Routing Channel-Patch Dependencies in Time Series Forecasting with Graph Spectral Decomposition
Authors: Dongyuan Li
Shun Zheng
Chang Xu
Jiang Bian
Renhe Jiang
Venue: ICLR 2026 Poster (2026)
Keywords: Time Series Forecasting
Channel Dependency
Graph Learning
Primary area: learning on time series and dynamical systems
TL;DR: We propose xCPD, a lightweight plugin that models channel-patch dependencies via spectral decomposition and routing, enabling accurate and generalizable forecasting with minimal overhead.

Abstract: Time series forecasting has attracted significant attention in the field of AI. Previous works have revealed that the Channel-Independent (CI) strategy improves forecasting performance by modeling each channel individually, but it often suffers from poor generalization and overlooks meaningful inter-channel interactions. Conversely, Channel-Dependent (CD) strategies aggregate all channels, which may introduce irrelevant information and lead to oversmoothing. Despite recent progress, few existing methods offer the flexibility to adaptively balance CI and CD strategies in response to varying channel dependencies. To address this, we propose a generic plugin xCPD, that can adaptively model the channel-patch dependencies from the perspective of graph spectral decomposition. Specifically, xCPD first projects multivariate signals into the frequency domain using a shared graph Fourier basis, and groups patches into low-, mid-, and high-frequency bands based on their spectral energy responses. xCPD then applies a channel-adaptive routing mechanism that dynamically adjusts the degree of inter-channel interaction for each patch, enabling selective activation of frequency-specific experts. This facilitates fine-grained input-aware modeling of smooth trends, local fluctuations, and abrupt transitions. xCPD can be seamlessly integrated on top of existing CI and CD forecasting models, consistently enhancing both accuracy and generalization across benchmarks. The code is available [https://github.com/Clearloveyuan/xCPD].

Published as a conference paper at ICLR 2026
ROUTING CHANNEL-PATCH DEPENDENCIES IN TIME SERIES FORECASTING WITH GRAPH SPECTRAL DE- COMPOSITION
Dongyuan Li
Shun Zheng²
Chang Xu
Jiang Bian²
Renhe Jiang'
The University of Tokyo
Microsooft Research Asia
{lidy, Jiangrh)ecsis. u-tokyo ac. ip
(shun.zhen) chanx, jiang
ABSTRACT
Time series forecasting has attracted significant attention in the field of AL. Pre- vious works have revealed that the forecasting performance by modeling each channel individually, but often suffers from poor generalization and overlooks meaningful inter-channel interactions. Con- versely, Channel- Deeenndet (CD) strategies aggregate all channels, which may in- troduce irrelevant information and lead to oversmoothing. Despite recent progress, few existing methods offer the ficxibility to adaptively balance CI and CD strategies in response varying channel dependencies. To address this, we propose a generic plugin xCPD, that can adaptively model the channel-patch dependencies from the perspective of graph spectral decomposition. Specifically, xCPD first projects mul- tivariate signals into the frequency domain using shared graph Fourier basis, and groups patches into Iow-, mid-, and high-frequency bands based their spectral energy responses. xCPD then applies a channel-adaptive routing mechanism that dynamically adjusts the degree of inter-channel interaction for each patch, enabling selective activation of sequuency-speciff experts. This facilitates fine- fine-graine -o-iggrinnded input- aware modeling of smooth trends, local fluctuations, and abrupt transitions. xCPD can be seamlessly integrated on top of existing CI and CD) forecasting models, consistently enhancing both accuracy and generalization across benchmarks. The code is available [https: //tpt/ggtttt///t/ com/ CeeeCleelllaaa /cl.a/rlooeeaaaaeee)
(CI) strategy improves
INTRODUCTION
Multivariate time series forecasting (MTSF), a fundamental task in time series analysis, has emerged as a key focus in artificial intelligence due to its broad applications energy, and environment. Early efforts primarily focused on adapting variety of model architec- tures-ranging from linear models (Zeng et al., 2023; Li et al., 2023) and convolutional networks (Liu et al., 2022a; Wu et al., 2023) to Transformers (Nie et al., 2023; Liu et al., 2024a) this task, establishing a strong foundation for the field. Whatever the model architecture, the key to success lies in accurate modeling of both temporal dynamics and inter- variable (a.ka., channel) dependencies. Building on this insight, recent works have been focusing on the channel strategy, giving rise to three distinct time series modeling paradigms: Channel Independence (CI), Channel Dependence (CD), and Channel Partiality (CP). CI methods (Zeng et al., 2023; Zhou et al., 2021) model each channel separately, capturing only intra-channel temporal dynamics while completely ignoring inter- channel interactions. This design enhances robustness and scalability but often overlooks valuable inter-channel patterns. CD methods (Zhang Yan, 2023; Liu et al., 2024a), on the other hand, jointly model all channels to capture global dependencies, but may suffer from oversmoothing and noise propagation from heterogeneous or weakly correlated variables. CP methods (Qiu et al., 2024; Chen et al., 2024; Hu et al., 2025b) attempt balance CI and CD by selectively modeling inter-channel relationships, adaptively adjusting the dependency structure based on contextual or learned relevance.
