frequency features, KAN enhances nonlinear representation capability, and a temporal gating control mechanism further improves the model's ability to distinguish normal and anomalous patterns. Extensive experiments on MSL, SMAP, and SWaT datasets demonstrate that our method significantly outperforms existing state-of-the-art approaches. Keywords: time-series anomaly detection, state-space model, Mamba, Fourier transform, Kolmogorov-Arnold Network

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper proposes Fourier-KAN-Mamba, a hybrid architecture integrating a Fourier layer for multi-scale frequency features, Kolmogorov-Arnold Networks (KAN) for nonlinear representations, the Mamba selective state-space model, and a temporal gating control mechanism to better capture complex temporal patterns and nonlinear dynamics in time-series anomaly detection. It claims that extensive experiments on the MSL, SMAP, and SWaT datasets show significant outperformance over existing state-of-the-art approaches.

Significance. If the empirical results hold after proper validation, the work could advance efficient long-sequence modeling for anomaly detection by combining frequency-domain extraction and nonlinear function approximation within state-space models, offering potential benefits for industrial monitoring and fault diagnosis applications.

major comments (2)
  1. [Abstract and §4] Abstract and §4: The central claim of significant outperformance on MSL, SMAP, and SWaT is asserted without any reported quantitative metrics, baseline models, error bars, statistical significance tests, hyperparameter search details, or data split information. This renders the empirical superiority unevaluable from the manuscript.
  2. [§4] §4: No ablation studies or internal variants (Mamba-only, Fourier-Mamba, KAN-Mamba, or temporal-gating-ablated) are presented. Without isolating the contribution of each proposed component, performance deltas cannot be attributed to the Fourier-KAN-Mamba integration rather than base Mamba efficiency, training choices, or dataset-specific adjustments.
minor comments (1)
  1. [Title and Abstract] The title refers to a 'Novel State-Space Equation Approach' but the abstract and description focus on an architectural combination; clarify whether a new state-space equation is derived or if the contribution is the hybrid model built on the standard Mamba equations.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive comments, which help strengthen the empirical rigor of our work. We address each major point below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Abstract and §4] Abstract and §4: The central claim of significant outperformance on MSL, SMAP, and SWaT is asserted without any reported quantitative metrics, baseline models, error bars, statistical significance tests, hyperparameter search details, or data split information. This renders the empirical superiority unevaluable from the manuscript.

    Authors: We agree that the abstract and Section 4 would benefit from explicit quantitative details to allow direct evaluation. The full manuscript reports results on the MSL, SMAP, and SWaT benchmarks with comparisons to prior methods, but we will revise the abstract to include key performance numbers and expand Section 4 to explicitly list all baseline models, report error bars or standard deviations across runs, include statistical significance tests (e.g., paired t-tests), detail the hyperparameter search procedure, and specify the train/validation/test splits used. revision: yes

  2. Referee: [§4] §4: No ablation studies or internal variants (Mamba-only, Fourier-Mamba, KAN-Mamba, or temporal-gating-ablated) are presented. Without isolating the contribution of each proposed component, performance deltas cannot be attributed to the Fourier-KAN-Mamba integration rather than base Mamba efficiency, training choices, or dataset-specific adjustments.

    Authors: We concur that ablation studies are necessary to attribute gains to the specific components. The current manuscript emphasizes the integrated model but omits systematic ablations. We will add these experiments in the revised Section 4, including results for Mamba-only, Fourier-Mamba, KAN-Mamba, and the full model without the temporal gating mechanism, to isolate each contribution. revision: yes

Circularity Check

0 steps flagged

No circularity: empirical architecture proposal with external benchmarks

full rationale

The paper introduces a hybrid model (Fourier layer + KAN + Mamba + temporal gating) for time-series anomaly detection and supports its claims via experiments on the public MSL, SMAP, and SWaT datasets. No derivation chain, first-principles equations, or predictions are presented that reduce by construction to fitted inputs or self-citations. The abstract and described contributions are self-contained engineering choices evaluated against independent external benchmarks; performance deltas are not forced by internal re-use of the same fitted quantities. This is the normal, non-circular case for an applied ML architecture paper.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 1 invented entities

The central claim rests on the empirical superiority of the proposed hybrid; the abstract does not enumerate free parameters or axioms, but standard ML practice implies several fitted quantities and domain assumptions about the datasets.

free parameters (2)
  • model architecture hyperparameters
    Number of layers, hidden dimensions, Fourier scales, KAN grid sizes, and gating thresholds are chosen or fitted to achieve the reported results on the three benchmarks.
  • training hyperparameters
    Learning rate, batch size, optimizer settings, and early-stopping criteria are adjusted to produce the claimed performance.
axioms (1)
  • domain assumption The MSL, SMAP, and SWaT datasets contain representative normal and anomalous patterns for the target industrial monitoring tasks.
    The paper evaluates only on these three public datasets and treats their anomaly labels as ground truth.
invented entities (1)
  • temporal gating control mechanism no independent evidence
    purpose: To improve distinction between normal and anomalous patterns by dynamically controlling information flow in the Mamba backbone.
    Introduced as an additional component on top of the Fourier-KAN-Mamba stack; no independent falsifiable prediction (such as a predicted effect size on a new dataset) is supplied.

