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REVIEW 3 major objections 5 minor 52 references

MillGNN: Learning Multi-Scale Lead-Lag Dependencies for Multi-Variate Time Series Forecasting

T0 review · 3 major / 5 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read A graph network that learns multi-scale lead-lag dependencies between time series can improve multivariate forecasting.

desk verdict A well-engineered new architecture for multi-scale lead-lag forecasting, but the empirical case lacks error bars and the sibling-pair filter makes the 'comprehensive' claim too strong. read the letter →

arxiv 2509.03852 v1 pith:T6VGRMRD submitted 2025-09-04 cs.LG cs.AI

Binqing Wu , Zongjiang Shang , Jianlong Huang , Ling Chen This is my paper
classification cs.LGcs.AI
keywords multivariatetimeseriesforecastinglead-lagdependenciesgraphneuralnetworksmulti-scalegroupingFFTcross-correlationdecay-awareattentionhierarchicalmessagepassing
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

Time series often influence each other after a delay: one sensor reading helps predict another sensor hours later. Most forecasting models either align all series at the same time step or model delays only between individual series, missing delays that operate between groups of series. MillGNN is a graph-neural-network forecaster that learns these lead-lag dependencies at several grouping scales at once. It builds a hierarchy of variate groups from similarity, then per-scale graphs whose edges are FFT-based cross-correlations refined by a decay-aware attention, and finally passes messages both within and across scales. On 11 datasets against 16 state-of-the-art methods, including traffic, electricity, and air-quality data, it reports the best forecasts overall, with the largest margins on data with strong spatial propagation.

What carries the argument

The central objects are the scale-specific lead-lag graph and the hierarchical message-passing scheme. A lead-lag dependency is a delayed effect where one series' past helps predict another series' future. Each scale keeps its own directed graph whose nodes are patches of variates or variate groups; edges are seeded by FFT-based cross-correlation (the 'statistical interpretability') and reweighted by a decay-aware attention mechanism with excitatory and inhibitory rates (the 'data-driven flexibility'). The hierarchical message passing then aggregates fine-grained messages within the same group at scale 0 and coarsened messages from parent groups at higher scales down to the variates, which i

What would settle it

Take a dataset where the only predictive relationship is a delayed effect between two variates that DTW clustering places in different groups. If MillGNN cannot learn that relationship while a version that allows cross-cluster edges can, the static grouping restriction is the limiting factor. More practically, compare test-time error on ETTh2 after re-estimating the lag distribution and updating the graphs versus keeping the training-time graphs fixed.

Watch

Extended reading notes

Core claim

MillGNN claims to be the first forecaster to learn and exploit multi-scale lead-lag dependencies for multivariate time series forecasting. The central claim is that delays exist not only between individual variates but also between variate groups of different sizes, and that capturing both yields material gains. The model computes FFT-based cross-correlations to seed a directed lead-lag graph at each scale, then reweights those edges with a decay-aware attention mechanism that models how influence weakens over time, and then propagates messages within each scale and from coarser groups down to individual variates. The reported result is that MillGNN outperforms 16 baselines on 11 real-world

Load-bearing premise

The model only searches for delayed effects between variates that the training set already clusters together as similar, and that cluster structure never changes at test time; any delay that crosses clusters or shifts over time is invisible to it.

Editorial extensions

If this is right

  • If the central claim holds, forecasting models should stop assuming simultaneous alignment: delays should be searched at both variate and group levels.
  • On datasets with strong spatial propagation, like traffic flow and air pollution, the hierarchical multi-scale lead-lag modeling is the reported source of gains; removing the hierarchy degrades accuracy.
  • The decay-aware reweighting is load-bearing: ablation shows that removing either the cross-correlation initialization or the dynamic weights hurts performance, with the largest drop on air-quality data.
  • The claimed O(SLN) complexity makes the multi-scale message passing affordable on large variate counts, supported by experiments on 862-variate traffic data.
  • The model's weaker result on ETTh2 indicates the approach assumes lag distributions stay similar between training and test.

