Are Global Dependencies Necessary? Scalable Time Series Forecasting via Local Cross-Variate Modeling
Authors: Kun Liu
Renjun Jia
Ruifeng Yang
Xirui Zeng
Yuqi Liang
Cen Chen
Venue: ICLR 2026 Poster (2026)
Keywords: Time Series Forecasting
Time Series Analysis
Deep Learning
Primary area: unsupervised, self-supervised, semi-supervised, and supervised representation learning
TL;DR: This work shows that local cross-variate dependency capturing is effective for dense time series and introduces VPNet, which reinterprets patch embeddings as a variate–patch 2D field to enable accurate, scalable forecasting with linear complexity.

Abstract: Effectively modeling cross-variate dependencies is a central, yet challenging, task in multivariate time series forecasting.
While attention-based methods have advanced the state-of-the-art by capturing global cross-variate dependencies, their quadratic complexity with respect to the number of variates severely limits their scalability. In this work, we challenge the necessity of global dependency modeling. We posit, through both theoretical analysis and empirical evidence, that modeling local cross-variate interactions is not only sufficient but also more efficient for many dense dependency systems.
Motivated by this core insight, we propose VPNet, a novel architecture that excels in both accuracy and efficiency. VPNet's design is founded on two key principles: a channelized reinterpretation of patch embeddings into a higher-level variate-patch field, and a specialized VarTCNBlock that operates upon it. Specifically, the model first employs a patch-level autoencoder to extract robust local representations. In a pivotal step, these representations are then re-conceptualized as a 2D field constructed over a "variates × patches" grid. The VarTCNBlock then applies depthwise 2D convolutions across this field to efficiently capture local spatio-temporal patterns (i.e., cross-variate and temporal dependencies simultaneously), followed by pointwise convolutions for feature mixing. This design ensures that the computational complexity scales linearly with the number of variates. Finally, variate-wise prediction heads map the refined historical patch representations to future ones, which are decoded back into the time domain. Extensive experiments demonstrate that VPNet not only achieves state-of-the-art performance across multiple benchmarks but also offers significant efficiency gains, establishing it as a superior and scalable solution for high-dimensional forecasting.

Published as a conference paper at ICLR 2026
ARE GLOBAL DEPENDENCIES NECESSARY? SCAL- ABLE TIME SERIES FORECASTING VIA LOCAL CROSS- VARIATE MODELING
Kun Liu', Renjun Jia", Ruifeng Yang', Xirui Zeng', Yuqi Liang", Cen Chen Fast China Normal University, Tongji University, SSek Data Group, Emoney Inc. 511659000288stu. eenttteeecncccmm wwn..edu.cc...... cenchen& dase ecnu. edu. en
ABSTRACT
Effectively modeling cross-variate dependencies is central, yet challenging, task in multivariate time series forecasting. While attention-basel methods have advanced the state-of-the-art by capturing global variate dependencies, their quadratic complexity with respect to the number of variates severely limits their scalability. In this work, challenge the necessity of global dependency modeling. We posit, through both theoretical analysis and empirical evidence, that modeling local cross-variate interactions is not only sufficient but also more efficient for many dense dependency systems. Motivated by this core insight, we propose VPNet, a novel architecture that excels in both accuracy and efficiency. VPNet's design is founded on two key principles: a channelized reinterpretation of patch embeddings into higher-level variate-patch field, and a specialized VarTCNBlock that operates upon it. Specifically, the model first employs patch-level autoencoder to extract robust local representations. In a pivotal step, these representations are then re- conceptualized as 2D field constructed over "variates x patches" grid. The VarTCNBlock then applies depthwise 2D convolutions across this field to efficiently capture local spatio-temporal patterns (i.e., cross-variate and temporal dependencies simultaneously). followed by pointwise convolutions for feature mixing. This design ensures that the computational complexity scales linearly with the number of variates. Finally, variate-wise prediction heads map the refined historical patch representations future ones, which decoded back into the time domain. Extensive experiments demonstrate that VPNet not only achieves state-of-the-art performance across multiple benchmarks but also offers significant efficiency gains, establishing it as a superior and scalable solution for high-dimensional forecasting. Code is available at this repository: http:///gtuubbcoomuuu/u//////
INTRODUCTION
Multivariate time series forecasting a cornerstone of data-driven decision-making, with critical ap- plications spanning wide range of domains from energy grid management and traffic flow prediction metcorology and finance (Granger & Newbold 2014 Martín et al.) [010 Qian et al. 2019 Chen ct al. 2001] Yin et al. 22011] Wu et al. 2023b). A key technical challenge is modeling cross-variate dependencies: the complex, vare-iarying interactions among many co-evolving series (Zhang & Yan 2023 Liu et al. 2024a). Effective modeling of these dependencies is crucial for accurate long-horizon
forecasting in high-dimensional regimes.
Recent progress has been driven by channel-fusion architectures (Zhao & Shen 2024). particularly Transformer-based designs that explicitly model global cross-variate interactions (e.g., iTransformer
(Liu et al. 2024al). These models attain strong predictive performance by searching for dependencies across all variates, but their expressivity comes at a steep computational and memory cost: the cost of channel-mixing attention grows quadratically with the number of variates, making such models impractical for systems with hundreds or thousands of variates. At the opposite end of the design spectrum, channel-independen models (including PatchTST (Nie et al. 2023), TimeMixer (Wang)
"Corresponding Author

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Published as a conference paper at ICLR 2026
Weather
Electricity
Electricity
Traffic
SolarEnergy
0.22
4.5
0,47
636
6TTh1
VPNet
TimcKAN
Timaht our
Transforer
Pathvoommer
RotchTST
Croveformer
MICN
6:30
TIEE
ETTma
Dünzar
Timesket
Traffic
Solar
Figure I: Benchmarking model performance on eight datasets (left) and variate correlation analysis on four high-dimensional datasets (right). For model performance, the average MAE across four prediction tasks on each dataset is used as the comparison result.
ct al. 2024a). Dliner (Zeng et al. 2023)) are highly efficient, yet by construction they forgo explicit vross-variate modeling and therefore struggle to exploit important inter-series predictive signals.
This accuracy-scalabiliyy tension raises a fundamental question:
Is searching for global dependencies necessary for accurate forecasting in dense, highh-dimensional systems? ?
In response, we formulate the Local Sufficiency Hypothesis: in many real-world dense systems, the dependency graph is sufficiently rich that an appropriately chosen finite local neighborhood will almost surely capture the key signals required for prediction. This makes exhaustive global search not only computationally unnecessary but also potentially noise-amplifying.
We support this principle via two complementary pillars of evidence. First, we provide a probabilissic analysís showing that under realistic dense-dependency regimes, bounded local neighborhood has high probability of including informative neighbors (Theorem I; Appendix C]. Second, we corroborate this theoretical insight empirically. Figure (right) displays correlation heatmaps for four high-ding nigh-dimensional benchmarks, which all exhibit the strong. dense inter- variate correlations that our hypothesis relies upon. This evidence collectively validates focus on developing an architecture centered on local, rather than global, interactions.
Guided by this Local Sufficiency Hypothesis, introduce VPNet, principled architecture that attains strong predictive accuracy while scaling linearly with the number variates. The model first employs patch-level autoencoder, a technique proven effective in recent literature, generate robust representations of local temporal patterns. Building on these representations, VPNet rests on two core ideas. (1) We reinterpret the patch embeddings as higher-level variate-patch field organized as a 2D grid over variates and temporal patches. This representation enables the model to treat cross-variate and temporal structure jointly at coarser, more robust abstraction level. (2) We design the VlloCk,, a lightweight module that applies efficient depthwise 2D convolutions over the variate- -ptach field to capture local spatio-temporal patterns. By restricting computation to local neighborhoods and using depthwise operations, VPNet achieves linear complexity with the number of variates. Finally, a variate-wise prediction head maps the refined historical patch representations to future ones, which are then mapped back to the time domain by the autoencoder's decoder. As previewed in (left,, this principled design allows VPNet to achieve state-of- of-the--rrrrrtt performance across eight forecasting benchmarks, consistently outperforming strong baselines while offering substantial efficiency gains (see (see/4.1) 4.1 for full analysis).
We summarize main contributions follows:
We formulate the Local Sufficiency Hypothesis, providing both theoretical and empirical evidence challenge the necessity of global dependency modeling in dense systems.
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We introduce the variate-patt patch field and the VarTCNBlock, novel representation and architecture that operationalize our hypothesis and enable the efficient capture of cross- variate dependencies with linear complexity.
We establish new hate-of-theear results across eight diverse forecasting benchmarks, demon- strating that VPNet simultaneously achieves superior accuracy and linear scalability, effec- tively resolving the critical uccuracy-efficiency trade-off.
2
PRELIMINARIES
This section fixes the notations and describes the patch-level overcomplete autoencoder used as input/output projection, so the Method section can focus on the novel components.
Notation and problem formulation. Let X BBxLXC denote minibatch of B multivariate time series samples, where is the look-back window length and C' the number of variates (channels). The forecasting objective is to predict the subsequent S time steps, denoted by Y BXSxC from the history X. We denote the model by fe and its prediction by Y fx
Let the patch length be E 24. For simplicity we assume L is divisible by and define the number of non-overlapping patches and patch index i E (1,..., P). the i-th temporal patch is denoted as Lbc. E R..
L For batch index b (I...., B}. variate index (I,.... C},
We adopt the absolute (MAE, I1) as the prediction loss due its robustness outliers:
B 8
pred
ΣΣΣ
PoLL -ace-Mo Yo.eell
BSC
Patch-level overcomplete autoencoder: To obtain robust, locally informative representations of non-stationary time series, we employ a patch-level overcomplete autoencoder (Liu et al. 2025b) the input and output projection module. A shared encoder Enc IRP TR. and shared decoder Dec ; f + I2 are defined by
Eb,c, = Enc(2 E R",
Tb.ct Dec(Es,o.) E R..
(2)
The encoder output is followed by Layer Normalization:
Eb,ck LayerNorm(eo.c.) E R.