pith-pipeline@v0.9.0 · 5468 in / 1554 out tokens · 29260 ms · 2026-05-17T20:11:17.246277+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

25 extracted references · 25 canonical work pages · 1 internal anchor

  1. [1]

    Cloud-based industrial cyber–physical system for data-driven reasoning: A review and use case on an industry 4.0 pilot line,

    A. Villalonga, G. Beruvides, F. Castano, and R. E. Haber, “Cloud-based industrial cyber–physical system for data-driven reasoning: A review and use case on an industry 4.0 pilot line,”IEEE Transactions on Industrial Informatics, vol. 16, no. 9, pp. 5975–5984, 2020

    work page 2020

  2. [2]

    Abnormal vibration signal detection of emu motor bearings based on vmd and deep learning,

    Y. Cui, W. Zhang, and Z. Wang, “Abnormal vibration signal detection of emu motor bearings based on vmd and deep learning,”Sensors, vol. 25, no. 18, p. 5733, 2025

    work page 2025

  3. [3]

    Vibration source inversion-based fault diagnosis: Approach and application,

    Z. Bi, X. Yu, Y. Huangfu, J. Yao, P. Zhou, Q. He, and Z. Peng, “Vibration source inversion-based fault diagnosis: Approach and application,”Journal of Sound and Vibration, vol. 597, p. 118818, 2025

    work page 2025

  4. [4]

    Vibration-based bearing fault diagnosis of high-speed trains: A literature review,

    W. Hu, G. Xin, J. Wu, G. An, Y. Li, K. Feng, and J. Antoni, “Vibration-based bearing fault diagnosis of high-speed trains: A literature review,”High-speed Railway, vol. 1, no. 4, pp. 219–223, 2023

    work page 2023

  5. [5]

    Unsupervised image anomaly detection and segmentation based on pretrained feature mapping,

    Q. Wan, L. Gao, X. Li, and L. Wen, “Unsupervised image anomaly detection and segmentation based on pretrained feature mapping,”IEEE Transactions on Industrial Informatics, vol. 19, no. 3, pp. 2330–2339, 2022

    work page 2022

  6. [6]

    Deep anomaly detection for time series: A survey,

    X. Jia, P. Xun, W. Peng, B. Zhao, H. Li, and C. Shen, “Deep anomaly detection for time series: A survey,” Computer Science Review, vol. 58, p. 100787, 2025

    work page 2025

  7. [7]

    Anomaly detection in univariate time-series: A survey on the state-of-the-art,

    M. Braei and S. Wagner, “Anomaly detection in uni- variate time-series: A survey on the state-of-the-art,” arXiv preprint arXiv:2004.00433, 2020

    work page arXiv 2004

  8. [8]

    Transformers in time series: A survey,

    Q. Wen, T. Zhou, C. Zhang, W. Chen, Z. Ma, J. Yan, and L. Sun, “Transformers in time series: A survey,” arXiv preprint arXiv:2202.07125, 2022

    work page arXiv 2022

  9. [9]

    Anomaly transformer: Time series anomaly detection with association discrepancy

    J. Xu, H. Wu, J. Wang, and M. Long, “Anomaly trans- former: Time series anomaly detection with associa- tion discrepancy,”arXiv preprint arXiv:2110.02642, 2021

    work page arXiv 2021

  10. [10]

    Mamba adaptive anomaly transformer with association discrepancy for time series,

    A. Z. Sellam, I. Benaissa, A. Taleb-Ahmed, L. Patrono, and C. Distante, “Mamba adaptive anomaly transformer with association discrepancy for time series,”Engineering Applications of Artificial 10 Intelligence, vol. 160, p. 111685, 2025. [Online]. Available: https://www.sciencedirect.com/science/ article/pii/S0952197625016872

    work page 2025

  11. [11]

    Decomposition-based multi- scale transformer framework for time series anomaly detection,

    W. Zhang and C. Luo, “Decomposition-based multi- scale transformer framework for time series anomaly detection,”Neural Networks, vol. 187, p. 107399, 2025

    work page 2025

  12. [12]

    Patchtrad: A patch-based transformer focusing on patch-wise re- construction error for time series anomaly detection,

    S.-M. Vilhes, G. Gasso, and M. Z. Alaya, “Patchtrad: A patch-based transformer focusing on patch-wise re- construction error for time series anomaly detection,” arXiv preprint arXiv:2504.08827, 2025

    work page arXiv 2025

  13. [13]