transportation, finance,
"Corresponding Author

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Published as a conference paper at ICLR 2026
Despite their success, existing CP methods fundamentally struggle capture fine- aieeggrined, patch-level, and seeuunccydecoupleed channel dependencies. First, current CP approaches operate primarily at the channel level, either by grouping entire channels via clustering (Qiu et al., 2024; Chen et al., 2024) or learning channel-wise attention weights (Hu et al., 2025b; Lee et al., 2025). This reliance on the entire channel embedding as the relational unit introduces coarse-grauularity bottleneck: they fail to model localized patch-level interactions or capture nuanced segment-specific patterns. For example, if Channel-A exhibits a smooth seasonal trend during segment T₁ but a sharp anomaly during segment T2, channel-level models generate single averaged dependency weight for the entire channel, failing to capture the distinct local interactions that trend and anomaly segments should establish with other channels. Second, while CD/CP models produce time-varying attention weights, these interactions are computed entirely the temporal domain, where low-, mid-, and high-frequency components are mixed within a single embedding (Huang et al., 2023; Donghao & Xue, 2024; Zhao & Shen, 2024). As a result, a high attention score between two channels may simultancously reflect meaningful low- ewwffreqeeccy seasonal dependence and irrelevant high-frequency noise component, without distinguishing between them. This frequency coupling forces the model to treat heterogeneous spectral behaviors single dependency signal, often introducing spurious correlations-espedee when high-frequency noise and low-frequency trends coexist. Consequently, temporal-domain dynamism alone is insufficient for capturing fine-grained and brequency-speciifee dependency evolution, leaving important dynamic cross-channel relationships entangled.
To address these fundamental limitations, we reformulate the basic modeling unit from an entire localized temporal segment within a channel, following the widely
channel to a channel-patee adopted patching paradigm in modern sequence modeling. We represent each channel-patch as node in graph, where edges denote the interactions (dependencies) between channel- patch pairs. This novel graph perspective enables unified and highly granular treatment of channel- patch dependencies (CPD), encompassing inter-channel patch, intra-channel patch, and patch-level channel interactions. Building on this foundation, we propose xCPD where "x" denotes graph spectral decomposition for Channel-Patee Dependencies. xCPD is generic and lightweight plugin that enhances existing forecasting models by adaptively routing CPD through the lens of graph spectral decomposition. Unlike prior methods, xCPD operates exclusively in the spectral domain, where it leverages a shared graph Fourier basis to embed channel- -pathh representations and subsequently groups them by their spectral energy responses. This design enables the model to distinguish between smooth, moderately varying, and highly fluctuating patterns (i.e., low-, mid- and high-frequency). A filters
dynamic fixture-off-xpertt routing mechanism further adaptively selects frequency-specie for each patch, enabling xCPD to capture time-varying dependencies while suppressing irrelevant frequency-couue coupled correlations. Finally, xCPD is fully model-agnostic and plug-and and-play, allowing seamless integration into a wide range of CI and CD backbones, including large foundation forecasters, without retraining. This provides practical and scalable solution for real-world forecasting pipelines.
RELATED WORK
Multivariate Time Series Forecasting (MTSF). The core challenge of MTSF lies in jointly modeling inter-channel dependencies and temporal dynamics (Li et al., 2023; Shang et al., 2024). Early approaches like ARIMA (Zhang, 2003) and Prophet (Triebe et al., 2021) focus exclusively on temporal patterns within individual series, lacking mechanism for inter-channel modeling. To address this, deep learning models have been widely explored (Zhou et al., 2022a; Yi et al., 2023b; Zhang et al., 2024). Specifically, RNNs (Hewamalage al., 2021; Lin et al., 2023) capture sequential patterns through recursive hidden states but suffer from vanishing gradients when modeling long-term dependencies. CNNs (Wu et al., 2023; Dai et al., 2024) offer efficient extraction of local features through convolutions, but their limited receptive fields constrain global context understanding, To overcome these limitations, Transformers (Wu et al., 2021; Zhou et al., 2022b; Liu et al., 2022b; Feng et al., 2024; Liu et al., 2025) use self-attention mechanisms to capture long-range dependencies, but their permutation invariance can compromise the temporal order, which is crucial for forecasting. Recently, MLPs (Zeng et al., 2023; Chen et al., 2023; Das et al., 2023; Hu et al., 2025a; Yu et al., 2024) have shown that simple dense connections across timesteps can achieve competitive performance without architectural complexity. Beyond temporal dynamics. GNNs (Yi et al., 2023a: Yang et al., 2023; Wang et al., 2024b; Cai et al., 2024) model inter-channel dependencies by treating multivariate series as graphs, capturing structured cross-variable dependencies more effectively.