(3)
We typically choose overcomplete latent dimension H > p provide redundant capacity for representing complex patch dynamics. The encoder and decoder are implemented as lightweight MLPs (e-g.. two linear layers with nonlinearity) and their parameters are shared across variates and patches, yielding universal patch basis. Applying the encoder LayerNorm to every patch produces the initial embedding tensor BBxXxxx!
To enforce reconstruction fidelity we include auxiliary reconstruction loss:
Lroe BCP b=
ΣΣΣ T Jo,colly
(4)
The total training objective balances prediction and reconstruction:
Liotal = Lired + d'eess
(5)
METHOD
Global cross-variate dependency search incurs prohibitive computational and memory costs, while purely channel-indepennden models lack sufficient expressive power. Grounded in the local-suficciency hypothesis, we introduce VPNet Sariate-Patell Network), a scalable architecture for multivariate time series forecasting that exploits localized structures without resorting full global mixing. As illustrated in Figure VPNet is built upon the patch-level overcomplete autoencoder described in Section 2 and follows eequence-to-sequene paradigm operating on patch embeddings. Its core
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past
Xiuture
Patchify
Channelize
Per-variable Flatten
Patch-Level Encoder Ep
2
DepthwiseConv
Predictor fa
Adaptive Normalization
Patchify
E stack(s, Rg. Pn)
BN+GELU
S 2x ".,,
X N
Predicted Patch
Representation
Pointwise-MLP
Patch-Level Decoder Dy
Patch-Level Decoder De
ResidualAdd
Reconstruction
Paramatet
Shared
(1)
Porameter
Decode
past
Xfuture
Eyec
Fpood
Figure 2: VPNet overall framework. Inputs are projected by the patch-level overcomplete autoencoder and channelization to produce ( which is processed by stacked VarTCNBlocks and finally decoded to produce the forecast.
pipeline consists of four stages: (1) input projection module that encodes series into patch embeddings via the patch autoencoder; (2) channelization step that reinterprets patch embeddings as variate-patch field to expose cross-variate structure; (3) stack of VarTCNBlocks that progressively transform the channelized field through efficient local convolution; and (4) an output projection module that decodes the refined patch representations into the final forecast.
Design Motivation: The Local Sufficiency Hypothesis. A core design principle of VPNet is that, in dense systems where many variates are mutually informative, small local neighborhood along the variate axis suffices with high probability to include an informative neighbor for any given target variate. We formalize this intuition the following theorem; the detailed concentration proof is deferred to Appendix C
Theorem 3.1 (High-probability capture of informative neighbors). Fix target variate. Suppose that among the remaining C variates, exactly variates belong to an information set C (l,...,, C} (i.e., these variates truly informative for predicting the target). Assume the variate ordering is random permutation. Let Wa denote a contiguous window of width is centered (or centered as close as boundary allows) on the target, and define the event
E W
(6)
meaning the window contains at least one informative variate. Then
kr )
Pr Pr(EE]] 2 - 1xexp
(7)
Sketch. See Appendix C.2 for the full derivation. Intuitively, under random permutation the expected number of informative variates inside We kr/(C 1); Chernoft/Poisson-sti bound on the tail yields the stated exponential lower bound.
Corollary 3.1 (practical kernel selection). To guarantee that We contains an informative variate with probability at least 1 suffices to choose
C-1
In ln
(8)
This provides a direct, interpretable guideline for initializing the variate-axis kernel width which then be fine-tuned empirically.
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3.1 STRUCTURE OVERVIEW
Input projection. Inspired by AdaPatch, we leverage a patch-level overcomplete autoencoder to handle potential distribution shifts and to extract local temporal patterns. Concretely, the input time series X E BxLC is partitioned into P = L/p non-overlapping patches of length p. Denote the i-th patch of batch and variate e by Tb,c, E IR.. A shared encoder Enc(.) maps each patch high-dimensional latent vector:
Cb,c, Enc(.),).) IR"
i= P.
Stacking these latents for all b, c, yields the initial patch representation tensor E TRRBB BxCxPx which serves as the input to the subsequent channelization.
Channelization and the Variate-Patch Field. This rein- terpretation is central to method and marks con- ceptual departure from prior TCN-based models. As illus- trated in the figure, while prior works such ModemTTN and TimesNet treat the time series as a 2D input with a single channel reshape it into 2D plane based periodicity to capture intra-series patterns, our approach innovatively treats each patch holistic unit to con- struct a high-dimensional Variate-Patch Field (VP-Field),
Embedding
Varlates
Specifically, the initial representation is permuted create the channelized variate-patch field ());
Patches
(10) ( = Permute(E) E BB
This permutation recasts the patch embedding dimension H as the channel dimension for a 2D operator, while the variates and patches P form wvo-dimensional spatial grid we term the variate-patch field. By applying convo- lution directly on this high-level semantic field, we can simultaneously and efficiently capture both cross-variate and temporal dependencies.
Figure 3: (a) Traditional Convolution (Yellow): Applying convolutional op- erations directly on the original time series. (b) Frequency-based 2D Con- volution (Green): Reconstructing inde- pendent variates into two-dimensional plane based on their freguency character- istics, (c) Our Method (Purple) ( -,,,...
Var TCrTCBBlocs. The channelized tensor (0) is then pro- cessed by stack of N VarTCNBlocks, which iteratively refine the patch representations. The abstract forward propagation through the stack is defined as:
(11) Each block operates on the channelized input ( E BBCHXXXP and produces an output of the same shape, progressively capturing more complex dependencies across both variates and patches.
(++ VarTCNBlock(""
Output projection. After N blocks the stack produces the context-aware representation Z xBXCxPxxx PexcxPxx..ee We apply a channel-independent prediction head Head(-) to map each variate's historical patch sequence future patch embeddings. Concretely, for each variate e flatten its history
Wbec vec (.
E "P..
(12)
and shared perivariate MLP Head !!P + R2
P produces predicted patch coefficients
Wh.r = Head( (u..) E HP,
(13)
which we reshape into predicted future patch embeddings 2 E BxCxP,x (here Pp is the number of predicted patches and the forecast horizon
Pp p). Finally, the shared decoder Dec(.)
maps each predicted patch embedding back to the time domain and we concatenate these patch-level reconstructions to obtain the final forecast
Concal Pe
P. Dec(2....)) BBSxC
(14)
(shared across variates) but operates on per-variate flattened patch histories; this design preserves parameter efficiency while allowing each variate to leverage mixed information aggregated by the VarTCN stack. The decoder is reused from the autoencoder (weight sharing) to regularize predicted embeddings and improve reconstruction fidelity.
Remarks. The prediction head is
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3.2 THE VARTCNBLOCK
The VarTCNBlock is the core computational engine of VPNet, designed to efficiently process the channelized variate-patch field z R RBxHxxXP Each block refines these representations by capturing dependencies across both variates and time, and its design is centered around residual connection that wraps two main components: a depthwise convolution for local spatial mixing and a pointwise feed-forward network for feature transformation.
Motivated by our probabilistic analysis that local interactions are sufficient for capturing salient signals in systems with dense dependencies, employ a depthwise 2D convolution over the variate-patch field. Unlike standard convolution, a
Depthwise Convolution for Local Mixing:
depthwise convolution applies a distinct 2D kernel W(h) E R X to each input channel Z (l) independently. This operation is formulated as:
Ydw = DWConv2D
(z() "AAA W()),, h h= H.
(15)
This step effectively aggregates information from a local neighborhood of size ki x kp on the variate- patch field, explicitly modeling temporally-local cross-variate dependencies. Crucially, as advocated in our introduction, this is achieved with a computational cost and parameter count that scale linearly with the number of variates C, making it exceptionally suitable high-dimensional forecasting.
Pointwise Feed-Forward Network for Feature Transformation: Following the spatial mixing, the resulting features Ydw are passed through normalization layer and an activation function before being processed by feed-forward network (FFN). The FFN is implemented with pointwise (1 x 1) convolutions and serves to mix information across the feature channels at each position on the suriate-patchl field. It follows inverted bottleneck structure, first expanding the channel dimension by a factor rg and then projecting it back. The full sequence of operations is:
Y act GELU (BN(UBBBN))) (BN(Yd")) Y fin = FFN(Y act
(16)
The FFN module itself consists of two pointwise convolutions, activations, and dropout for regular- ization. The output of this entire sequence is then added the block's ' original input via residual connection:
(1+1)) ( Yffl
(17)
This residual design is essential for stabilizing the training of deep models by allowing gradients to flow more freely.
EXPERIMENTS
In this section, we conduct a comprehensive empirical evaluation validate the efficacy and efficiency of VPNet. We begin by benchmarking VPNet against diverse suite of state-of-the the-art models on several public datasets to establish its overall performance (Section 4.2). Subsequently, we perform series of detailed ablation studies to dissect the model and verify the contributions of its design principles, particularly the local cross-variate convolution mechanism (Section [43).. Finally, analyze the model's 's practical properties, focusing its computational and memory efficiency (Section 4.5)
4.1 EXPERIMENTAL SETUP
Datasets. We evaluate VPNet on eight widely-used public benchmarks for long- term time series forecasting: Weather, Truffic, Electricity, Solar-Energy, and four ETT datasets (ETTh), ETTh2, ETTm1, ETTm2). Detailed statistics for each dataset are provided in Appendix focus the nigh-dimensional datasets (Electricity, Traffic) which contain hundreds of variates, making them ideal for assessing the model's ability to handle cross-variate dependencies. All datasets partitioned into training, validation, and testing sets following 6:2:2 ratio for ETT datasets and 7:1:2 for the others. This aligns with prior benchmarks set by (Zhou et al. 2021 Liu et al.) 2022).
We particularly
Baseline Details. To provide robust comparative analysis, we evaluate VPNet against carefully curated collection of strong baselines that together represent the principal modelling paradigms for long horizon multivariate time series forecasting. This collection comprises KAN based models

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exemplified by TimeKAN (2025), MLP centric architectures exemplified by TimeMixer (2024a),
TiDE (2023) and DLinear (2023). Transformer variants designed for long sequence forecasting
exemplified by iTransformer (2024a), PatchTST (2023), Pathformer (222)), Crossformer (2023) and
FEDformer (002)) and convolutional approaches exemplified by MICN (2023) and TimesNet (2023a).