Reading between the lines

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

  • A testable extension: monitor the learned time-varying attention weights as a lag-drift alarm; when the lag distribution shifts, re-estimating cross-correlations or re-clustering online could recover accuracy.
  • Because the hierarchy is frozen after training, any important delayed relation between two variates that DTW clustering marks as dissimilar is structurally invisible; a synthetic experiment injecting a lead-lag pair across clusters would reveal this limitation directly.
  • The top-K lag selection in the lead-lag graph can be read as a lag-specific form of channel attention, suggesting that combining it with time-aligned channel-mixing models might generalize beyond the reported baselines.
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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 / 5 minor

Summary. MillGNN is a graph-neural-network method for multivariate time series forecasting whose stated contribution is to learn lead-lag dependencies at multiple grouping scales. The method builds a hierarchical clustering of variates from training-set DTW similarities, initializes lead-lag graphs at each scale using FFT-based cross-correlation between sibling groups, refines those graphs with a decay-aware attention mechanism, and then propagates messages within and across scales via hierarchical aggregation and update operations. The paper reports experiments on 11 public datasets against 16 baselines, with ablations, a hyperparameter study, an efficiency study, and a case study on China-AQI.

Significance. The idea of explicitly modeling lead-lag dependencies at multiple grouping scales is a meaningful extension of single-scale lead-lag work such as LIFT and VCformer, and the combination of statistically motivated FFT cross-correlation with learnable decay-aware attention is a plausible and interpretable design. The hierarchical message-passing scheme with duplication is a sensible attempt to keep multi-scale propagation efficient, and the ablations in Table 4 attribute concrete value to each component. However, the central claim of 'comprehensive' multi-scale lead-lag modeling is weakened by a structural restriction in the candidate graph construction, and the stated complexity bound is not established for the configurations actually run. The empirical case is also presented without error bars or significance tests, despite several improvements over strong baselines being small. With targeted revisions, the core idea could be a solid contribution, but in its current form the manuscript overclaims both capacity and empirical support.

major comments (3)
  1. [Sec. 4.2.1, Eq. (10), Sec. 4.1] The candidate filter in Sec. 4.2.1 restricts C^s to group pairs sharing the same parent at scale s+1. Since A^s = C^s ⊙ Λ^s, all cross-cluster pairs have zero weight at every scale. Combined with the train-time-only DTW hierarchy (Sec. 4.1), any lead-lag relationship between variates or groups in different branches of the fixed hierarchy is structurally unreachable, including the long-range 'regional blackout' scenario motivating Sec. 1. The ETTh2 degradation (Sec. 5.2, Fig. 4) is a symptom of this frozen skeleton. This undercuts the 'comprehensive multi-scale lead-lag dependency' claim. Please either generalize candidate selection beyond siblings (e.g., top-K over all pairs or a learned sparsity) or explicitly reframe the contribution and provide evidence about whether cross-cluster lead-lag effects are needed for the reported gains.
  2. [Sec. 4.3.2, Proposition proof] The O(SLN) proof assumes a uniform group size k and a single top-level group (S = log_k N + 1). The hyperparameter settings in Sec. 5.1 contradict this: for Traffic, Electricity, PEMS04/08, and China-AQI, scale-2 group counts are 1–3 and scale-1 counts are 5–25, so the top level is not a single group and per-group sizes are not a constant k. Consequently, the claimed linear-in-N complexity is not established for the configurations actually used. Please provide a complexity proof for general trees (e.g., bounded branching) or report actual edge counts and runtimes for the large datasets, and state the complexity under the reported hyperparameters.
  3. [Tables 2–3, Sec. 5.2] All reported results appear to be single runs without standard errors or significance tests. Several headline gains are small: for example, ETTh1 average MSE is 0.431 versus 0.433 for FilterTS, and on Weather the average MAE of MillGNN (0.275) is not better than TimeMixer (0.271). Because baseline numbers are a mix of cited and re-run values, the claim of consistent superiority over 16 SOTA methods is not statistically supported. At minimum, report mean±std over multiple seeds and significance tests for the key comparisons, and clearly document which numbers are re-run under identical protocols.
minor comments (5)
  1. [Sec. 4.2.2, Eq. (9)] The description of the inhibitory rate is counterintuitive: increasing β_i decreases the magnitude of the negative term exp(-β_i Δ), i.e., it weakens inhibition. Please clarify the intended semantics or rename the quantity to avoid confusion.
  2. [Sec. 4.2.1 and Sec. 4.3.1] Minor language issues: 'patifying' should be 'patchifying'; 'coarsen-grained' would be more standard as 'coarse-grained'.
  3. [Table 2] The 'Count' row is difficult to interpret and appears to be a concatenation of counts for multiple metrics. Please explain its meaning or remove it.
  4. [Fig. 7] The caption lists 'Selected Lags (p0): 1,2,3,4,8' but the main text describes K selected lags. Clarify whether these are patch offsets or absolute time steps, and define p0/p1 consistently.
  5. [Sec. 5.1] The hyperparameter search uses NNI with up to 20 trials on the validation split; please report the validation protocol and how overfitting to the validation set is avoided. Also, the paper would benefit from a code-availability statement.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular construction: lags, graph weights, and forecasts are distinct and externally benchmarked; self-citations are non-load-bearing.