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Channel Dependency Strategy. In MTSF, dependency modeling strategies fall into three categories: CI methods (Zhou et al., 2021; Jin et al., 2024: Liu et al., 2024b) treat channels independently, capturing only intra-channel temporal patterns. While this design enhances robustness by avoiding inter-channel noise, it fails to use valuable inter-channel relationships. CD methods (Liu et al., 2024a; Woo al., 2024; Gao al., 2024) jointly model all channels to exploit inter-channel dependencies, but introduce spurious correlations that degrade performance in noisy or heterogeneous settings. CP methods (Huang al., 2023; Donghao & Xue, 2024; Zhao & Shen, 2024; Lee al., 2025) balance CI and CD strategies by selectively modeling inter-channel interactions based on intrinsic channel relationships. Recent approaches like DUET (Qiu et al., 2024) and CCM (Chen et al., 2024) employ clustering or sparse attention to adaptively identify relevant dependencies. However, although recent methods such as TimeFilter (Hu et al., 2025b) and PCD (Lee et al., 2025) introduce time-varying and even patch-level interactions, they still operate entirely in the temporal domain. As a result, their dependency scores remain ceqquency-coupled a single attention mixes cow-frequency trends, mid- mud-frequency variations, and high- tighffreuuency noise into one unified correlation signal. This prevents the model from distinguishing which frequency component responsible for the observed interaction, limiting its ability to capture frequency-aware and fineggriined patch-level dependency patterns that evolve over time. In contrast, xCPD performs dependency modeling in the spectral domain the channel-patch level, enabling fine-grained and interpretable modeling of frequency-specifee interactions that evolve over time. Specifically, xCPD differentiates itself from prior CP methods across three critical dimensions: modeling granularity (channel-patch vs. channel), modeling domain (spectral vs. temporal), and adaptivity (frequency-speciis routing vs. mixed attention). Furthermore, as lightweight and fully aodel-ag plugin, xCPD can be seamlessly integrated into wide range of forecasting architectures without any backbone retraining, substantially enhancing its practical efficiency, deployment flexibility, and scalability in real-world forecasting systems.
PRELIMINARIES
Multivariate Time Series Forecasting. Let X =xxx,...... [x1,....,.c E RCXT denote multivariate time series, where C' represents the number of channels (i.e., variates) and T denotes the lookback horizon. MTSF aims to build predictive model fmodd: R that takes historical observations X as input and generates future predictions X [ai..... #c] Rox"", where T' is the forecasting horizon. In this study, rather than building Smodet, we develop a generic plugin model fylugini RxT' RRC that post-proe post-processes the predictions from existing models. Given base model's 's output X Smoda (X),
our plugin refines predictions to obtain improved forecasts X = fplugin (Ximolee).
Graph Signal Processing. Given a graph g = (V,E) with node set V and edge set E, let A and D denote the adjacency matrix and the degree matrix, respectively. The normalized Laplacian graph is 1/2 defined as L = I D D-12 AD 1/2, the graph Fourier basis, and A diag(Ai...... An) is a diagonal matrix containing the eigenvalues.
11/2, Its eigendecomposition is L UAUT, where U represents
METHODOLOGY
We propose xCPD, a lightweight plugin module designed to enhance multivariate forecasting models. As shown in Figure 1, xCPD comprises three key components: Spectral Channel-Patcl Embedding (Section 4.)), Channel-Patch Grouping (Section 4.2), and Channel-Patce Routing (Section 4.3).
4.1 SPECTRAL CHANNEL-PATCH EMBEDDING
As shown in Figure 1(A), we first construct channel-patch graph to model fine-grained peteh-level dependencies, where each patch from every channel forms a "channel-patch" node. We then apply graph spectral decomposition obtain spectral embeddings that capture spectral dependencies.