Together these baselines span complementary inductive biases and computational tradeoffs, thereby providing a stringent benchmark for assessing VPNet's capacity to capture interactions across variates and to scale high dimensional settings.
Implementation Details. All experiments were implemented in PyTorch and conducted on a single NVIDIA 1 40GB GPU. Following standard long-term forecasting protocols, we use a fixed input sequence length of L = 96 to predict future horizons of S E {96, 192, 336, 720). For comprehensive list of all model hyperparameters and training configurations for each dataset, please refer to Appendix
4.2 MAIN RESULTS ON LONG-TERM FORECASTING
Table Table|rreprte the long-term forecasting results, with averages summarized in the main paper for clarity. VPNet achieves the best overall performance, outperforming all baselines on most datasets and metrics.
On high-dimensional datasets (Weather, dependencies are dense, VPNet demonstrates substantial gains. For example, on Electricity, it reduces MSE by 9.0% % compared with iTransformer (0. (0.162 vs. 0.178), and on Traffic, it achieves 26 % improvement over TimeKAN (0.421 vs. 0.572). These results strongly support our proposed Local Sufficiency Hypothesis.
Electricity, and Truffic), where cross-variate
On the ETT benchmarks with lower dimensionality, VPNet remains competitive, frequently ranking first or second. For instance, it outperforms all baselines on E77m2 and ETTh2, and is only marginally behind Ti on ETThl. This consistency across both high-dimensional and low-dimensional regimes highlights the robustness and general applicability of VPNet without requiring dataset- specific adaptations.
Table 1: Long-term forecasting results. We average the results across 4 prediction lengths: (96, 192, 336, 720). The best performance is highlighted in red, and the second-bes underlined. Full results can be found in Appendix K
Madels
vitten
YireKAN
Pabloran
PrahiST
MBCN
TIDE
Pillforaar
Meuie
MSE
MAE
MSE
MAE
MSSE
MAR
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAA
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
MAE
Weather
10.230
0.263
11243
0.272
0.2772
0.266
0.320
0.268
0.271
0.320|00259
0.266
0.315/0.309
00.30
Solas Solas-Enrgy
(0.004
0.207 (1.276 0,310 012 0,250 (1.233 0.202
(1241 11250)
0.333 0.4166 0.442 0.283 0.35M 0.347 0.41 0.417|o.410 01374
0.401
(0.325
0.385
Electricity
(0.162
0 10.197 0,286 0.182 0,273 (0 0 0.270 0.269)
0,25
0.216
0,318 0.264 0.334 O. 96 0.309 0.252
0.34:jo. 193
0.206|0.225
0.319/0.214 0.327
Triliic
[0.421
0.273 (0.572 0,372 485 0.258 (1 424 222 (1.501 0.209)
0.529 0,341 0.667 0.426 O. 0.593 0.356 0.761 0.473j0.620 0.336
0.376
ETT1+
[0.434
0.427 (0.426 0,431 447 0,440
(0.455
(.420) 0.507 0.472
0.522 0.475 0.481 (0.541 0.507)o.458
450j0.446
0.458j0.406 0.84
ETTh2
10.356
39391 0:009 3-3 0.395 30.38 0.407
(0.374 0.395 0.391 0.912
(0611 0.550j0.414 0.127/0.563
0.1990..39
RTTml
[0.376
0.382
11..222.8380 383 0.308 0.181 0.306/00.00 410 0.410
0.386 0,2
0.406 0.513 0.495
0.419 0:419|0 0.66)004044
0.08/004444
FTTm2
(0.270
0.312|0.277 0,322 2.5 0.323 jo 288 o. 333 221 (.304)02200
0.38 O 757 0.611
O. 0.353 0 0.402 io 0.404|o.201 0.338jo
0.402|0.305
549
4.3 ABLATION STUDIES
To dissect our model and validate its core design principles, we conduct a series of targeted ablation studies.
Effectiveness of Cross-variate Convolution. To quantify the impact of our local cross- variate mechanism, we conduct an ablation on the kernel size k. We evaluate the model with range of kernel sizes: k2 E 31, 7, 17, 27). where k, = represents the channel-indepennent baseline. As shown in Table 2 performance improves dramatically when moving from k, 1 to k. = 3, confirming that local variate mixing is crucial. However, we observe that further increasing the kernel size to 7, 17, and beyond offers diminishing returns and can even slightly degrade performance. This result provides strong empirical validation for our sufficienf hypothesis: for dense dependency systems, compact local receptive field across variates already contains sufficient auxiliary predictive
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signals. attempting to model wider, quasi-global interactions, validating the probabilistic motivation outlined in our introduction.
Table 2: Ablation study the varistv kernel size k.. We report the average MSE over all prediction lengths for each benchmark. The case K2 = is the channel-ine -ineeellddeeeedentt baseline. Lower is better.
VPΝET
ETTh1
ETTh2
ETTml
ETTm2
ECL
Traffic
Weather
Solar
k =
0,437
0.362
0.375
0.282
0,184
0,443
0.254
0.224
ky 3
0.435
0.362
0.378
0.275
0.171
0.431
0.250
0.203
ko 7
0.434
0.355
0.375
0.273
0.167
0,435
0.248
0.204
k 17
0.162
0.423
0.239
0.204
h 27
0.160
0.422
0.243
0.204
Effeet of variate Reordering. Our model's reliance on local convolutions suggests that the ordering of variates could influence performance. To investigate this, conduct rigorous experiment across all cight benchmark datasets. We fix the model configuration to have a small cross- variate kernel size (Rgg 3) and stack of two VarTCNBlocks, resulting in an effective receptive field of 5 across the variate axis. This constrained setup is designed to be highly sensitive to the local variate neighborhood. We compare the model's performance under four distinct ordering strategies: Original Ordering. Random Ordering, Degree Ordering, and Spectral Ordering (see Appendix | for details).
The results, summarized in Table 3 are counter-intuitive yet illuminating. The model exhibits surprising degree of robustness to the variate ordering, as all four strategies yield remarkably similar performance. This finding suggests that correlation based sorting methods, which operate by grouping highly similar variates, may not provide the most effective dependency signals. The dependencies captured by VPNet appear to be more complex than simple instantaneous correlations, implying that other factors, such as time-lagged relationships, may play a more critical role.
Table 3: Ablation study on variate reordering strategies. We report the average MSE across four prediction lengths 96, 192, 336, 720 for each of the eight benchmarks, with a fixed input length of 96. All models use a fixed configuration (ki, 3, N layers). Lower is better.
VPNET
ETThl
ETTh2
ETTml
ETTm2
ECL
Traffic
Weather
Solar
Original
0.435
0.362
0.378
0.275
0,171
0.431
0.247
0.205
Spectral 0.436
0.366
0.374
0.275
0.174
0.428
0.246
0.209
Degree
0.436
0.365
0.378
0.276
0.173
0.433
0.247
0.209
Random 0,439
0.362
0.378
0.275
0.171
0.431
0.247
0.205
4.4 LOCAL MODELING MECHANISMS ANALYSIS
To further investigate the efficacy of local cross-variate modeling, we implemented two Transformer- based variants: LANet (Local-Windas Attention) and SANet (Sparse Attention). The quantitative results across eight benchmark datasets are summarized in Table As shown in the table, while VP- Net generally outperforms the attention- based variants (particularly on dense datasets like Electricity and Traffic), the variants achieve competitive performance on datasets with different characteristics, prompting a deeper analysis of the trade-offs between TCsed and Attention- approaches.
Structural Prior vs. Content-Based Addressing. The fundamental distinction lies in the depen- dency capture mechanism. Transformer-basee variants rely on content- based addressing, dynamically computing attention weights (Softmax (Q (QKT)) to learn instance-specift relationships. Theoretically, this offers a higher representational ceiling by capturing arbitrary dependencies without geometric constraints. Conversely, VPNet (TCN-based) enforces a strong inductive bias through fixed convo- lutional kernels, treating the multivariate input topological grid. While this "static" modeling appears less flexible, our results suggest that this strong structural prior acts as an effective regular- izer. It leads to superior optimization stability and generalization on dense datasets, whereas fully data-driven attention mechanisms often struggle with optimization or require larger data regimes to converge effectively.
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Robustness and Variate Ordering. In the Vense-Dependence Scenario (e.g., datasets such as Traffic and Electricity with dense correlations and high information redundancy), VPNet exhibits pronouneed robustmess to variate ordering. This is because even under random permutations of the variables, each local neighborhood can still cover multiple subsets of correlated variates. As a result, in such settings, VPNet serves robust and efficient modeling strategy.
In contrast, in the Sparse-Dependence Scenario (e.g., Hetero- Mix or datasets with sparse dependency structures), variate ordering has a much more significant impact on VPNet's performance. In these cases, if the correlation structure among variates is ignored and the ordering is randomly permuted, a single local receptive field is likely to contain many irrelevant or weakly correlated variates, which reduces the concentration of informative signals. By adopting correlation-aware or structure- aware ordering strategies, one can "compress" " more useful dependencies into each receptive field, substantially improving the local signal-to-noise ratio and allowing the convolutional kernels to enhance VPNet's performance without increasing the number of parameters.
Table 4: Long-term forecasting performance comparison for horizons of {96, 192, 336, 720). We compare three variants: VPNet (Raw), LANet (Local), and SANet (Sparse). The input sequence length is fixed to 96. The best results are highlighted in bold.