full rationale

I walked the paper's claimed derivation chain. The lags are computed from the input windows via FFT-based cross-correlation (Eq. 2, Sec. 3.3) and materialize as binary initial lead-lag graphs C^s (Eq. 5, Sec. 4.2.1). The dynamic decay weights Λ^s are trained model parameters obtained from learned embeddings and attention (Eqs. 6-9), and the final adjacency is A^s = C^s ⊙ Λ^s (Eq. 10). This is graph construction, not a fitted quantity renamed as a prediction. Forecasts are produced by hierarchical message passing (Eqs. 11-12) and a linear projection (Eq. 13), then evaluated against 16 external baselines on 11 public datasets. The paper's self-citations (GAGNN, Ada-MSHyper, ST-Hyper, etc.) are used for dataset configuration or as related work; none carries a load-bearing inference. The Sec. 4.2.1 restriction of candidate lead-lag pairs to sibling groups under a frozen DTW hierarchy is a structural assumption and a genuine limitation (cross-cluster lead-lag dependencies are unreachable, and the ETTh2 drift discussion in Sec. 5.2 illustrates when it fails), but it is not circular: it does not make any predicted output equivalent to an input or fitted value by construction. No step reduces to its own inputs, so the paper receives a circularity score of 0.

Assumptions & free parameters 6 free parameters · 6 assumptions · 0 invented entities

The central claim rests on per-dataset tunable hyperparameters (scales, group sizes, patch lengths, number of lags, embedding dimension), the standard FFT cross-correlation identity, the domain assumption that lead-lag relations live inside a static train-time similarity clustering, the top-K lag truncation, the exponential decay modeling choice, and idealized uniform-group-size assumptions in the complexity proof. No invented entities are introduced; variate groups are aggregations of observed variates, and excitatory/inhibitory rates are learned scalars rather than new entities with independent evidential handles.

free parameters (6)
  • number of grouping scales S = searched {1,2} or {1,2,3}; optimal 2 on China-AQI (Fig. 5a)
    Sets hierarchy depth; performance rises then declines with S, so it is tuned per dataset.
  • group sizes at scales 1 and 2 = scale 1 from {1,2,3} or {5,10,15,20,25}; scale 2 from {1,2,3}
    Defines which pairs become lead-lag candidates; tuned per dataset.
  • patch length p_s = searched in {2,4,8,12,24}
    Sets the granularity of lead-lag edges per scale.
  • number of selected lags K = searched in {4,6,8,10,15,20}; optimal 5 on China-AQI (Fig. 5b)
    Top-K cross-correlation lags define the initial binary graph C^s in Eq. 5.
  • embedding dimension d_e = optimal 7 on China-AQI (Fig. 5c)
    Dimension of the excitatory/inhibitory rate embeddings in Eq. 7.
  • network dimensions (hidden size d, query/key dim d_a, number of layers) = not reported
    Unspecified in the paper; required for exact reproduction.
assumptions (6)
  • standard math Wiener-Khinchin theorem: FFT-based circular cross-correlation in Eq. 2 estimates true cross-correlations
    Section 3.3; standard result, but no zero-padding note, so circular wraparound is a silent implementation risk.
  • domain assumption Lead-lag dependencies occur mainly between temporally similar variates, so candidate pairs are restricted to those sharing a parent group
    Sections 4.1 and 4.2.1; load-bearing: cross-cluster lead-lag is excluded by construction.
  • domain assumption The train-time grouping (DTW similarity plus spectral clustering) remains valid at test time
    Section 4.1; the paper's own ETTh2 analysis (lag-distribution drift, Fig. 4) shows a concrete failure mode.
  • domain assumption The top-K highest cross-correlation lags include the operative lags, with small K sufficient
    Section 4.2.1; K is tuned per dataset and Fig. 5b shows strong sensitivity to it.
  • ad hoc to paper Exponential decay in lag with excitatory/inhibitory rates (Eq. 9) adequately models lead-lag dynamics
    Section 4.2.2; a modeling choice specific to this paper, validated only by ablations on 3 datasets.
  • ad hoc to paper Uniform group size k and a single top-level group in the complexity proof
    Section 4.3.2 proof; contradicts the actual searched group-size ranges reported in Section 5.1.