Channel-Patel as Node. We first partition the output X nmddel E CRC from a time series model
into contiguous, non-overlapping temporal patches X patch E (CX(NxP), where N = [ is the number of patches, and P is the patch length. Subsequently, we apply Linear(-) to transform each patch from length P to hidden dimension d, obtaining the embedded tokens eemh e RR Nxxd as:
X patch Patching( (
emb Linear X Patch)
(I)
By flattening the dimensions to n = C x N, we obtain channel-patch node embeddings as X emb e Raxd For the embedding matrix of the 1-th input Xemb,t, construct weighted similarity
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Published as a conference paper at ICLR 2026
(C) Channel Routing
mmmmi
Rocting Metwork
Channel Patch
Node Emheediing
11
Pasion
Embeedling
Select Nil--rre, illgh-fraj Filtare
(Ros)
Channet Patch Filtering with Moß
Adaptive Graph Learning
Channel Patch
(B) Spectral Channel Patch Grouping
Dependenccy Flugin. xCP
Channel-PPin
Low
Time Series Model
Alid
Alligh
Resporse
Energy
Fiih
Linear
High
Comsenkation
Transformer
Presuesce Band Partition
Spectral Node Gromping
(A) Spectral Channel-Pated Embeedding
Normalization
Channel
Timesttpss
Channel Putch Node
GrephhSpecttrll Graph Spectral Rarbedding
Figure 1: The proposed XCPD plugin consists of three modules: (A) spectral channel-pated embed-
ding, (B) channel-patee -aannel-patch grouping, and (C) channel-pater routing with Mixture-of -f--xeerts (MoE).
graph using a somila-similarity formulation: At, = cos (x;
X. Cosine similarity yields
affinity values bounded within [-1, 1] and automatically introduces self-connections through A 1. Because it is invariant to scaling. the resulting graph is unaffected by magnitude differences across channels, which is especially important in MTSF where variables operate at different physical scales.
Graph Spectral Embedding. Time series channels often exhibit complex dependencies that are more discemible in the spectral domain, where correlated channels tend to share similar spectral patterns (Cao et al., 2020). Motivated by this, we learn spectral embeddings for channel-patel nodes effectively capture their underlying spectral dependencies. However, batch-wise spectral decomposition yields inconsistent Fourier bases across batches, resulting in incomparable embed- dings (Streicher al., 2023). To overcome this, we introduce shared graph Fourier basis that ensures a consistent spectral domain across all timesteps. This shared basis is learned to closely approximate the eigenbasis of individual batches, minimizing their differences as bounded in Theorem 4.1.
Theorem 4.1 (Shared Graph Fourier Basis). Let L' be normalized Laplacian graph derived from trix A the t-th timestep where E (I....,"') Define the average Laplacian as Love=t Σt" L', U denotes the shared Fourier basis that approximates each individual eigenbasis Us from the decomposition of L', satisfying IU'-UR where C depends on the eigen-gap of Long and R is an orthogonal rotation matrix.
with rigen-decompositiee =
C Lavgl
Theorem 4.1 establishes the theoretical Foundation for our shared Fourier basis, ensuring that all batches are mapped to a consistent and comparable spectral domain. Importantly, because the basis is learned from the normalized Laplacian, whose eigenvalues lie within the bounded interval [0,2], the resulting spectral operator is both scale-invariant and numerically stable across timesteps. Following Theorem 4.1, we construct the shared basis U and compute spectral embeddings as:
x spst
emb.t) T embtt E Igaxd
where F(-) denotes the graph Fourier transformation, and sps denotes nodes spectral embeddings. For notational simplicity. we omit the time index f hereafter, e.g., we write x instead of X spe..
4.2 SPECTRAL CHANNEL- GROUPING
As shown in Figure 1(B), develop spectral grouping mechanism that classifies nodes by their dominant frequencies (low/mid/high) through energy analysis. We then construct ego-graphs connect- ing nodes with similar frequency properties capture meaningful frequency-awai dependencies.
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Frequency Band Partition. We employ learnable parameters T1, 72 [1, with T1 72 define adaptive boundaries. To ensure differentiabilityy we compute soft membership weights as follows:
Sigmoid(m(T - -)), aj
high Sigmoid(m(j - T2)), mid = aj
high
where j (I,...,y)) n) indexes frequencies and temperature m controls sharpness. oj E [0, form a normalized convex partition, ensuring stable band membership for each frequency component.
Spectral Node Grouping. X captures feature-level frequency responses (Cao et al., 2020), but does not directly reflect node-specific firequeccy intensities. We address this by defining spectral energy function to quantify how strongly each node responds to different frequency components as:
Theorem 4.2 (Spectral Energy Response). Given spectral node embeddings X and graph Fourier basis U. spectral energy response of node for frequency j is S.,, = ||Ut,, x 1121. The orthonormal- ity of U ensures energy preservation between spatial and spectral domains: Σك Su, || 13. confirming spectral energy response retains all information from original channel-patch embeddings.