DataSets
Weather
Solar Energy Electricity
Traffic
ETTh2
ETTml
ETTm2
Metric
MSE MAE|
MSB MAB MSE MAE|
MSE MAE
MSE
MAE|
MSE
MAE
MSB
MAB|
MSE
MAE
96
0.115 0.192 0.177 0.178 0.113 0.224 0.384 0.258 0,3374 0,386/0.284 0.284 0.330 0,313 0.342
0.169 0.244
VPNett
192
0.207 0,239 0.6 0.206 151 0.239 01,406 0.266 0.428 0.418 0.357 0.357 0,376 0.362 0.371 0.233 0.288
336
0.258
0,281 00.16 0.219 06 0.257 01.429 0.275 0.464 0.439 0,392 0.406 0.385 0.389 0.291 0.327
720
00.30
0.333 (0.228 0.225
(.. 0.284 (0,466
0.294
0.471 0.465 0.465 (0.391 0.418 (1.444 0.424
0.424|0.387
00387 0.387
Avg
00.238
0.261 00.204 0.207 00.162
0.251 0..42 0.273 0.434 0.427 (0.356 0,383
0.3766 0.382 |0.270 0.312
96
0.157
0.115
0.196
(0.144
0.230
0.392
0.257 0.377 0.388/0.288
00.288 0.330
0.330 0.313 0.340 (0.177 0.344//0.177 0.249
LANet
192
0.211
0.245
0..19 0.207
0.155 0.243
0.2443 0.413 0.265 0.430 0.420 0.356 0.356 0.377 0.369 0.372 00,243 0.294
336
0.277
0.293 0.216 0,222
0.173
0.259 0.429 0.272 0.468 0.441 (0.393 0.407 0.396
0.394 0,3300 0.333
720
0.353
0.342
00.255
0.229
(0.208
0.291
(0.462
0.293 0.485 0.468
0.468 0.415 0.431
00.456
0.428
0.399 0.389
Avg
|0.250 0.26%% (0.205
0.214
10.171 0.256 0.256/0.424 0.272 (.440 0..33 0.386|0.3844
0.384
|0,278 0.316
0.1558
0.199/0.118
0.197
(0.149
0.233
0.259
0.376
0.383
00.244
0.329
0.313
0.340
0.5 0.248
SANet
192
0.213
0.246 0.194 0.206 0.162 0.245 0.413 0.267 0.430
0.420
0.354
0.377
0.371
0.241 0.293
336
0.277
0.291 0..217 0.222 0.175 0.259 0.429 0.274 0.466
0.392
0.398
0.393
0.301
720
0.361
0.345 0.227 0.230
(0.207 0.285 0.4465 0.295 08446
0.473 0.430
0.428
( 0,392 0.388
Avg
(.2252
1.269/0.200 0.214 10.173 0,256/0..25 0.274 0..40 0.431
0.386)0.384 0.383
(0.277 |0.87
0.316
4.5 MODEL EFFICIENCY ANALYSIS
For practical deployment in high-dimensional forecasting, computational and memory efficiency as critical as predictive accuracy. We therefore evaluate efficiency on the two datasets with the largest number of variates, Electricity (C 321) and Traflic (C' = 862), comparing VPNet with leading baselines including iTransformer, PStchTST, TimeMixer, Crossformer, and Pathformer. Our analysis considers the joint trade-off among accuracy (MSE), training time per batch, and GPU memory
usage.
Figure 4 summarizes the results, revealing a clear accuracy- efficiency frontier. VPNet consistently achieves the most favorable balance, delivering state-of-the-art accuracy at competitive computational cost. On both datasets, VPNet attains the lowest MSE, demonstrating the effectiveness of its local cross-variate modeling.
A direct comparison with iTransformer illustrates the difference in scaling behavior, iTransformer achieves the fastest training times due to its simple design, but its reliance on variate-wise attention induces quadratic memory scaling. As the number of variates grows from 321 to 862, its peak memory nearly doubles (+99%, from 2174MB to 4376MB). In contrast, VPNet's memory footprint increases by only 67% (from 3308MB to 5520MB), consistent with its linear complexity in the variate dimension.

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Published as a conference paper at ICLR 2026
Electricity Efficiency
Traffic Efficiency
0.3
0.75
Mersory
mm
0.0
0.30
200
100
400
600
200
4000
800
1000
Training Time / ittr)
Training Time (ms iter)
Figure 4: Efficiency comparison on Electricity (C = 321) and Traffic (C = 862). We report mean squared error (MSE), training time per batch, and peak GPU memory usage for VPNet and baseline models.
At the opposite extreme, global dependency models such as Crossformer and Pathformer incur prohibitive costs in both computation and memory, rendering them impractical for large-scale use. Conversely, TimeMixer achieves strong efficiency but sacrifices accuracy due to its channel- independent design, underperforming on complex, high-dimensional datasets.
VPNet establishes itself at the accuracy-efficiency Pareto frontier. By grounding its design in The Local Sufficiency Hypothesis, it delivers both superior forecasting accuracy and scalable efficiency, making it practical solution for real-world, scage forecasting tasks.
RELATED WORK
Channel Independence. Channel independence has emerged as simple yet effective strategy for multivariate time series forecasting. The core idea is to model each variate independently, thereby avoiding the "negative transfer" that may arise from noisy spurious cross- variate correlations. PatchTST (Nie et al., 2023) exemplifies this paradigm by combining a patching strategy with a channel-independent Transformer architecture. Similarly, lightweight models such TimeMixer and DLinear (Zegg et al. (203)) have shown that accurate univariate forecasting can achieve strong performance with high computational efficiency. However, these methods often underperform in high-dimensional settings where cross-variate dependencies are critical.
Channel Fusion. In contrast, channel fusion aims explicitly capture dependencies across vari- ates (Liu et al.] 2025a). Representative approaches include Crossformer (Zhang & Yan (023).
Pathformer (Chen et al.) 2024), iTransformer (Liu et 2024a). and CARD (Wang et 2024b).
While such methods generally outperform channel-independen models on high-dimensional datasets, their computational and memory costs grow rapidly with the number of variates, limiting scalability. To address this challenge, we propose the local sufficiency hypothesis. By focusing capturing local cross-variate dependencies, our proposed method reconciles the trade-off between modeling expressiveness and computational cost. It achieves new state-of-the-an (SOTA) performance while ensuring that computational complexity scales linearly with the number of variates, making it practical and effective solution for high-dimensional forecasting tasks.
6 CONCLUSION
In this work, we revisit the long-standing assumption that global dependency modeling is indispens- able for high-dimensional time series forecasting. We formalize the Local Sufficiency Hypothesis, which posits that local cross-variate interactions are often sufficient to retain predictive power while avoiding the inefficiencies of global mixing. Building on this principle, we introduce VPNet, a new architecture that leverages variate-patch field representation and the VarTCNBlock to model local dependencies with linear scalability. Through comprehensive evaluation on eight public benchmarks, VPNet achieves new state-of-the-ane results. These findings demonstrate that focusing local sufficiency provides principled and scalable solution to the critical accuracy-efiicienee trade-off in multivariate forecasting.
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ACKNOWLEDGMENTS
This work was supported by the Guizhou Provincial Program on Commercialization of Scientific and Technological Achievements (Qiankchezhongyindi [2025] No. 006) and Seek Data Group, Emoney Inc.
ETHICS STATEMENT
This research is based on publicly available, anonymized datasets commonly used for benchmarking in the time series forecasting community. The work focuses foundational modeling techniques for general forecasting tasks, such as predicting electricity consumption and traffic patterns. We do not foresee any direct negative societal impacts ethical concerns arising from our methodology or its applications. Our research adheres to the principles of ethical academic conduct.
REPRODUCIBILITY STATEMENT
To ensure the full reproducibility of our results, we provide the following resources.
Code The complete source code for VPNet, along with scripts to run all experiments reported in this paper, is provided in the Supplementary Materials. The code is also available at this repository: httpgggthth..c/uuauuuu/u//
Data All eight datasets used in our evaluation (Weather, Traffic, Electricity, Solar-Enne Energy, and ETT benchmarks) are publicly available and can be downloaded from the dataset links provided in the official TimeMixer (Wang et al. 2024a) source code.
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A LLM USAGE STATEMENT
In the preparation of this manuscript, we utilized Large Language Models (LLMs) as writing assistants. The use of these tools was strictly limited improving the clarity, grammar, and overall style of the text. No part of the core scientific contributions, including the formulation of the hypothesis, the design of the model architecture, the implementation of the code, the generation of experimental results, or the analysis of those results, was produced by LLMs. All suggestions from these tools were critically reviewed, edited, and manually incorporated by the authors to ensure the final text accurately reflects
work and ideas.
B IMPLEMENTATION DETAILS
Dataset Details. This section provides a detailed description of the public benchmark datasets used for the empirical evaluation of our model multivariate time series forecasting. For data preprocessing, splitting, and normalization. we adhere to the standard protocols established in widely- recognized previous works (Zhou et al., 2021: Wu et al., 2021). A summary of the key statistical properties of each dataset is presented in Table The evaluation suite includes several standard benchmarks. The ETT (Electricity Transformer Temperature) collection contains data from two electricity transformers with 7 variates, recorded hourly (ETTh1, ETTh2) and 155minute intervals (ETTml, ETTm2) from 2016 to 2018. The Electricity (ECL) dataset contains the hourly power consumption of 321 clients from 2016 2019. The Weather dataset comprises 21 meteorological indicators from Germany, collected every 10 minutes during 2020. The Traffic dataset documents hourly road occupancy rates from 862 sensors in the San Francisco Bay Area from 2015 2016. Finally, the Solar-Enee Eolery dataset records solar power generation from 137 photovoltaic (PV) plants at 10-minute intervals during 2006.
Experiment Details. All experiments were implemented in PyTorch and conducted on a single NVIDIA A100 40GB GPU. For model optimization, we employ the Adam optimizer (Kingma & 4 and batch size of 32. To remain consistent with prior works, we use a fixed look-bace window of L = 96. For the core model hyperparameters. select the number of VarTCNBlocks N from (1, 3). the patch length p from {8, 16), and the
Ba 2017) with an initial learning rate of x 1
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hidden dimension H from (64, 128, 256) based validation set performance for each dataset. Mean Squared Error (MSE) and Mean Absolute Error (MAE) are used as the primary evaluation metrics. For baselines where the experimental settings align with main study, directly report the results from MixeMixer (Wang et al. 2024a). In other cases, we reproduced the baseline results using the benchmark framework from the Time-Series Library
Table 5: Statistics of the benchmark datasets.
