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REVIEW 5 major objections 4 minor 42 references

Auto-Regressive Moving Diffusion Models for Time Series Forecasting

T0 review · 5 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read ARMD replaces Gaussian noise with a deterministic slide from future to past, and reports state-of-the-art forecasting across 12 of 14 benchmark settings.

desk verdict ARMD is a clever deterministic sliding-window iterative forecaster wearing a diffusion costume, and the costume does not fit: the borrowed DDIM reverse update is not a valid inverse of the forward sliding chain. read the letter →

arxiv 2412.09328 v1 pith:S2SJHQMJ submitted 2024-12-12 cs.LG cs.AIstat.AP

Jiaxin Gao , Qinglong Cao , Yuntian Chen This is my paper · ORCID
classification cs.LGcs.AIstat.AP
keywords diffusionmodelstimeseriesforecastingauto-regressivemovingaverageslidingmultivariatedeterministicsampling
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

This paper argues that standard diffusion-based forecasting is misaligned with the continuous, sequential nature of time series because it corrupts the data into Gaussian noise and conditions on history. It proposes ARMD, a diffusion model whose forward process deterministically slides the future series toward the historical series, so that intermediate states are meaningful intermediate windows. The reverse process, a linear devolution network, reverses this slide and makes forecasting an unconditional, deterministic sampling procedure. Across seven multivariate benchmarks, ARMD wins 12 of 14 settings against diffusion-based baselines and also outperforms strong non-diffusion forecasters, while cutting training and inference time by more than tenfold.

What carries the argument

The central mechanism is a sliding-based forward process that replaces noise addition with a deterministic shift: $X^t_{1-t:T-t} = \mathrm{Slide}(X^0_{1:T}, t)$, and the evolution trend $z_t$ is defined post hoc by Equation (3) as the discrepancy between the slid window and the future target under the DDPM scaling. The reverse sampling step adapts the DDIM update (Equation 8) to this trend and removes the stochastic noise term, making sampling deterministic. The other load-bearing component is the linear devolution network $R(.)$, which predicts the distance $D$ from the current window to the target and combines it with the input through a time-dependent weight $W(t)$.

What would settle it

Train the same linear devolution network to predict the future series directly in a single step, bypassing the iterative sampling; if single-step prediction matches or beats the multi-step ARMD results on identical benchmarks, then the diffusion or iterative-refinement machinery is not what drives the reported gains. Alternatively, apply the forward slide to a synthetic series and check whether the DDIM update exactly recovers the previous sliding state at an intermediate step; any mismatch means the reverse process is not a faithful inverse.

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Extended reading notes

Core claim

ARMD is introduced as the first continuous sequential diffusion-based time series forecasting model. The future series $X^0_{1:T}$ is the initial diffusion state, the historical series $X^T_{-T+1:0}$ is the final state, and each intermediate state $X^t_{1-t:T-t}$ is obtained by sliding the future window $t$ steps toward the past. The reverse process adapts the DDIM sampling update, replacing predicted noise with a predicted evolution trend $\hat{z}(t,\theta)$ that is defined through the DDPM interpolation formula. The devolution network is a linear module that estimates the distance from the current window to the target and combines it with the input via a time-dependent weight, initialized from the DDPM coefficients $\bar{\alpha}_t$ and trained end-to-end. The paper reports that this design aligns diffusion with the natural evolution of time series, yielding a faster, more stable, and more accurate forecaster than existing noise-based diffusion models.

Load-bearing premise

The sampling update borrowed from DDIM is assumed to correctly reverse the deterministic sliding forward process, even though the forward process is not noise-based and the trend term $z_t$ is defined after the fact rather than derived from a generative model.

Editorial extensions

If this is right

  • If ARMD's claims hold, diffusion-based forecasting no longer requires stochastic sampling: forecasts are deterministic and reproducible, and sampling steps can be skipped to reduce inference cost.
  • The reported tenfold reduction in training and inference time would make diffusion-based forecasting practical for long series and high-dimensional multivariate data.
  • The ablation results indicate that the sliding-based intermediate states, the distance-based linear devolution, and the absence of sampling noise each contribute to the reported performance gains.
  • The alignment of diffusion states with actual time-series evolution suggests a design principle: the diffusion path should match the semantics of the data domain rather than being a generic noise-corruption schedule.
  • The unconditional formulation removes the need for a separate conditioning network, simplifying the forecasting pipeline while improving stability.