Based on Theorem 4.2, we classify the nodes by their maximum spectral energy response as:
exp(E a Saa)) (0,1, E (E exp (1 ay S 1, denotes (low, mid, high}, (4)
Group(i) = arg max
where at are the soft membership weights and S the spectral energy response of Theorem 4.2.
Following Huang et al. (2023), we decompose the graph G (V,E) into n ego-graphs {9) to reduce noise. Each ego-graph G; (V,E) is centered node i. We construct i-th ego-graph adjacency matrices A using the k-Nearest Neighbor (k-NN) method on similarity matrix as:
{ Apg if 00
A = K-oNNNoom(( (X emb
( (x emmb ()ammb)),cc))) a),
A
(4) Pg
=
otherwise,
(5)
where p.q E (1.....n), n), a is a hand-tuned scaling factor that determines the number of neighbors k aa n|, and Norm(-) applies row-wise normalization. Within each ego-graph G. create frequency-specific subgraphs by grouping neighboring nodes based on their spectral labels, enabling frequency-aware message passing, as illustrated in Figure
4.3 SPECTRAL CHANNEL-PATCH ROUTING
As shown in Figure I(C), we introduce a Mixture of Experts (MoE) routing module that dynamically selects rreueecc-sppcciff filters for each node capture diverse spectral dependencies. This adaptive selection enables tailored graph learning of channel-patch relationships.
Channel-Patch Filtering with MoE. Different time series exhibit heterogeneous spectral depen- dencies. To address this, our routing network assigns each ego-graph to one of three specialized frequeney filters, which construct adjacency matrices by emphasizing relevant spectral components while suppressing irrelevant dependencies. We design three specialized filters as follows:
Low-Frequency Filter constructs adjacency matrices using wwwfrreuency spectral components, capturing smooth, slowly evolving patterns and long-term regularities in time series, such as seasonal cycles and periodic trends (Yi et al., 2024).
Mid- hid-Frequency Filter constructs adjacency matrices using mid-free uud-freuenccy spectral components, modeling moderate dynamics and local fluctuations in cemi-stationary processes, such shifting traffic patterns and regional weather transitions (Yu et al., 2025).
fiigh-Freeuecc Filter constructs adjacency matrices using high- frequency spectral components, detecting rapid changes and discontinuities in volatile processes, such as system anomalies, market disruptions, and abrupt state transitions (Galib et al., 2024).
Unlike classical MoE with fixed Top- K expert assignment, our Dynamic MoE (DyMoE) selects a variable number of experts per input. For each ego-graph G, we compute routing scores for filters as:
(a;; Linear.(a:.) Softplus(Linear,, (æx)) R3
(6)

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Published as a conference paper at ICLR 2026
where z; is the i-th row of X Linear, and Linear, compute deterministic and stochastic compo- nents, respectively, é a noise scale parameter. DyMoE sorts (xx,) in descending order and selects the minimum number of top-ranked filters whose cumulative probability exceeds the threshold T:
8 = min L [1,2,3] =imin+
Σ Pm Pmm2Tj g=1
2
p = Softmax((((m:)),
where Ps denotes the g-th largest element. For each ego-graph Gs this produces an adaptive expert set V (G.) = {Filter, 1 1 q S s}, where s E (1,2,3) denotes the number selected experts.
Adaptive Graph Learning. Using the selected filters (GG) from DyMoE, we construct sparse adjacency matrix A' for each ego-graph G, by filtering irrelevant dependencies (edges) as:
G. = -V U (ilter,(G()), q-1
A' k-NN(C,),
(8)
where Filter, (Gi) extracts edges based on specific frequency bands (low, mid, or high), and k-NN constructs a sparse adjacency matrix by retaining only the k nearest neighbors for each central node. Then, the graph learning layer adaptively captures the hidden relationships among time series as:
COMB(h (f [GG ((-1) vy E ((x)))
(9)
where é E fl,..., L} denotes the layer index, (u,) denotes the neighborhood of node vs h corresponds the i-th row of X and h! denotes the embedding of node v; the é-th layer. (GC3(.) and COMB(.) (-) are the functions used to aggregate neighborhood information and combine ego- and weighhor-embeediiee embeddings as:
Xia al. (2026). After layers of propagation, we obtain the final node
( = [n 1OT E CCT where h = Linear (1! D)
(10)
4.4 PREDICTION AND OPTIMIZATION
Prediction. xCPD produces the final prediction via a gated dual-path residual correction:
predict
mimdde
+ ((gNN) dUNN + (ggin) O ōLin
(11)
where óGNN = W proj H (1) is the GNN-path correction capturing crois-variable spectral dependencies,
5rin fin (inn"Xmodei (mmodel) is the Linear- path correction preserving CI refinement, and JGNN It.n E R are learnable per-variable gate parameters with (--) denoting the sigmoid. This residual formulation ensures that when gate values approach zero, xCPD degrades to the backbone prediction midee
Optimization. The training objective of xCPD comprises three components. First, we minimize the Mean Squared Error (MSE) loss CUSE between predictions Xirddict and the ground truth X as:
LMSR
pediic
X
(12)
To encourage diverse expert usage, add two regularization terms. Let R., (Sffmax((((et denote the routing probability of the expert j for node We define the entropy loss Eimrryy to prevent all experts from having low confidence scores (Huang et al., 2024), and the balance loss LBlannce to prevent the model from using only few experts repeatedly (Shazeer et al., 2017):
Std R..) 72 Mcan R...) + 8
Fatropy
Ra, log( (R,, + 5)), LRalance
(13)
where is a small constant for numerical stability, Std(.) and Mean(-) calculate the standard deviation and mean, respectively. Then, the overall training objective can be formulated as follows:
L LASE + HLenoopy + &
(14)
where hyperparameters μ and balance the regularization terms.
