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REVIEW 3 major objections 7 minor 42 references

FLDmamba: Integrating Fourier and Laplace Transform Decomposition with Mamba for Enhanced Time Series Prediction

T0 review · 3 major / 7 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read FLDmamba claims that adding Fourier-filtered step sizes and an inverse Laplace output layer to Mamba captures multi-scale periodicity, transient dynamics, and noise robustness, yielding state-of-the-art long-term time series forecasting…

desk verdict A plausible incremental architecture for Mamba-based forecasting, but the SOTA claim rests on a contradictory benchmark protocol and needs a re-run before it is taken seriously. read the letter →

arxiv 2507.12803 v1 pith:KQRT4HZI submitted 2025-07-17 cs.LG cs.AI

Qianru Zhang , Chenglei Yu , Haixin Wang , Yudong Yan , Yuansheng Cao , Siu-Ming Yiu , Tailin Wu , Hongzhi Yin This is my paper
classification cs.LGcs.AI
keywords timeseriesforecastingMambastatespacemodelsFouriertransformLaplacelong-termpredictionnoiserobustness
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

FLDmamba claims that Mamba-based time-series forecasting can be made state-of-the-art by adding two transform stages: a Fourier filter that cleans and separates multi-scale periodic patterns from noise, and an inverse Laplace output layer that represents predictions as damped oscillations, capturing both transient fluctuations and long-term cycles. The paper reports that this design outperforms 12 Transformer-, MLP-, and Mamba-based baselines on 60 of 72 (83.3%) task/dataset combinations across nine real-world benchmarks, while preserving Mamba's linear-time complexity. The practical stake is that long-term forecasting could get both higher accuracy and better efficiency than current attention-based models, and better robustness to noisy data.

What carries the argument

The load-bearing object is the FMamba-Mamba (FMM) block with a Fourier-filtered step size. In standard Mamba, the input-dependent step size $\Delta$ gates information flow through the hidden state; FLDmamba computes $\Delta_F$ by taking the FFT of $\Delta$, multiplying by a learnable kernel $\widetilde{W}$, and applying the inverse FFT, then uses $\Delta_F$ to discretize the state matrices $\bar{A}_F$ and $\bar{B}_F$. The other load-bearing object is the inverse Laplace transform output layer, which writes the prediction as a sum of damped cosines $A_n e^{-\sigma_n t}\cos(\omega_n t + \varphi_n)$, directly representing both transient and periodic structure. An RBF kernel smooths the input before both encoders, and the two encoders run in parallel with their outputs summed, in a residual-block style.

What would settle it

Re-run the 12 baselines under the paper's own protocol (lookback windows of 96/192/336/720, hidden sizes searched over {128,...,2048}, same learning rate schedule) and compare against FLDmamba on the same nine datasets; if the win rate drops below the reported 60 of 72, the claim fails. Alternatively, train FLDmamba with the Fourier filter disabled but every other component intact: if the performance gap does not widen as claimed, the filter mechanism is not doing the work.

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Extended reading notes

Core claim

The paper's central claim is that FLDmamba, by filtering the selective state-space step size $\Delta$ through the Fourier transform and by decoding through an inverse Laplace transform, overcomes three known weaknesses of Mamba for time-series: it captures multi-scale periodicity, it models transient dynamics, and it resists data noise. Concretely, the Fourier-filtered step size $\Delta_F$ replaces the raw $\Delta$ when discretizing the state matrices $\bar{A}_F$ and $\bar{B}_F$, so the selective state space only propagates frequency components that a learnable kernel marks as relevant. The output layer then parameterizes the forecast as $\hat{Y}(t) = \sum M_n A_n e^{-\sigma_n t}\cos(\omega_n t + \varphi_n)$, whose cosine factors encode periodicity and whose exponential factors encode transient decay. The paper further claims that the RBF kernel contributes noise robustness and that the parallel FMamba-Mamba block keeps the overall complexity at $O(BLVN)$.

Load-bearing premise

The state-of-the-art claim depends on the baseline numbers taken from S-Mamba and Moirai papers being comparable to FLDmamba's own runs; if those baselines were tuned or evaluated under a different protocol, the 60-of-72 advantage could be inflated.