Reading between the lines

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

  • A natural extension the paper does not take: the same sliding-diffusion idea could transfer to other ordered-sequence domains such as language or video, where past and future have a clear temporal ordering, though the paper does not demonstrate this.
  • A reader should be cautious that the state-of-the-art claim is inseparable from the strength of the linear backbone; the paper's ablations do not isolate whether the iterative diffusion step itself adds accuracy over a single direct linear prediction.
  • Because the forward process is deterministic and sampling noise is removed, ARMD does not naturally provide calibrated probabilistic forecasts; a user requiring uncertainty quantification would need to extend the method, which the paper does not address.
  • The post hoc definition of the trend term $z_t$ means the diffusion is effectively a deterministic interpolation schedule; the paper's framing as a true denoising process is an interpretation rather than a derived generative model.
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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

5 major / 4 minor

Summary. The paper proposes Auto-Regressive Moving Diffusion (ARMD), a deterministic diffusion-style model for multivariate time series forecasting. The forward process slides a future window toward a historical window over T steps, and a residual variable z_t is defined so that the state X^t satisfies a DDPM-like scaling equation. A linear network is trained to predict z_t from the current state, and sampling is performed by iterating a DDIM-style update, starting from the historical series and ending at the predicted future series. The paper claims state-of-the-art results on seven benchmarks, reports a large speed-up relative to noise-based diffusion forecasters, and identifies ARMD as the first continuous sequential diffusion-based TSF model.

Significance. If the method and its theoretical framing were sound, ARMD would be a valuable contribution: it is simple, fast, deterministic, and the released code would facilitate reproduction. The reported 10x+ training and inference speed-ups over diffusion baselines are practically relevant, and the empirical wins in Table 1 are substantial. However, the paper's central claim to be a diffusion model is currently not supported by the derivations: the forward process in Eqs. (2)-(3) is a deterministic re-labeling, and the reverse update in Eq. (8) is not a valid inverse of the forward process even with a perfect network. The experimental evidence also lacks error bars and uses an asymmetric tuning protocol. These issues are load-bearing for the paper's novelty and for its "state-of-the-art diffusion model" claim.

major comments (5)
  1. [Forward Diffusion (Evolution) of ARMD, Eqs. (2)-(3)] Equation (3) defines z_t as the residual that makes Eq. (2) an identity: z_t is a deterministic function of X^t and X^0, so the forward process contains no stochastic diffusion and no learned representation of noise. The training target in Eq. (7) is therefore a deterministic regression target rather than a denoising objective. This does not by itself invalidate the method, but it means the claimed analogy to DDPM and the "continuous sequential diffusion" novelty must be justified by a derived forward/reverse pair, not by an algebraic re-parametrization.
  2. [Sampling/Forecasting of ARMD, Eqs. (8)-(9)] Equation (9) is not the inverse of Eq. (1) even with an oracle predictor. Substituting \hat z(t,\theta)=z_t and \sigma_t=0 into Eq. (9) yields X^{t-1}_{rec} = \sqrt{\bar\alpha_{t-1}} X^0 + \sqrt{1-\bar\alpha_{t-1}} z_t, whereas the true forward state is X^{t-1} = \sqrt{\bar\alpha_{t-1}} X^0 + \sqrt{1-\bar\alpha_{t-1}} z_{t-1}. For sliding windows, z_t \neq z_{t-1} in general, so the update cannot invert the forward process even with a perfect network. Thus Algorithm 2 is not a principled DDIM/DDPM denoising chain. The authors need to derive a valid reverse transition from the forward process, or prove that the iterative refinement converges to X^0; otherwise the diffusion-based framing and the "first continuous sequential diffusion" claim are unsupported.
  3. [Supplement, Experiment Details] The supplement states that ARMD's sampling steps are selected by grid search on the validation set from {1,2,3,4,6,8,12}, while all other diffusion baselines use the default sampling steps of 100. This is an asymmetric comparison: ARMD is tuned on validation, whereas the baselines are not given the same tuning opportunity. Since Tables 1 and 2 report no error bars or confidence intervals despite averaging over 10 sampling runs, the claim of "significantly outperforming" existing models is not statistically supported. Please report standard deviations or intervals and use a comparable tuning budget for all methods.
  4. [Table 2 and experimental protocol] The comparison with non-diffusion TSF models in Table 2 is mostly sourced from the iTransformer paper rather than from a common experimental protocol. Different data splits, normalization, and hyperparameter choices can change MSE/MAE materially. The authors should either re-run all baselines under the same protocol or clearly annotate the source of each number and justify comparability.
  5. [Ablation Studies, Table 4] The claim that the linear backbone outperforms the Transformer backbone is weakened by Table 4: the Transformer backbone achieves better MSE on ETTh2, ETTm2, and Stock, and is close on Solar and Exchange. The aggregate "11 out of 14" count does not establish robustness without significance testing or an analysis of when the linear backbone is preferable. Please temper the claim or provide statistical support.
minor comments (4)
  1. [Notation, Eqs. (2)-(3)] The vector addition in Eq. (2) is ambiguous because X^t_{1-t:T-t} and X^0_{1:T} have different index ranges. Please specify the alignment convention used in the implementation.
  2. [Equation numbering] The main text refers to "Equation (14)" for the DDPM forward formula, but Eq. (14) appears in the supplemental material; please renumber or cite the supplement explicitly.
  3. [Table 3] Table 3 reports training and inference time in seconds but does not state whether training time is per epoch, per 2,000 iterations, or a full run; please clarify the measurement and include units for both phases.
  4. [Supplement, Hyper-parameters] The hyper-parameters b, c, and d in Eq. (5) are selected by grid search per dataset, but the selected values are not reported. Please provide the chosen values and a sensitivity analysis.