    Online time series anomaly detection with state space gaussian processes,

    C. Bock, F.-X. Aubet, J. Gasthaus, A. Kan, M. Chen, and L. Callot, “Online time series anomaly detection with state space gaussian processes,”arXiv preprint arXiv:2201.06763, 2022

    work page arXiv 2022

  14. [14]

    Joint selective state space model and de- trending for robust time series anomaly detection,

    J. Chen, X. Tan, S. Rahardja, J. Yang, and S. Ra- hardja, “Joint selective state space model and de- trending for robust time series anomaly detection,” IEEE Signal Processing Letters, 2024

    work page 2024

  15. [15]

    Time-ssm: Simplifying and unifying state space models for time series forecasting,

    J. Hu, D. Lan, Z. Zhou, Q. Wen, and Y. Liang, “Time-ssm: Simplifying and unifying state space models for time series forecasting,”arXiv preprint arXiv:2405.16312, 2024

    work page arXiv 2024

  16. [16]

    Interpretable recurrent varia- tional state-space model for fault detection of com- plex systems based on multisensory signals,

    M. Ma and J. Zhu, “Interpretable recurrent varia- tional state-space model for fault detection of com- plex systems based on multisensory signals,”Applied Sciences, vol. 14, no. 9, p. 3772, 2024

    work page 2024

  17. [17]

    KAN-AD: Time series anomaly detection with Kolmogorov-Arnold Networks,

    Q. Zhou, C. Pei, F. Sun, J. Han, Z. Gao, D. Pei, H. Zhang, G. Xie, and J. Li, “Kan-ad: time series anomaly detection with kolmogorov-arnold networks,” arXiv preprint arXiv:2411.00278, 2024

    work page internal anchor Pith review arXiv 2024

  18. [18]

    A comprehensive survey of deep transfer learning for anomaly detection in industrial time series: Methods, applications, and directions,

    P. Yan, A. Abdulkadir, P.-P. Luley, M. Rosenthal, G. A. Schatte, B. F. Grewe, and T. Stadelmann, “A comprehensive survey of deep transfer learning for anomaly detection in industrial time series: Methods, applications, and directions,”IEEE Access, vol. 12, pp. 3768–3789, 2024

    work page 2024

  19. [19]

    Time series anomaly de- tection in vehicle sensors using self-attention mech- anisms,

    Z. Zhang, Y. Yao, W. Hutabarat, M. Farnsworth, D. Tiwari, and A. Tiwari, “Time series anomaly de- tection in vehicle sensors using self-attention mech- anisms,”IEEE Transactions on Intelligent Trans- portation Systems, vol. 25, no. 11, pp. 15964–15976, 2024

    work page 2024

  20. [20]

    Tmanomaly: Time-series mutual adversarial net- works for industrial anomaly detection,

    L. Zhang, W. Bai, X. Xie, L. Chen, and P. Dong, “Tmanomaly: Time-series mutual adversarial net- works for industrial anomaly detection,”IEEE Trans- actions on Industrial Informatics, vol. 20, no. 2, pp. 2263–2271, 2024

    work page 2024

  21. [21]

    Time-series-based anomaly de- tection in industrial control systems using generative adversarial networks,

    C. Han and G. Gim, “Time-series-based anomaly de- tection in industrial control systems using generative adversarial networks,”Processes, vol. 13, no. 9, p. 2885, 2025

    work page 2025

  22. [22]

    A data-efficient active learning architecture for anomaly detection in industrial time series data,

    D. Holtz, C. Kaymakci, D. Leuthe, S. Wenninger, and A. Sauer, “A data-efficient active learning architecture for anomaly detection in industrial time series data,” Flexible Services and Manufacturing Journal, pp. 1– 32, 2025

    work page 2025

  23. [23]

    Timeseriesbench: An industrial-grade benchmark for time series anomaly detection models,

    H. Si, J. Li, C. Pei, H. Cui, J. Yang, Y. Sun, S. Zhang, J. Li, H. Zhang, J. Hanet al., “Timeseriesbench: An industrial-grade benchmark for time series anomaly detection models,” in2024 IEEE 35th International Symposium on Software Reliability Engineering (IS- SRE). IEEE, 2024, pp. 61–72

    work page 2024

  24. [24]

    Anomalydetectioninsmartmanufacturing: Anadap- tive adversarial transformer-based model,

    M. Orabi, K. P. Tran, P. Egger, and S. Thomassey, “Anomalydetectioninsmartmanufacturing: Anadap- tive adversarial transformer-based model,”Journal of Manufacturing Systems, vol. 77, pp. 591–611, 2024

    work page 2024

  25. [25]

    Unsupervised sig- nal anomaly transformer method: Achieving bearing life anomaly detection without the need for failure samples,

    P. Yu, M. Ping, J. Ma, and J. Cao, “Unsupervised sig- nal anomaly transformer method: Achieving bearing life anomaly detection without the need for failure samples,”Engineering Applications of Artificial In- telligence, vol. 136, p. 108940, 2024. 11

    work page 2024