Dataset
Dim
Series Length
Dataset Size
Frequency
Information
ETTh1
7
[96, 192. 336, 720)
(34465, 11521, 11521)
Hourly
Temperature
ETTh2
7
[96, 192, 336, 720)
(34465, 11521. 11521)
Hourly
Temperature
ETTml
[96, 192, 336, 720)
(8545, 2881, 2881)
min
Temperature
ETTm2
(96, 192, 336, 720)
(8545, 2881, 2881)
min
Temperature
Electricity
321
(96, 192, 336, 720)
(18317, 2633, 5261)
Hourly
Electricity
Weather
196. 192, 336. 720)
(36792, 5271, 10540)
10 min
Weather
Traffic
862
(96, 192, 336, 720)
(12185, 3509)
Hourly
Transportation
Solar- Energy
137
196, 192. 336, 720)
(36601, 5161, 10417)
10 min
Electricity
C PROOFS AND PROBABILISTIC DESIGN RULES
We formalize the probabilistic statement used in the introduction and provide proofs.
Table 6: Probability Analysis for Different Parameters
Info. Variate Ratio
Probability for &
(r/C)
k=1
k=3
k=7
k=17
10%
0.095
0.259
0.799
0.999
30%
0.259
0.593
0.996
1.000
50%
0.393
0.777
1.000
1.000
70%
0.503
0.878
1.000
1.000
90%
0.593
0.931
1.000
1.000
Theorem C.1 (Hypergeometric exact probability for zero hits). Let C 2 2 be the total number of variates and suppose that for fixed target variate there are exactly r "informative" variates among the other C - variates (i.e.. 0 < C 1). Consider selecting subset of k distinct variates uniformly at random from the C non- moonttarget variates (this models the set of variates falling into a fixed contiguous window under uniformly random permutation). Let X be the number of informative variates in the chosen subset. Then
C C-1
k
Pr =
(18)
www.:iiimeeeieeseeeeeeeeeee Library
Table Confidence Intervals for Different Parameters
k=7
k=17
Info. Variate Ratio
(r/C)
95%
99 %
95%
99%
10%
[0, 2]
[0, 3]
[0, 4]
[0,5]
30%
[0.5]
[0. 6]
[1, 9]
[0. 10]
50%
[1.6]
[0. 71
[3, 14]
[2, 15]
70%
[2.
[1, 7]
[7, 16]
[6, 17]
90%
[]
[4.
[12. 17]
[12, 17]
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Proof. There are 1 equally likely ways choose a k-subset from the C 1 non-target variates. The number of choices that contain zero informative variates is the number of ways to choose all elements from the
r non-informative variates, which is (C-1-T). Dividing the favorable count by the total count yields equation 18
Corollary C.1 (Exponential upper bound). Under the same notation as Theorem probability satisfies
C.1
the zero-hit
P=|X= - (1-0-1)
exp
kr -0-1)
(19)
Proof.
Starting from equation 18 we write the ratio form
Or|X = 0] Prod -1 C--1---
(= C--1-1
For each factor we have
C-1-r-i
r
C- $ --- C-1
C 1-1-i
because the denominator
C - 1- i is most C - for 0. Taking the product yields the first
inequality in equation 19 The second inequality follows from (1 - x) S e applied to x c- and exponentiation to the power k:
Corollary C.2 (Design rule for at-least-one hit). we require that a randomly chosen window of width k contuins at least one informative variate with probability at least 1 8, i.e.
Pr(X 2 1
then it suffices to choose l satisfying
k 2
In 5
(20)
Proof. From Corollary
C.I we have
Pr x 2 1] = Px|X 21 121-p exp
-- -
kr
Requiring 1 expl wepppeppp(C------ (C > 1 is equivalent to
< 6, which rearranges
to equation 20
denote the hypergeometric mean. Standard concentration bounds for the hypergeometric distribution (which be derived from Hoeffding's inequality or by coupling to an appropriate binomial distribution) give that for any
Concentration around the mean. Let =[ -ं
0<E<> <
E²p 2
Pr X < (1 =)
s exp
(21)
corresponding upper tail bound holds:
Pr x Z +l+el (1+))
< exp
3
These inequalities quantify that once moderate, the number of informative variates inside a random window concentrates tightly around μ.
D EMPIRICAL ANALYSIS OF DATASET CHARACTERISTICS
To empirically validate the Local Sufficiency Hypothesis, we analyzed the intrinsic properties of all benchmark datasets. We focus on three key aspects: (1) Granger Dependency Density, measured by the ratio of significant Granger Causal edges (p < 0.01, lag=3) after differencing and global mean removal; (2) Information Redundancy, quantified by Principal Component Analysis (PC1 variance and Effective Dimension Compression Ratio); and (3) Correlation Strength, measured by the distribution of pairwise Pearson correlation coefficients. The quantitative results are summarized
in Table g
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Table 8: Statistical analysis of dataset characteristics. Granger Density indicates the ratio of significant causal pairs. PC1 Var denotes the variance explained by the first principal component. Compression Ratio is the total number of variates divided by the number of components required to explain 95% variance. Correlation metrics show the percentage of variate pairs exceeding absolute correlation thresholds. Bold highlights the highest values indicating extreme density.
Granger
PCA (Redundancy)
variate Correlation (Pearson
(r)
Dataset
Density
PC1 Var
95% Comps
Comp. Ratio
Avg Ir]
0.3
0.5
0.7
Solar
51.87%
91.77% %
4/ 136
34.00x
0.9167
100.0%
100.0%
100.0%
Traffic
69.45%
57.67%
202 / 862
4.27x
0,5638
89.92%
66.84%
25.67%
Electricity
89.19%
54.68%
86 321
3.73x
0.4893
68.29%
46.37%
32.49%
Weather
64.52%
42.44%
9/21
2.33x
0.2956
35.71%
25.24%
20.95%
ETTh1
64.29%
34.39%
5/7
1.40x
0.2221
19.05%
9.52%
9.52%
ETTh2
52.38%
43.10%
5/7
1.40x
0.3246
42.86%
28.57%
4.76%
ETTml
47.62%
34.57%
5/7
1.40x
0.2243
19.05%
9.52%
9.52%
ETTm2
30.95%
43.08%
517
1.40x
0.3245
42.86%
28.57%
4.76%
Analysis.
Large-scale datasets with high redundancy. The Solar, Traffic, and Electricity datasets exhibit distinct characteristies of dense regimes. They possess high effective dimension compression ratios (> x) and strong variate correlations. Notably, Electricity maintains exceptionally high causal density (89.19%) even after removing global trends, suggesting ubiquitous local interaction network. This justifies the superior performance of VPNet on these datasets, as local kernels can efficiently aggregate the redundant and dense information.
Lower-dimensional datasets with weaker cross-variate dependencies. Conversely, the ETT datasets show significantly lower redundancy (Compression Ratio 2 1.4x) and weaker correlations (strong correlations |r] 0.5 are generally 10%).
E EVALUATION ON VARIATE ORDERING USING A HETEROGENEOUS COMPOSITE DATASET
To examine how variate ordering affects local modeling in datasets with heterogeneous dependency patterns, we constructed a composite dataset named Hetero-Mix by concatenating the variates from three benchmark
Electricity, and Weather- along the channel dimension. Unlike homogeneous datasets where dense correlations reduce sensitivity to ordering. Hetero-Mix brings together variables with distinct statistical characteristics, making it suitable for testing how ordering influences local receptive field models.
We evaluated VPNet and local-attention variant (LANet) under two ordering conditions: (1) Clustered, where variates are grouped according to their dataset of origin; and (2) Shuffled, where all variates are randomly permuted. As shown in Table both models achieve better accuracy under Clustered setting. This demonstrates that when correlations vary substantially across groups of variables, aligning the input ordering with the underlying dependency patterns improves the effectiveness of localized modeling.
F STRESS TEST UNDER EXTREME NOISE AND SPARSE DEPENDENCIES
To rigorously evaluate model robustness in scenarios dominated by sparse or long-range cross- variate dependencies, conducted stress test using the Electricity dataset injected with 10x Gaussian noise (10 X Gnoise). This setup creates an extremely low signal-to-noise ratio (SNR) regime, effectively disintegrating dense local correlations. To compensate for the high noise level and capture dispersed signals, we adjusted the VPNet configuration to use larger receptive field (Kernel Size=27, Layers=2).
We tested VPNet using two variable orderings: (1) Clustered, where variates follow their original dataset indexing; and (2) Shuffled, where the ordering is randomized. Table shows that VPNet
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Table 9: Performance comparison on the Hetero-Mix dataset. "Clustered" preserves the original grouping [Traffic + Electricity + Weather), while "Shuffled" randomly permutes variates. The results show that both models benefit from topology-aware ordering (Clustered).
VPNet
LANet
Ordering Setting
Horizon
MSE
MAR
MSE
MAE
96
0.345
0.264
0.347
0.264
192
0.368
0.274
0.369
0.274
Clustered (Ordered)
336
0.390
0.286
0.387
0.284
720
0.435
0.313
0.432
0.314
Avg
0.385
0.284
0.384
0.284
96
0.349
0.270
0.352
0.268
192
0.369
0.278
0.371
0.275
Shuffled (Random)
336
0.393
0.288
0.389
0.286
720
0.440
0.317
0.433
0.314
Avg
0.388
0.288
0.386
0.286
performs consistently better with the original ordering. This indicates that when the signal is weak, variable orderings that place moderately correlated variates closer together help local models extract meaningful structure more effectively.
Additionally, the large-kernel configuration allows VPNet approach the behavior of broader- peceptive-field models, suggesting that increasing the receptive field is practical adaptation strategy when dependency patterns become weak or diffuse.
Table 10: Stress test on Electricity + 10x Noise. The model configuration fixed at [Layers=2, Kernel=27]. "Ordered" denotes structural prior-based clustering, while "Shuffled" denotes random permutation. Bold indicates the best performance.