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Cite this review

Pith. "Pith review of MillGNN: Learning Multi-Scale Lead-Lag Dependencies for Multi-Variate Time Series Forecasting." pith.science (2026). https://pith.science/paper/T6VGRMRD

@misc{pith2026250903852,
  author       = {Pith},
  title        = {Pith review of: MillGNN: Learning Multi-Scale Lead-Lag Dependencies for Multi-Variate Time Series Forecasting},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/T6VGRMRD}},
  note         = {Machine review of arXiv:2509.03852}
}
read the original abstract

Multi-variate time series (MTS) forecasting is crucial for various applications. Existing methods have shown promising results owing to their strong ability to capture intra- and inter-variate dependencies. However, these methods often overlook lead-lag dependencies at multiple grouping scales, failing to capture hierarchical lead-lag effects in complex systems. To this end, we propose MillGNN, a novel \underline{g}raph \underline{n}eural \underline{n}etwork-based method that learns \underline{m}ult\underline{i}ple grouping scale \underline{l}ead-\underline{l}ag dependencies for MTS forecasting, which can comprehensively capture lead-lag effects considering variate-wise and group-wise dynamics and decays. Specifically, MillGNN introduces two key innovations: (1) a scale-specific lead-lag graph learning module that integrates cross-correlation coefficients and dynamic decaying features derived from real-time inputs and time lags to learn lead-lag dependencies for each scale, which can model evolving lead-lag dependencies with statistical interpretability and data-driven flexibility; (2) a hierarchical lead-lag message passing module that passes lead-lag messages at multiple grouping scales in a structured way to simultaneously propagate intra- and inter-scale lead-lag effects, which can capture multi-scale lead-lag effects with a balance of comprehensiveness and efficiency. Experimental results on 11 datasets demonstrate the superiority of MillGNN for long-term and short-term MTS forecasting, compared with 16 state-of-the-art methods.

Figures

Figures reproduced from arXiv: 2509.03852 by the authors.

Figure 1
Figure 1. Illustration of lead-lag dependencies at different [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Framework of MillGNN. scales is transformed into {𝑿¯ 𝑠 } 𝑆−1 𝑠=0 = {𝑿¯ 0 ,𝑿¯ 1 , · · · ,𝑿¯ 𝑆−1 }, and 𝑿¯ 𝑠 ∈ R 𝑁 𝑠 ×𝑃 𝑠 ×𝑝 𝑠 , where 𝑃 𝑠 = 𝐿/𝑝 𝑠 and 𝑝 𝑠 are the patch number and patch length of scale 𝑠, respectively. At this stage, we construct a hierarchy of time series {𝑿¯ 𝑠 } 𝑆−1 𝑠=0 and their corresponding similarity graphs {𝑫 𝑠 } 𝑆−1 𝑠=0 on multiple grouping scales. When 𝑠 = 0, each variate forms an individual … view at source ↗
Figure 3
Figure 3. Illustration of hierarchical message aggregation. [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: Lag distributions of the training and test datasets [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]
Figure 6
Figure 6. Figure 6: Results of the efficiency study. Efficiency compar [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]
Figure 7
Figure 7. Figure 7: Example of multi-scale lead-lag dependencies on [PITH_FULL_IMAGE:figures/full_fig_p009_7.png]

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Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.