Circularity Check

2 steps flagged · score 6.0 of 10

The ARMD 'diffusion' is a reparameterized direct forecaster: Equation (3) defines z_t as the residual that makes Equation (2) hold, so predicting z_t is equivalent to predicting X^0, while the DDIM sampling step is imported without a valid inverse.

  1. self definitional [Section 'Forward Diffusion (Evolution) of ARMD', Equations (2) and (3)]
    "X t 1−t:T −t = Slide(X 0 1:T , t) = √¯αtX 0 1:T + √ 1 − ¯αtzt, (2) ... Given that each time step of X t 1−t:T −t is deterministic, zt can be calculated as: zt = ( r 1 ¯αt X t 1−t:T −t − X 0 1:T )/ r 1 ¯αt − 1. (3)"

    The intermediate state X^t is already defined by Eq. (1) as a sliding window of the target series X^0. Equation (2) is not an independent diffusion relation; it is an identity because Eq. (3) defines z_t as the residual that makes it hold. Thus the 'evolution trend' is constructed from the very future series the model must forecast, rather than being an independent noise or trend variable. Substituting Eq. (6) into the loss Eq. (7) shows that |z_t - z_hat| is a fixed rescaling of |X^0 - X_hat^0|: training to predict the trend is exactly training to regress the future directly. The claimed continuous sequential diffusion mechanism is therefore a reparameterized direct forecaster, not a generative diffusion process.

  2. ansatz smuggled in via citation [Section 'Sampling/Forecasting of ARMD', Equations (8) and (9)]
    "The method follows the sampling approach from DDIM (Song, Meng, and Ermon 2020), replacing the predicted noise ϵθ(xt, t) with the predicted evolution trend ˆz(t, θ)."

    DDIM's update rule is derived for a single noise realization that satisfies x_t = sqrt(alpha_t)x_0 + sqrt(1-alpha_t)eps for every t in the deterministic ODE. In ARMD, z_t changes with t by Eq. (3), so even an oracle predictor z_hat(t,theta)=z_t does not invert the forward slide: Eq. (9) yields sqrt(alpha_{t-1})X^0 + sqrt(1-alpha_{t-1})z_t, whereas the true next state is sqrt(alpha_{t-1})X^0 + sqrt(1-alpha_{t-1})z_{t-1}. The sampling loop is asserted by citation to DDIM rather than derived from the forward process; the diffusion interpretation is an imported ansatz, not a consequence of the paper's own equations.

full rationale

The empirical SOTA claim is not circular: it is checked against external benchmark datasets and independent baseline implementations, so the raw MSE/MAE results carry independent content. The circularity is in the framework itself. Equation (3) defines the 'evolution trend' as the residual that makes Eq. (2) hold, so the forward process is an identity rather than a generative diffusion step; the training objective (7) reduces to direct regression of the future series. The reverse sampler then imports DDIM's equation with z_t substituted for epsilon, without proving that the imported update inverts the sliding forward process; for an oracle predictor it provably does not. These two moves make the 'first continuous sequential diffusion-based TSF' claim a relabeling of a deterministic sliding-window regression, although the numerical results could still be valid. Because the claimed diffusion mechanism is partially circular while the benchmark evidence is independent, a score of 6 is appropriate. Note also that ARMD's sampling steps are tuned on the validation set while baseline diffusion models use a default of 100 steps; that is a comparison-fairness concern, not a circularity.