Electricity Noise (Ordered)
Electricity Noise
[SeutricttynnnnchSeeeeellee (Shuffled)
Horizon
MSE
MAE
MSE
MAE
96
0.137
0.232
0.148
0.239
192
0.153
0.248
0.161
0.252
336
0.174
0.270
0.178
0.270
720
0.199
0.295
0.224
0.312
Arg
0.166
0.261
0.178
0.268
G VISUALIZATION OF LEARNED DEPENDENCIES AND RECEPTIVE FIELDS
To better illustrate the dependencies learned by VPNet and verify its capability to capture local cross-variate interactions, we conducted gradient-based saliency analysis. We computed the absolute gradients of the prediction output with respect to the input (jag/ (X)), averaged over 64 samples for stability.
Methodology. We present three complementary visualizations: (1) a saliency heatmap show- ing the spatio-temporal receptive field, (2) global variate importance summarizing cross-variate contributions, and (3) temporal importance reflecting how historical information is utilized.
Analysis.
The visualizations reveal dataset-specifie dependency patterns, summarized as follows:
ETThI: Predominantly Auto-Regressive Behavior. As shown in Figure S when pre- dicting variate 2 at the first step, gradients concentrate almost entirely on the variate's own history. Neighboring variates contribute minimally. This aligns with our earlier statistical findings indicating weak cross-variate correlations in ETTh1. The temporal importance plot also shows periodic spikes, demonstrating that VPNet captures seasonal patterns even when cross-variate signals are limited.
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Electricity: Discovery of Dataset-Specific Couplings. In Figure 6 when predicting variate 200, the model assigns substantial importance to the history of variate 182, rather than relying only reggrssggsss.... This indicates that VPNet can detect meaningful cross-variate relationships within its receptive field and exploit them for prediction.
Traffic: Utillizaiinn of Local Neighborhood Clusters. In the Traffic dataset (Figure 7 the prediction of variate 400 activates a wide band of nearby variates. This reflects collective neighborhood influence, where VPNet aggregates information from group of highly correlated variates to infer near-future behavior, which is consistent with the strongly correlated structure observed Traffic.
In summary, these visualizations demonstrate that VPNet adapts naturally to the dependency charac- teristics of each dataset: it behaves regreselgyyyy when cross variate signals are weak (ETTh1), identifies meaningful pairwise couplings when present (Electricity), and aggregates rich local neigh- borhoods when the data exhibit strong spatial correlations (Traffic).
Salency Mas Reecetivv Feld)
410
Mum man Wall Mall
Figure 5: Interpretability Analysis on ETTh1 Visualization for predicting variate at the first step. The saliency heatmap and variate importance indicate that the model primarily relies the variate's own temporal history, with minimal cross-variate contribution. This is consistent with the weak inter-variate correlations observed in ETThl. The temporal importance curve also exhibits clear periodicity.
H ADDITIONAL EVALUATION ON GIFT-EVAL BENCHMARK
H.1 EXPERIMENTAL SETUP AND IMPLEMENTATION
We acknowledge the recent emergence of comprehensive benchmarks aimed at standardizing the eval- uation of forecasting models (Liu et al. 2024b Fan et al. 2025), such as the FEV-benchmark (Shchur et al., 2025) and GvFT-Eval (Aksu et al.) (024). To further assess the generalization capability of VPNet a unified evaluation environment, we extended our experiments to the GIFT-Eval benchmark.
To ensure compatibility with the benchmark protocol, we integrated VPNet into the g uonts frame- work and added probabilistic projection head to support both point and probabilistic forecasting.
H.2 PERFORMANCE COMPARISON
We compared VPNet against ITTansformer, a state-offthe the-art baseline under the GIFT-Eval setup. The evaluation metric is the Mean Absolute Percentage Error (MAPH).
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Salleccy Map Zooomed 166-232) Sarpe
19
W
1:00
3.000
Figure 6: Interpretability Analysis on Electricity. Visualization for predicting variate 200. The global variate importance plot shows that variate 182 contributes more significantly than the target variate itself, demonstrating that VPNet successfully identifies strong and meaningful cross-variate dependencies within its receptive field.
Sarieccy Map iooureed 246-432) repe Les.
Maxi Inporaate
04000
08000
E00
Figure 7: Interpretability Analysis on Traffic. Visualization for predicting variate 400. The heatmap exhibits strong activations across wide group of highly correlated variates, indicating that the prediction relies on the collective information from multiple related variables rather than any single one. This highlights VPNet's ability effectively utilize locally correlated patterns in datasets with rich cross-variate structure.
The results presented Table II show that VPNet achieves competitive performance under standardized evaluation pipeline, outperforming ITransformer 17 out of 33 datasets/settings. These findings indicate that VPNet maintains strong robustness and generalization capability across diverse forecasting tasks, further supporting the effectiveness of its locality-driven modeling strategy.
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Table 11:
Performance comparison GIFT-Eval benchmark (Metric: MAPE). Best results
highlighted in bold. VPNet achieves superior performance in 17 settings, demonstrating robustness even under univariate input conditions.
Dataset Setting
VPNet (Ours)
I'Transformer
Winner
bitbrains fast_ /t/sst
2.178
1.689
l'Transformer
bitbrains_fast fasttafsttttttt faaafasssttrrage//shooort
3.272
2.962
l'Transformer
bitbrais_mmd/ST/
1.522
1.268
ITrrasformer
bitbrains_mdd/H//horn
3,280
2.535
ITrrasformer
bzitobbsappliicatoon/////nngg
0.054
0.065
VPNet
0.043
0.047
VPNet
bizitobs
crtiiobs_upplcaatoo//lOSShoooo
0.038
0.041
VPNet
0.463
0.537
VPNet
hizitobs_12c/557mmedium
0.461
0.567
VPNet
bizitobs_1lcc/5T//hon
0.148
0.174
VPNet
bizitobs_l2c//Hllnng
0.664
0.670
VPNet
Dizitobs_2cc/H/mmelll
0.602
0.586
ITransformer
bizitobs_l2c/H/siin
0.643
0.707
VPNet
bizitobs_serrvicee/0///ooo
0.420
0.250
ITransformer
biziobssserrvice//SS/meiii
0.307
0.189
ITransformer
bizitobs_sette
htitbbbs_servicc/////hoorrr
0.147
0.161
VPNet
ettl/15T/long
0.797
0.787
ITransformer
ettl/L5T/medium
0.735
0.749
VPNet
ett e/5//15T/short
0.641
0.622
ITrrasformme
ettl/D/short
1.697
1.623
I'Transfommer
ettl/H/short
0.511
0.506
ITTransformer
ett////hort
0.882
0.770
ITransformer
eu22/15T/logg
0.150
0.156
VPNet
et12/15T/medium
0.144
0.152
VPNet
cu2//.5T/short
0.139
0.162
VPNet
ett2/D/short
0.368
0.464
VPNet
ett2/H/long
0.220
0.219
ITransformer
ctt2/H/mediumm
0.207
0.199
ITransformer
ett2/H/short
0.138
0.139
VPNet
ett2/W/short
0.159
0.292
VPNet
jena_
heau_weather/100//hrrr
0.519
0.416
I'Tranffrmme
eena_wcather/D/short
0.887
1.120
VPNet
sena_weather////hhort
1.427
1.320
ITrrnsformer
I VARIATE ORDERING STRATEGIES
To investigate the impact variate arrangement on docality-based architectures, we consider four ordering strategies. Each strategy reflects a distinct principle for structuring the variate dimension:
Original Ordering. variates are preserved the order provided by the dataset. This ordering reflects any implicit structure imposed during data collection (c.g., spatial layout of sensors or industry grouping of assets). serves as a natural baseline.
Random Ordering. variates are permuted uniformly at random. This destroys any pre- existing adjacency structure and thus provides neutrality test. If a model still performs well under random ordering, it suggests robustness to locality assumptions.
Degree Ordering. variates are ranked by their aggregate similarity (e.g., total correlation strength with others). The intuition that highly connected variates are globally influential, and placing them adjacently increases the likelihood that local operators can capture their dependencies.
Spectral Ordering, variates are arranged by the coordinates the Fiedler vector of the graph Laplacian built from pairwise similarities. This spectral seriation seeks embed
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variates onto line such that strongly related variates appear contiguously. It provides principled way of linearizing high- digh-dimensional dependency structures for local processing.
EFFECT OF LOOK -BAC WINDOW SIZE
model capable of capturing long-range temporal dependencies is expected benefit from longer historical contexts (Zeng et al., 2022; Nie et al., 2023). To examine VPNet's ability to leverage historical information, we conduct an ablation study on the input sequence length L. We evaluate VPNet against two competitive baselines, iTransformer and TimeMixer, on the high-dimensional Electricity and Traffic datasets. The prediction horizon is fixed S 96, while the input length varies as L E [48, 96, 144, 192, 240, 336}. Results are reported in Figure g
All three models generally improve as the look-back window increases, with lower MSE at larger However, their behaviors differ in how performance scales with context. iTrrnsformer and TimeMixer exhibit gradual and consistent improvements across the full range of input lengths, indicating steady reliance longer histories. In contrast, VPNet reaches its best performance with substantially shorter contexts: its error decreases rapidly when increases from 48 to 144, after which further gains marginal. On Electricity, VPNet achieves MSE of 0.127 at L 144, outperforming TimeMixer even at I, = 336 (0. (3.133). These results suggest that VPNet is able to extract the most relevant predictive patterns from moderate-lenge length histories, highlighting its efficiency in utilizing contextual information without requiring excessively long sequences.
Electricity Dataset (Preck Length
Datavet Length
What
vie
Tre
0.55
0.52
0.38
0.25
Irput
Length
Figure 8: The effect of look back window size on forecasting performance (MSE). We compare VPNet, ITransformer, and TimeMixer the Electricity (left) and Traffic (right) datasets with fixed prediction horizon of = 96.
K
FULL RESULTS
Due the space limitation of the main text, we place the full results of all experiments in the following Table 12
L SHOWCASES
For qualitative assessment, we visualize the forecasts representative variate from the test set for each dataset (Figures 9 10
14 15 16). These visualizations illustrate that VPNet's
predictions consistently align more closely with the ground truth, adeptly capturing complex dynamics where other models falter.