Complexity. xCPD has time complexity of Onnkd+ Lnkd) and space complexity of O nk). where n is the number of nodes, is the number of neighbors, d is the embedding size, and the GNN layers. These lightweight operations ensure scalability large multivariate time series.

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Published as a conference paper at ICLR 2026
Table 1: Long-term forecasting results on real-world datasets in terms of MSE and MAE. The lookback horizon is set 96 (512 for TSMixer) and forecasting horizons are (96, 192, 336, 720). The better performance in each setting is shown in bold. The best results for each row are underlined.
Moddl Metric
TSMixer MSE
xCPD
DLincar
xXPD
PatchTST MAE
+xCPD
TimnsNet MSE
xCPD
MAE
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MSE
MAE
MAE
MSE
MAE
96
0.361
0.392
0.357
0.388
0.384
0.397
0.381
0.392
0.414
0.419
0.402
0.404
0.384
0.402
0.378
0.39g
ETTm2 ET1ml ETTh2 kTThl
0.404
0.418
0.405
0.433
0.430
0.421
0.460
0.445
0.443
0.436
0.429
0.423
336
0.420
0.431
0.402
0.411
0.479
0.456
0.472
0.444
0.501
0.466
0.482
0.451
0.491
0.469
0.478
0.456
720
0.163
0.472
0.458
0.461
0.505
0.503
0.497
0.485
0.500
0.488
0.494
0.482
0.521
0.500
0.509
0.491
96
0.274
0.270
0.336
0.314
0,369
0.302
0.357
0.302
0.348
0.274
0.340
0.374
0.330
0.368
192
0.339
0.385
0.321
0.381
0.426
0.459
0.414
0.428
0.388
0,400
0.372
0.385
0.402
0.414
0.389
0.402
336
0.361
0.405
0.356
0.3M2
0.576
0,530
0.547
0.511
0.426
0.433
0.408
0.420
0.452
0.452
0.435
0.439
0:445
0:470
0,430
0.458
0.795
0.765
0.623
0.451
0.446
0.417
0.462
0.468
0.448
0.455
96
0.285
0.339
0.280
0.334
0.345
0.373
0.329
0.362
0.329
0.367
0.318
0.358
0.538
0.375
0.329
0.367
192
0.327
0.365
0:321
0.360
0.381
0.391
0.371
0.381
0.367
0.385
0.352
0.374
0.574
0.387
0.368
0.376
0.356
0.382
0,254
0-17%
0.415
0,415
0
0.403
0.399
0.410
0.380
0,394
0.410
0.411
0,408
0.410
720
0.419
0.414
0.417
0.413
0.473
0,450
0.457
0.436
0.454
0.439
0.437
0.428
0.478
0,450
0.467
0,441
96
0.163
0.252
0.160
0.250
0.193
0,292
0.188
0.284
0.175
0.259
0,163
0.250
0.187
0.267
0.185
0.266
0.216
0.290
0221
0.288
0.284
0,361
0.276
0.351
0:241
0.502
0.230
0,297
0.249
0,309
0,245
0.303
336
0,268
0.324
0.266
0.324
0.394
0.429
0.376
0.418
0.305
0,343
0.281
0.334
0.521
0.351
0.313
0.344
720
0.420
0.422
0.366
0.387
0.556
0.523
0.522
0.501
0.402
0.400
0.388
0.390
0.408
0.403
0.388
0.385
Weather Solar- Ring Exchange
96
0.089
0.209
0,083
0.203
0.083
0,206
0.081
0.202
0.088
0.205
0.083
0,201
0.107
0.234
0,103
0.228
192
0.195
0.315
0.173
0.296
0.153
0.286
0,150
0L283
0.176
0.299
0.168
0.294
0.226
0.344
0.221
0.335
0.343
0.421
0.304
0.399
0.289
0.266
0.397
0.297
0.367
0,448
0.439
720
0.898
0.710
0.835
0.685
0.793
0.671
0.751
0.651
0.901
0.714
0.885
0.702
0,964
0.746
0.948
0.725
96
0,201
0.238
0.196
0.2.0
0.290
0.378
0.284
0L372
0.234
0.286
0.229
0.280
0.250
0.292
0.244