Assumptions & free parameters 3 free parameters · 5 assumptions · 1 invented entities

The model relies on a handful of tuned hyperparameters (b, c, d, sampling steps) and a trainable step-weight schedule, plus several domain assumptions about the validity of treating deterministic sliding as diffusion and borrowing DDIM updates. The only invented latent is z_t, which is a data-derived residual with no external evidence.

free parameters (3)
  • b, c, d in Equation (5) = not reported, grid-searched per dataset
    Selected from {1,1.5,2}, {-1,-0.5,0.5,1}, and {0.3,0.5,1} per dataset; these balance the linear distance prediction D and input X^t.
  • Sampling interval / number of steps = chosen from {1,2,3,4,6,8,12} via grid search
    Selected on the validation set for each dataset; this controls the number of iterative refinement steps and affects inference quality.
  • Weight schedule W(t) = initialized from alpha_bar_t and updated during training
    A trainable scalar function of the diffusion step t that controls the weighting in Equation (5); its trained values are not reported.
assumptions (5)
  • domain assumption The forward process can be written as X^t = sqrt(alpha_bar_t) X^0 + sqrt(1 - alpha_bar_t) z_t with z_t defined as in Equation (3).
    This is an identity rather than a physical law; it defines z_t to fit the equation. The paper uses this to treat sliding as diffusion but provides no independent justification that z_t acts like noise.
  • domain assumption The DDIM sampling update (Equation 8) is a valid reverse process for the deterministic sliding chain.
    The paper borrows the DDIM update without proof, and because the forward process is deterministic and non-stochastic, the usual derivation of the reverse step does not apply.
  • domain assumption The connection to ARMA justifies the linear devolution network and the sliding chain.
    The supplemental connection is analogical; no equations from ARMA are used in the model derivation.
  • domain assumption Intermediate windows created by sliding are informative targets, and the network trained on them generalizes to synthetic windows at inference.
    The paper adds small deviations to mitigate distribution shift but provides no theoretical guarantee of generalization.
  • ad hoc to paper The diffusion schedule alpha_bar_t, and the deviation schedule equal to it, is a suitable choice for weighting and loss scaling.
    The schedule is taken from DDPM without adaptation to time series; its role is to scale the target z_t and the update, and the paper does not study sensitivity to this choice.
invented entities (1)
  • Evolution trend z_t
    purpose: Acts as the training target replacing noise and as the quantity predicted by the devolution network.
    z_t is defined by Equation (3) as a scaled residual between the slid window and the future series. It is a reparameterization of the data, not an independently observed or predicted quantity.

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Cite this review

Pith. "Pith review of Auto-Regressive Moving Diffusion Models for Time Series Forecasting." pith.science (2026). https://pith.science/paper/S2SJHQMJ

@misc{pith2026241209328,
  author       = {Pith},
  title        = {Pith review of: Auto-Regressive Moving Diffusion Models for Time Series Forecasting},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/S2SJHQMJ}},
  note         = {Machine review of arXiv:2412.09328}
}
read the original abstract

Time series forecasting (TSF) is essential in various domains, and recent advancements in diffusion-based TSF models have shown considerable promise. However, these models typically adopt traditional diffusion patterns, treating TSF as a noise-based conditional generation task. This approach neglects the inherent continuous sequential nature of time series, leading to a fundamental misalignment between diffusion mechanisms and the TSF objective, thereby severely impairing performance. To bridge this misalignment, and inspired by the classic Auto-Regressive Moving Average (ARMA) theory, which views time series as continuous sequential progressions evolving from previous data points, we propose a novel Auto-Regressive Moving Diffusion (ARMD) model to first achieve the continuous sequential diffusion-based TSF. Unlike previous methods that start from white Gaussian noise, our model employs chain-based diffusion with priors, accurately modeling the evolution of time series and leveraging intermediate state information to improve forecasting accuracy and stability. Specifically, our approach reinterprets the diffusion process by considering future series as the initial state and historical series as the final state, with intermediate series generated using a sliding-based technique during the forward process. This design aligns the diffusion model's sampling procedure with the forecasting objective, resulting in an unconditional, continuous sequential diffusion TSF model. Extensive experiments conducted on seven widely used datasets demonstrate that our model achieves state-of-the-art performance, significantly outperforming existing diffusion-based TSF models. Our code is available on GitHub: https://github.com/daxin007/ARMD.

Figures

Figures reproduced from arXiv: 2412.09328 by the authors.

Figure 1
Figure 1. The comparison between (a) existing diffusion [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Illustration of the diffusion process in ARMD. During the forward diffusion process, the future series is progressively [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. The distributions of 10 different predictions made [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: Prediction showcases of ARMD. pendencies between these tokens. The available code can be found at https://github.com/yuqinie98/patchtst. DLinear (Zeng et al. 2023): DLinear employs a linear model to outperform most of Transformer-based models in TSF. Their publicly ava…

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