M FUTURE WORK
Our findings open several promising avenues for future research. The surprising robustness of VPNet variate ordering suggests that the model captures complex, dynamic relationships that
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transcend static correlations. A deeper theoretical investigation into the nature of these time-lagged dependencies would be a valuable contribution. Another compelling direction is the development of architectures with adaptive locality. While our work shows the efficacy of a fixed local neighborhood, models that can learn to dynamically adjust the scope of their receptive field for different variates or layers could unlock further performance gains. Finally, the variate variate-patch field representation itself may prove to be generalizable concept, and exploring its application to other multivariate sequence modeling tasks (e.g., spatio-temporal forecasting) is a promising direction for future research.
Table 12: Performance comparison on the long-term forecasting task for prediction horizons of (96, 192, 336, ]20} and their average. The input sequence length is fixed to 96. The best and second- best results on each dataset in each metric are highlighted in bold red and underlined blue fonts, respectively.
Models
VPNet (Ours)
TimcKAN
TimeMlier
(TImisfommer
Puthfonner
PuccTTT
Croseamer
MICN
TIDE
TimesNet
Diisear
HEIMfamer
3023
20:3
A02
Metric
MSB
MAE
MSB
MAE
MSB
MAE
MSB
MAB
MSC
MAB
MSC
MAB
MSC
MAE
MSE
MAB
MSC
MAL
MSB
MAIl
MSE
MAD
MSE
MAD
96
0
0.157
0.192
0,162 0 308 0.163 0.20%/0.1144
0.214
0.156
0.192
0,16
0.227
0,195
0.271
0.199 0.26
0.202 0.261
0.172 0.220
0.195
0.25:/..217
0.217 0.296
Weather
192
0.207
0.239
0.207
0.249 0.208 0.250
0.254
0.206
0.2400.234
0.240
0.234
0.265 0.209 0.277
0.239
0.299
0.242 0.298
0.299
0.219 0.261
0.237 0.295
0.276 0.336
336
0.258
0.263
200 0.251 0.287
0:278
0.296
0.255
0.287 0.284
0.301
0.273 0.330
0.285
0.336 0.267
0.351
0.339 0.360
720
00.330
0.333
0.338
0.3 0.339 0.311
0.347
[0.340
0.336
0.356
0.349
0.349
0.401
00.351
0.388
(.355 0,380
0.366
0,359
(03315
0,382
(0,403 0.428
Avg
(0.238
0.261
(0.243
0.278
[0.230
0.26
00.26
0.286|0.264
0.320|o.268
0.321
(0:27] 0.320f0.299
0.257
|0.265
0.315
(.315|0.509
(0.309 0.300
Solar-Eaengy
96
10.177
0.178
(0.234
0.290/0
0,259lo.205
0.257
10,202
0.225|0.265
10.265
0.32:30.232
0.302|0.25
0.325
0.399/0.372
0.355 (0.290
(0.286 0.33//o.666 0.341
0.378
192
4.19
0.206
0.274
0:322
0:222
0.233
0.251
0.235
0.245
0,288
0.5320.371
0.371
0.410.0.278 0.354 (.330
0.399 0.370.
0.320
0.35e 0.291 0.337
336
0.216
0.219
0.242
0.273
0.272
0.272
0.301
0.339
0.51 00.20
0.575 0.366
0.43000.400 o. 350
0.330..335
0.415
0.354 0.416
0.228
0.225
0.295
0.223
0.285
0.285
0:349
0.275
0.2255
0,256
0.295
0.336
0.526
0.542 0.299 0.379 0.37700030000000 0.420 0.420
0.383 0.357
0.413
0.380 0.437
|Ag 10.204
0.207
(.2266.3100
0.280/..33
0.252
[0.241
0.250|0.287
0.333
0.442|0.283
0.355/p..47
0.417|0
0.394
|o.330
0.400
[0.328 0.383
Electrieity
96
0..35
0.26600.2660..555
0.24770..234
0.240
10.145
0,236/0,
0,190
0.296102219 0,314
180
0.293/0.257
0.18
0.202
10.210 0.232/0.210
0.302
0.302/0.193 0.308
192
0.151
0.239
0.182
0.273
0.266
0.253
0.167
0.256 0.199
0.304 0.231
0.182 0.302 (1236
0.189
0.330
0.33
0.144
0.322
0.210 0.300 0.201 0.315
836
0.166 0.257
0.2970.288...11
0.277
0.178
0.250
0.186
0.275 0.217
0.319 0:246
0.537 0.198
0.513
0.223 0.214 0.329
720
0.196
0.2844
0.236 0.200.3.3200
0.317
0.231
0,
0.258
0.352 0,280 0.363
(12217 0.330
0.260
0.323
0.200
0,300
0.255
0.350
(1.246 0.35
|Avg
0..62
0.251
(o.197
0.286|0.182
0.273
0.270
Jo.182
0.269
0.2216 0.221 0.316
0.334 1916 0.30>
0.30>|0.222
0.344
0.50d
|0.22|
0.319/0.214 0.327
96
00.384
0.258
0.550
0.39900377
0.268
104179
0.28310.526
0.347 00664
0.429/0.577
0.350
0.693/0.593
0.521
(0.650
1..96 0.366
(0.587
192
0.406
0.266
0.550:0.365
0.473
0.296
0.417
0.276
0.484
0.2920.522
0.332..666 0.665
0.431 0,589
0.356
(1756
0,474
0.617
0.336
0.598
0.370
0.804 0.373
836
0.429 0.275
0.559
0.468
0.296
0.433
0.253
0.503
0.299 0.517
0.334
0.674
0.420
0.594
0.355
(.762
0.477
0.629
0.336
0.605
0.573
0.621 0.383
0.466
0.294
0.6000 0.506
0:537
0.322 0,552
0.352
0.683
0.434 0.6613 0.361
0.430
0.640
0.350
0.645
0.304
0.626 0.382
Avg 00.421 0.273
0.29%
0.282
00,501
0.529
0.341
0.341
06667
0./26/0..933
0.426
0.366
0.475
0,336
(.625
0.383
0.383
96
00.374
0.386
(0.374
0.37
0.405
10.390
0. 0.39000.460
0,447/0.423
0.445|0.4426
0,446
0.479
0.464|0.344
0.384
0,400
00399
10.395 0,43000300 0:424
0.413
192
0:428
0.418
0.416
0.122 0.119
0.121 0,441
0.436
00437
0.419 0.477
0.129
0.471
0.474
0.154
0.464
0.525 0.492
0.436
0.429
0.496
0.641
0.469 0.470
3,36
0.464
0.439
00.45
0.443 0.484
0.45%
0.458
0.497
0.445 0,546
0.496
0,570
0.546
0.493
0.487
(1566 0.515
0.491
0.469
0.489
0.467
0.530 0.499
720
0.47)
0.465
0.463
0.463
0.582
0.401
0..49
0.461
0.544
0.517
0.653
0.621
0.552
0.526
0.594
0.558
0.521
0,500
0.511
0.510
(1.5% 0.544
Avg
01.434
0.427
(0.426
0.431
(0.444
0.440/o.454
0.447
10:455
p.429|0.507
0.47
(0.29
0.52:
00.475
0.481
[0.54] 0.507
0.507|a.458
0.458
0.450
|0.661
(0.498 0.458|0.499
0.458
0.484
96
10.284
0.330
0.300
0:341
10:077
0,549
10.290
0. 0.33610.
0.355/0.745
0.584
(0.372
0.424
0.400
0.440/0
(4340
0.354
10,340
(0.35% 3.797
192
0.357
0.39900396
0.396
0.39200380
0.400
0.372
0.365
0.405
0.656
0.192
0.492 0.528
0,509
0.402
0.414
0.482
0.479
0.129 0.439
836
0.302
0.406
0.423
0.435
0.386
0.444
0.43
00.402
0.477
0:416 0.427
9.436
1.043
0.731 006007 0 0.555
(3.43
0.571
0.452
0.452
00.591
0.541
(1.496 487
003991
0.418
0.473
0.465
46500.112
0:434
0.645
0.440
0.444 0:436
0.450
1.104
0.763
00.824 0.655 0.994
0.679
0.482
0,468
0.661
0.463 0.474
AAg
|0.356
0.383
10339
2.45
0.395
0.407
Jo.374
0.395
0.412|0.942
0.68:|0.574
0.684
(0.574 0.531
[061:
0.550/o.414
0.427
[.42700.563
|0.563
0.519/0.437 0.449
96
10.313
0.34
0.36660..2000
0.337
0.308
00.318
0.369/0.352
0.352
0,37e/0.104
00.104
0.387
0,387
0,387/0.338
0,535
|0.3-6
(0,379 0,374/0099.999 0.419
0.374
192
0.362
0.371
0.363
0,380 0.361
0.381
0,390
0.593
0.365
0.372
0.374
0.387
0.450
0.451
00.403
0.408
(1396
0,404
0.374
0.387
0.382
0.391
0.442 0.441
336
ML385
0.389
0,300
4040 0:46
0.420
0.401
0.391
0.421
0.414
0.532
0.51. 0.446
0.431
[1425
0.425
0.411
0.4415
0.415
0.445 0.459
720
0.444
0.424
0.460
o 443 0.54
0:44
0.491
0.459
0..66
0.432
0.462
0.449/0. 0.666
0.580
00.489 0.462
0.487 0.461
0.478
0,450
(1.475
0.451
(0.545 0.490
Avg
(0.376
0.382
6.336 0.398| 0.396/0.410
0.430
|0.382
0.386
00402
0.106|0.513
0.195|0.123
0.422/0.419
0.419/0.400
0,406
|0.900
0.408
|0.448
0.452
96
10.169
0.244
0.1174
0.25500017
0,25
0.264
10.168
0.247|0.185
1813
0.270 0.366
1917
0.296|0.207
0.005|0.187
0,267
(0.193
0.293
(0.203 0.287
192
0.233
0.288
0.239
0.299
0.29
0.309
0.234
0.291
0.255
0,314 0.414
0.492
0.284 0.361 0.290
0,364
0.249
0,309
0.284
0.361
0.269 0.328
336
0.291
0.327 0.301
0.39
0.311
0.548
0.297
0.33:
0.309
0.347 0.597
0.542 0.381 0.429
(.377
0.422
0.921
0.351
0.382
0.429
0.325 0.306
720
0.587
0.987
1,395 0.396
0.396 0.407
0.336
0.385
0.412
0:404 1.730
0.:549
0.522
p.588 0.504
0.408
0.408
0.555 0.525
(.422 0.415
Avg
j0.270
0.312
(02277 0.323 0.323/0.288
0.532
(0.271
0.31
0.290
0.334
(..57
0.611
|0.35 0.10000.022..0
0,404|0.291
(02291 0,335
(0.34
0.400
(0.305 00.300
0.349
22

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Published as a conference paper at ICLR 2026
-00
-1.