Editorial extensions

If this is right

  • Existing Mamba-based forecasters can adopt the Fourier-filtered step size and the inverse Laplace output layer without changing asymptotic running time, since both additions cost at most $O(BLN\log L)$.
  • FLDmamba could serve as a drop-in replacement for Transformer-based long-term forecasters in settings where quadratic attention cost is a bottleneck, if the reported win rate holds up.
  • The reported noise-robustness advantage suggests that deployment on noisy sensor streams could see smaller forecast degradation under data corruption.
  • The damped-cosine output representation gives an interpretable decomposition of a forecast into periodic cycles and decaying transients, which could aid post-hoc analysis.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The paper does not analyze which frequency bands the learned Fourier kernel actually selects; visualizing $\widetilde{W}$ per dataset would reveal whether the gain comes mainly from denoising or from periodicity extraction.
  • The damped-cosine decoder is reminiscent of exponential-sinusoidal decomposition, and a natural extension the authors do not explore is to impose sparsity on the number of terms or to allow the decay rates to vary over time.
  • Because the baseline numbers are quoted from other papers, the decisive external test is a from-scratch re-implementation of all baselines under the identical protocol; until that is done, the 60-of-72 claim should be read as conditional on those quoted numbers.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 7 minor

Summary. The manuscript proposes FLDmamba, a state-space model for long-term multivariate time series forecasting. The architecture combines three ideas: an RBF kernel for input smoothing; a Fourier-filtered discretization step inside a 'FMamba' layer; and an 'inverse Laplace transform' output head that parameterizes a sum of damped cosines. The authors report MSE/MAE on nine benchmark datasets and claim state-of-the-art performance in 60 of 72 cases (83.3%) against 12 Transformer-, MLP-, and SSM-based baselines, along with ablations, robustness tests, efficiency measurements, and hyperparameter studies. Code and data are promised via a GitHub URL.

Significance. If the empirical claims survive a properly controlled comparison, the paper would contribute a competitive and efficient Mamba-based forecasting architecture, and the release of code/data would aid reproducibility. The main strengths are the breadth of benchmarks, the complexity analysis showing linear-time scaling, and the explicit ablation of each component. However, the significance is currently limited by two issues: the 'inverse Laplace transform' is not a mathematical inversion (coefficients are learned), and the headline SOTA claim relies on quoted baseline numbers under an inconsistent tuning protocol without error bars. These are fixable but require substantial revision rather than minor edits.

major comments (3)
  1. [Section III-B5, Eq. (8)] The claimed ILT derivation is not an inversion. Eq. (8) presents Ŷ(t)=Σ A_n e^{-σ_n t} cos(ω_n t+φ_n), which is the form a true inverse Laplace transform would take for first-order poles; however, the authors then state that A_n, σ_n, ω_n, and φ_n are 'directly parameterized as learnable functions of Y(t) from the previous layer.' No constraint links these parameters to the residues β_n and poles μ_n of the Laplace-domain signal, so the module is a learned parametric basis expansion, not the inverse Laplace transform. Consequently, the paper's claim that Laplace analysis enables transient-dynamics capture is not established by the derivation; the 'w/o ILT' ablation demonstrates only that the damped-cosine head helps, not that an ILT is being performed. Please either derive an actual inversion or rename and justify the module as a Laplace-motivated head.
  2. [Section IV-A and Table I caption] The comparison protocol is internally inconsistent. Section IV-A states that 'we modify the hidden dimensionality of all compared algorithms within [128, 256, 512, 1024, 2048] to achieve their reported best performance, which is consistently observed at 1024,' while the Table I caption says 'All baseline results are obtained from [20] and [30].' Numbers quoted from prior papers cannot reflect the authors' hidden-size sweep. Additionally, Section II declares Moirai ([30]) 'out-of-scope for our baselines,' yet Table I credits [30] as a source. Because the 60/72 SOTA claim is the paper's central empirical contribution, the authors must rerun all baselines under the same data splits, lookback/horizon, normalization, and seeds (or clearly state the results are quoted and justify comparability).
  3. [Tables I and III; Figures 4-10] No uncertainty quantification is provided for any table or figure. Many of FLDmamba's reported advantages are small (e.g., ETTm1-96 MSE 0.318 vs S-Mamba 0.333; ETTm2-96 MSE 0.173 vs 0.179; Electricity-336 MSE 0.182 vs S-Mamba 0.176; Electricity average 0.170 vs 0.170). Absent error bars or a significance test, the claim that FLDmamba wins 60/72 cases could reflect seed noise. Please report mean±std over multiple runs and state whether differences are statistically significant.
minor comments (7)
  1. [Section IV-E] The paragraph preceding Figure 6 contains a long run of corrupted unicode sequences ('/uni0000001c/uni00000019/...') that makes the text unreadable; please repair the encoding.
  2. [Algorithm 1 and Section III-A] Algorithm 1 declares the output as Ŷ ∈ (B,L,V), but the problem statement in Section III-A defines predictions as (B,H,V) with horizon H; clarify how the ILT head maps to H time steps.
  3. [Section IV-A] The baseline taxonomy is inaccurate: DLinear is a linear/MLP model, not a Transformer-based method, and TimesNet is a 2D-variation CNN, not an MLP-based model.
  4. [Section III-B5] The text uses ξ_n for the decay parameter while Eq. (8) uses σ_n; standardize the notation.
  5. [Section IV-D] The statement that FLDmamba has '2.1× higher noise tolerance than iTransformer' is not defined; specify the metric and computation.
  6. [Section IV-G and Figure 9] Training time units (seconds per epoch?) and memory sizes are not stated in the figure or text; please specify.
  7. [Introduction, Contributions] The sentence 'FLDmamba achieves state-of-the-art (SOTA) performance on of tasks' is incomplete; please correct to 'on the majority of tasks' or similar.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: FLDmamba's forecasts are produced by a trained parametric composition; the Laplace-based output form is an explicit modeling ansatz, and benchmark claims rest on external comparisons rather than on the model's own outputs.