0.287
0.231
0.258
0,227
0.259
0.312
0.389
0.310
0.260
0.296
0.318
0.314
336
0.246
0.271
0.245
0.277
0.353
0.415
0.344
0.406
0.290
0,315
0.281
0.307
0.319
0.330
0.313
0.322
720
0.247
0.273
0.245
0.273
0.356
0.413
0.348
01.407
0.289
0.317
0.283
0,312
0.338
0.337
0.328
0.330
0.149
0.198
0,143
0.192
0.194
0,252
0.177
0.239
0.218
0.167
0,218
0.172
0.220
0.212
192
0.201
0.248
0.190
0.240
0.238
0.299
0.227
0.280
0.225
0,259
0.218
0.242
0.219
0.261
0.209
0.250
336
0,264
0.291
0.340
0.277
0.281
0.333
0.272
(L317
0.278
0.297
0.270
0.258
0.280
0,306
0.270
0.285
0,320
0.336
0.313
0.330
0.345
0,381
0.337
0.370
0.348
0.338
0.365
0.359
0.357
Electricity
96
0.142
0.237
0.138
0,230
0.194
0.277
0.174
0.262
0.181
0.273
0,170
0.261
0.168
0.272
0.153
0.254
192
0.154
0.248
0.144
0.242
0.194
0.281
0,180
0.268
0.188
0.279
0.176
0.265
0.184
0.289
0,170
0.258
0.163
0.26-1
0.156
0.257
0.207
0.296
0.195
0.293
0.204
0.296
0.196
0.198
0.300
0.280
720
0,208
0.300
0.200
0.294
0.242
0.330
0.240
0.326
0.246
0.328
0.233
0.308
0.220
0.320
0.198
0.303
96
0.376
0.264
0.371
0.259
0.650
0.397
0.636
0.386
0.462
0.298
0,448
0,278
0.593
0.321
0,549
0,303
Traffic
0.397
0.277
0.275
0.574
0.356
0,300
0.453
0.617
0.336
0.325
336
0.413
0.290
0.410
0.287
0.605
0.374
0,586
0.362
0.482
0.307
0.464
0.295
0.629
0.336
0.561
0.332
720
0.444
0.306
0.439
0.303
0.645
0.395
0,630
0.387
0.517
0.326
0.502
0,305
0.640
0,350
0,573
0.344
5 EXPERIMENTS
Benchmarking Datasets. For long-term forecasting, we experiment with 9 popular benchmarks in various domains (Zhou et al., 2021; Wu et al., 2021; Lai et al., 2018), including weather, electricity, and traffic. For short-term forecasting, we use two univariate datasets: M4 (Makridakis et al., 201 8), covering diverse domains and frequencies, and Stock (Chen et al., 2024), containing 1,390 stock price series over 10 years with high variability and non-stationarity. Details are provided in Appendix B.
Backbones and Baselines, xCPD is a into arbitrary forecasting models to improve their performance. To ensure comprehensive evaluation, (CI) mod-
strategy that can be seamlessly integrated
we select four recent state-of-the ohtthe-art forecasting backbones: two Channel-Independder els-DLinear (Zeng et al., 2023) and PStchTST (Nie et al., 2023) -add two Channel-Dependent (CD) models--TMMixr (Chen et al., 2023) and TimesNet (Wu et al., 2023). These backbones span three representative architectural paradigms: linear, transformer-based, and convolution- models. In addition, we compare xCPD against four best-performing methods: LIFT (Zhao & Shen, 2024), CCM (Chen et al., 2024), PCD (Lee et al., 2025), and PRReg (Han et al., 2024). CCM and PCD are CP-based baselines. We exclude PCD and PRReg from some experiments, since PCD is transformer-specific and PRReg requires specific optimization. More details refer to Appendix C.2.