(a)VPNet
(b)TimeKan
(c))imeMixer
Figure 9: Showcases from ETThl by different models under the input-96 coput-96-preeeee 10----6eee-----9 settings.
148
-150
-37
-.5
+15
-223
(a)V
(b)TimeKan
(c)TimeMixer
Figure 10: Showcases from ETTh2 by different models under the input-96-predict-6 settings.
(a)t
(b)TimeKan
(c)mimeMixer
Figure 11: Showcases from ETTml by different models under the input-96- predict-96 settings.
input-96-predict-96
32-
32
-4
(a)t
(b)TimeKan
(c)TimeMixer
Figure 12: Showcases from ETTm2 by different models under the 9nput-96-predict-9 settings.
23

---

Published as a conference paper at ICLR 2026
012
012
M Mil
(a)VPNet
(b)TimeKan
(c))imeMixer
Figure 13: Showcases from Weather by different models under the input-96 9o---ddcctr---- settings.
33-
15
13
(a)VPNet
(b)TimeKan
(c) imTemeiieer
Figure 14: Showcases from Solar-Energy by different models under the input-96-predict-6 settings.
1.4
-35
(a)t
(b)TimeKan
(c)mimeMixer
Figure 15: Showcases from Electricity by different models under the couut-96--redicc--e settings.
(a)VPNet
(b)TimeKan
ic)TimeMixer
Figure 16: Showcases from Traffic by different models under the input-96--re settings.
24


Reviews:
[{"review_id": "PPn04YaShs", "reviewer": "Reviewer_WGdL", "summary": "This paper tackles multivariate forecasting by questioning the need for global cross-variate attention and proposes VPNet, which targets local cross-variate interactions for better scalability. The authors offer theoretical and empirical support that dense dependency systems can be modeled effectively with locality. They also propose a specialized VarTCNBlock—depthwise 2D convolutions to capture joint spatio-temporal (cross-variate + temporal) structure, which haslinear complexity in the number of variates. Experiments across multiple datasets claim state-of-the-art accuracy with notable efficiency gains.", "questions": "1. The title in the manuscript is different from that of the submission page. I think it should be revised. \n2. I wonder how we define the \"appropriately chosen finite local neighborhood\" in Local Sufficiency Hypothesis. The variates naturally do not have orders and are permutation-invariant. Although the ablation study provides some results on the effect of variate reordering. I feel that might not be valid on all datasets. Some orders are better than others on some datasets. However, I do not observe some patterns/insights on the results - e.g., which order we should use for which dataset. \n3. I am curious how performance may change if we vary the input length. Because in my view, the variate locality may heavily depend on the input length. For example, if we really have a short input length, it may not be able to capture the dependencies.", "limitations": "", "rating": 4, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "1. The motivation to reduce the complexity of capturing cross-variate dependencies is sound. The authors investigate whether we really need to capture global dependencies considering all variates. \n2. The authors tried to provide some theoretical analysis for this local sufficiency hypothesis, which has some merits. \n3. The efficiency of the proposed method seems good compared with other baselines, even iTransformer (which only considers cross-variate dependencies without explicitly considering time dependencies)", "weaknesses": "1. I feel that the theoretical analysis of the proposed Local Sufficiency Hypothesis is not very convincing. My concern is mainly on the orders of variates. Please see the later question I raise on the \"finite local neighborhood\".\n2. The experiments are not comprehensive enough in my opinion. There are several more comprehensive benchmarks proposed since 2025, e.g., fev-benchmark [1] and Gift-Eval [2]. I would suggest to have some results on those benchmarks and investigate how it compare with other baselines. \n\nReferences:\n\n[1] fev-bench: A Realistic Benchmark for Time Series Forecasting\n\n[2] GIFT-Eval: A Benchmark For General Time Series Forecasting Model Evaluation", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "xqqt4HHbGX", "reviewer": "Reviewer_XMH1", "summary": "The paper challenges the common assumption that multivariate forecasting requires global cross-variate modeling. It proposes the Local Sufficiency Hypothesis—that, in dense dependency regimes, a small local neighborhood across variates is enough for accurate prediction. Building on this, the authors introduce VPNet, which (i) encodes sequences into patch embeddings with a lightweight overcomplete autoencoder, (ii) channelizes them into a 2D variate–patch field, and (iii) processes this field with stacked VarTCNBlocks, yielding linear complexity in the number of variates. Extensive experiments on eight benchmarks report state-of-the-art (SOTA) or competitive results with strong efficiency.", "questions": "1. Can VPNet **learn** variate orderings or local neighborhoods (e.g., via learned permutations, dynamic kernels, or dilation) rather than relying on fixed layouts?\n2. Do the authors have stress tests where inter-series dependencies are sparse/heterophilous or dominated by long-range cross-variate relations?\n3. The paper gives a probabilistic guideline for kernel selection; can the authors show empirical calibration of that rule (Eq. 20) on real datasets?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The Local Sufficiency Hypothesis is a useful lens to revisit the global-vs-local trade-off, with a simple probabilistic argument that motivates local kernels.\n2. The variate–patch field + VarTCNBlocks pipeline is conceptually clean and easy to implement. The channelization step makes the locality idea actionable.\n3. The proposed design keeps computation linear in the number of variates while still modeling cross-variate signals—addressing a pain point of global attention models.\n4. Evaluation spans 8 common datasets (Weather, Traffic, Electricity, Solar-Energy, ETTh1/2, ETTm1/2) with standardized horizons and protocol; results are competitive or SOTA.", "weaknesses": "1. The theory assumes random variate permutations when quantifying the probability that a local window contains an informative neighbor. Real systems often have structured (and non-exchangeable) dependencies; the proof does not address such structure nor provide guarantees beyond dense regimes.\n2. While the paper claims robustness to different orderings in one setting, the practical definition of “local neighbors” depends on variate ordering. More systematic analysis (beyond a small-kernel, shallow-stack case) is needed, including learned or data-driven orderings and lag-aware neighborhoods.\n3. The strongest gains appear on high-dimensional datasets (Electricity/Traffic). On lower-dimensional ETT, improvements are smaller and sometimes only “competitive,” suggesting locality helps most in dense regimes; the paper could better quantify when global modeling is still useful.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "DjwKfRUI5u", "reviewer": "Reviewer_vhBd", "summary": "This paper proposes VPNet, a multivariate time-series forecasting framework that models localized cross-variate interactions via depthwise 2D convolutions on a variate–patch representation. The authors argue that global dependency modeling is not necessary and introduce the Local Sufficiency Hypothesis as theoretical justification. Experiments on eight public datasets show improvements over recent baselines.", "questions": "1. The Local Sufficiency Hypothesis is motivated theoretically under random or dense-dependency assumptions.  Could the authors provide empirical analyses of variable correlation / Granger causality / covariance decay across datasets to verify that real-world dependencies are indeed predominantly local?\n2. Can the authors identify conditions under which VPNet fails or requires adaptation? A discussion on when local modeling is insufficient would help clarify the scope of applicability.\n3. Since VPNet essentially enforces locality through convolutional receptive fields, how does it compare to local-window attention or strided/sparse attention variants on the same datasets?\n4. The model claims to capture local cross-variate structure. Can the authors provide patch-level saliency / variable importance / receptive field visualization to show what dependencies the model actually learns?\n5. The patch autoencoder is claimed to improve robustness to non-stationarity. Do the authors have shifted-train/test evaluations (e.g., seasonal shift, regime shift in traffic/energy data) to demonstrate this effect explicitly?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 2, "presentation": 2, "contribution": 3, "strengths": "1. The paper provides a clear and well-motivated perspective questioning the necessity of global dependency modeling in multivariate forecasting. By framing the problem through the Local Sufficiency Hypothesis, the work contributes a conceptual shift that has relevance for both theory and practical model design.\n2. The proposed Variate–Patch Field representation is intuitive yet effective, allowing variable-wise and temporal patterns to be captured jointly using simple depthwise 2D convolutions. This design reduces computational complexity from quadratic to linear with respect to the number of variables while preserving modeling capacity.\n3. The empirical evaluation is thorough, covering diverse datasets with significantly different dimensionalities and dynamics. The model consistently outperforms competitive baselines, particularly in high-dimensional settings where efficiency matters most.\nEmpirical results are strong and consistent across multiple datasets.", "weaknesses": "1. Insufficient empirical support for the Local Sufficiency Hypothesis.\nThe central assumption that cross-variate dependencies are predominantly local is not thoroughly validated using real-world correlation or interaction patterns. The justification relies more on theoretical reasoning than on dataset-driven evidence, which weakens the persuasiveness of the hypothesis.\n2. Lack of discussion on scenarios where locality may be insufficient.\nWhile the model performs well on the evaluated datasets, it is unclear how it behaves in settings with strong long-range or sparse inter-variable dependencies. Without an analysis of potential failure cases or applicability boundaries, the generality of the proposed approach remains uncertain.\n3. Limited interpretability analysis regarding learned dependency structure.\nGiven that the core contribution concerns how the model captures local cross-variate relationships, the paper would benefit from visualizations or interpretability studies (e.g., patch importance, receptive field inspection). These insights are necessary to substantiate the claims about the model’s learned dependency patterns.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)