full rationale

The derivation chain is self-contained in the sense required here. The forecast Y-hat is computed by a learned pipeline: RBF smoothing, an FFT-filtered step-size Delta_F in the FMamba encoder, standard Mamba layers, an FMM block, and finally a damped-cosine output layer (Eq. 8). Eq. (8) is not derived from the ground-truth Y(1); it is a parametric family obtained by stipulating first-order poles in a Laplace-domain kernel and then directly parameterizing A_n, xi_n, w_n, and phi_n as learnable functions of Y(t). That is an explicit modeling ansatz, not a hidden circularity: the coefficients are fit to minimize MSE on real data, and the prediction is a nonlinear function of the input through trained parameters. The Fourier and Laplace components are not renamed versions of the target; they are inductive biases. The paper's self-citations (e.g., refs. 6, 10, 26) are contextual and not load-bearing for the main derivation. The SOTA claim relies on Table I numbers quoted from refs. 20 and 30, and the experimental setup text says both that all compared algorithms were tuned in [128, 2048] and that all baseline methods were evaluated using their predefined settings. That is internally inconsistent and raises a fairness/reproducibility concern, but it is a correctness issue about the baseline comparison, not a reduction of the model's predictions to its inputs by construction. No fitted parameter is presented as an independent prediction, no uniqueness theorem is imported from the authors' prior work, and no ansatz is smuggled via citation. The unusual inserted fragments in the text (e.g., the incomplete contribution sentence and the unicode sequences in the Figure 6 caption) do not alter the derivation chain. Therefore no circular step is identified.

Assumptions & free parameters 5 free parameters · 5 assumptions · 0 invented entities

The model is a deep learning architecture with standard learned weights; the only non-standard tuning choices are the hyperparameters above and the ILT component count, which is unspecified. No new physical entities are introduced.

free parameters (5)
  • Hidden size = 1024
    Selected via hyperparameter study; all model variants scaled to 1024 for 'reported best performance', but this tuning is not reported for baselines.
  • Number of FLDmamba layers = 2
    Chosen from {1,2,3,4,5} based on ETTh1 validation performance.
  • Learning rate = 5e-6
    Chosen from the hyperparameter study; optimal in Figure 8.
  • ILT component count M
    Eq. (8) uses M singularities but the paper never states M or how it is set, leaving the capacity of the damped-cosine output layer unspecified.
  • RBF kernel bandwidth
    The RBF smoothing step is described qualitatively; the kernel width is never given, so the amount of smoothing is a hidden tuning choice.
assumptions (5)
  • standard math Convolution theorem for Fourier transforms
    Used in Eq. (2) to write the kernel integral operator as F^{-1}(W * F(x)).
  • standard math Partial fraction expansion of Laplace-domain kernels with first-order poles
    Used to obtain Eq. (8); valid only for the stipulated K_phi(s) form.
  • domain assumption Finite damped-sinusoid representation of target series
    Eq. (8) assumes the forecast can be expressed as a sum of M damped cosines; no justification is given for real-world data.
  • domain assumption RBF smoothing preserves relevant signal
    RBF kernel is applied before modeling; the paper does not analyze the bias introduced by smoothing.
  • standard math Mamba's selective SSM discretization is valid for this setting
    The model reuses Mamba's A, B, C discretization in Eq. (4)-(6).

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Cite this review

Pith. "Pith review of FLDmamba: Integrating Fourier and Laplace Transform Decomposition with Mamba for Enhanced Time Series Prediction." pith.science (2026). https://pith.science/paper/KQRT4HZI

@misc{pith2026250712803,
  author       = {Pith},
  title        = {Pith review of: FLDmamba: Integrating Fourier and Laplace Transform Decomposition with Mamba for Enhanced Time Series Prediction},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KQRT4HZI}},
  note         = {Machine review of arXiv:2507.12803}
}
read the original abstract

Time series prediction, a crucial task across various domains, faces significant challenges due to the inherent complexities of time series data, including non-stationarity, multi-scale periodicity, and transient dynamics, particularly when tackling long-term predictions. While Transformer-based architectures have shown promise, their quadratic complexity with sequence length hinders their efficiency for long-term predictions. Recent advancements in State-Space Models, such as Mamba, offer a more efficient alternative for long-term modeling, but they cannot capture multi-scale periodicity and transient dynamics effectively. Meanwhile, they are susceptible to data noise issues in time series. This paper proposes a novel framework, FLDmamba (Fourier and Laplace Transform Decomposition Mamba), addressing these limitations. FLDmamba leverages the strengths of both Fourier and Laplace transforms to effectively capture both multi-scale periodicity, transient dynamics within time series data, and improve the robustness of the model to the data noise issue. Our extensive experiments demonstrate that FLDmamba achieves superior performance on time series prediction benchmarks, outperforming both Transformer-based and other Mamba-based architectures. To promote the reproducibility of our method, we have made both the code and data accessible via the following URL:{\href{https://github.com/AI4Science-WestlakeU/FLDmamba}{https://github.com/AI4Science-WestlakeU/\model}.

Figures

Figures reproduced from arXiv: 2507.12803 by the authors.

Figure 2
Figure 2. This diagram illustrates the architecture of FLDmamba, showcasing the individual components and their integration. [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Ablation study of FLDmamba on prediction perfor [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 5
Figure 5. Performance comparison of robustness By comparing the performance of these variants against our full method, FLDmamba, we can isolate the individual contribution of each component to overall performance. The results presented in [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figures from the paper (5 more)
Figure 6
Figure 6. Figure 6: Long-term prediction with the lookback length from [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]
Figure 7
Figure 7. Figure 7: Case study of FLDmamba in terms of transient dy [PITH_FULL_IMAGE:figures/full_fig_p009_7.png]
Figure 8
Figure 8. Figure 8: Hyperparameter study of FLDmamba. AutoFormer RLinear iTransformer S-Mamba Ours 0.35 0.40 0.45 0.50 MSE ETTh1 0 20 40 60 80 Training Time AutoFormer RLinear iTransformer S-Mamba Ours 0.35 0.40 0.45 0.50 MSE ETTh1 0 1000 2000 3000 4000 5000 GPU Memory (MB) AutoFormer RLi…
Figure 9
Figure 9. Figure 9: Model efficiency comparison on ETTh1 and ETTh2. The batch size is 32. [PITH_FULL_IMAGE:figures/full_fig_p010_9.png]
Figure 10
Figure 10. Figure 10: Case study of FLDmamba in terms of multi-scale periodicity. [PITH_FULL_IMAGE:figures/full_fig_p010_10.png]

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.