Implementation Details. For fair evaluation, we adopt Time-Series-Librar (Wu et al., 2023) to reproduce all baselines and report their performance with optimal hyperparameters. All experi- ments are conducted using PyTorch on a single NVIDIA RTX A 80GB GPU. We use Adam optimizer (Kingma & Ba, 2014) with learning rates seleeted from (5c-4, 1c-4). The reported results are averaged over five runs with different random seeds. For dynamic expert allocation, the threshold is selected from (0.0, 0.5). Experimental setups and hyperparameters are provided in Appendix D.
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5.1 LONG-TERM FORECASTING PERFORMANCE
As shown in Table 1, we evaluate long-term time series forecasting performance using mean squared error (MSE) and mean absolute error (MAE) in 9 real-world datasets. The results demonstrate that integrating xCPD consistently improves predictive accuracy over the original backbone models on most datasets and forecasting horizons, encompassing 144 experimental settings in total. The improvements are most pronounced in datasets with complex, high-dimensional channel interactions. Specifically, xCPD achieves great MSE reductions on the electricity dataset (321 variables) and on the traffic dataset (862 variables), averaged over 4 backbones (with additional backbones evaluated in Appendix C.3). These gains stem from xCPD's ability to (i) model fine-grained channel-pade dependencies through ego-graph representations and (ii) suppress irrelevant connections via spectral filtering, thereby improving inter-channe relationship modeling. Since DLinear and PatchTST report results with lookback horizon 336, we provide the same setting in Appendix C.4 to further demonstrate xCPD's effectiveness. We provide more performance analysis in Appendix C.5.
As shown in Table 2, we further compare xCPD with two best-performing baselines: LIFT (Zhao & Shen, 2024) and CCM (Chen et af., 2024). LIFT improves forecasting by identifying leading indicators and leveraging their signals to guide lagging channels. CCM dynamically clusters channels based on similarity, balancing between CI and CD strategies. xCPD consistently outperforms both baselines by modeling fine-grained channel- -ptch dependencies and explicitly incorporating frequency-donee domain dependencies. Notably, xCPD maintains its performance ad advantage even under non-stationary conditions in Appendix C.6, showing the robustness of our shared basis in Theorem 4. 1.
Table 2: Averaged long-term forecasting results with horizons {96, 192, 336, 720} and look-back horizon are 512 for TSMixer and 96 for DLinear on 9 datasets in terms of MSE and MAE The better performance shown in bold.
Model Metric
TSMixer
+LIFTT
CCM
.XCPD
Dilincar
.IIFT
CCM
xCPD
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
ETTh1
0,412
0.428
0.407
0.424
0.413
0.428
0.401
0.416
0.456
0.452
0.451
0.448
0.451
0.451
0.445
0.435
ETTh2
0.355
0.401
0.351
0.397
0.351
0.399
0.345
0.392
0.559
0.515
0.553
0.511
0.552
0.511
0.507
0.479
ETTml
0.347
0.375
0.342
0.372
0.351
0.380
0.343
0.371
0.403
0.407
0.399
0.414
0.397
0.403
0.389
0.395
ETTm2
0,267
0.322
0.264
0.319
0,258
0.319
0.251
0.312
0,350
0.401
0.346
0.398
0,306
0.369
0.340
0.388
Exchange
0.414
0.378
0.411
0.349
0.396
0.351
0.411
0.348
0,380
Solar-Eng.
0.231
0.259
0.229
0.257
0.234
0.263
0.228
0.287
0.330
0.401
0.327
0. 0.398
0.327
0.399
0.322
0,393
Wenther
0.234
0.268
0.231
0.266
0.225
0.263
0.221
0.259
0.265
0.317
0.262
0.314
0.262
0.309
0.253
0
Blectricity
0.262
0.165
0.260
0.158
0.255
0.208
0.297
0.293
0.197
0.287
Traffle
0.408
0.284
0.405
0.282
0.396
0.284
0.394
0.281
0.625
0.383
0.620
0.350
0.614
0.378
0.606
0.372
In Table 3. we evaluate xCPD against baselines in a unified training setting. Following PRReg (Han et al., 2024), we reproduce two backbones (Linear and Transformer) with both CI and CD variants, where baselines are in- corporated as weighted regularization terms added to the primary forecast- ing loss. Table 3 shows xCPD con- sistently outperforms competing base- lines across all settings, demonstrating its superior adaptability to diverse forecasting architectures.
Table 3: Comparison of baselines under a general setting.
CD
C1
-PRRag
+LIITT
+PCD
+CCM
+xCPD
ETTN
Transformer
Linear
0.861
0.402
0.655
0.345
0.539
0.342
0.528
0.345
0,621
None
0,518
0.342
0.512
0.338
Linear
0.404
0.354
0.311
0.316
None
0,310
0.305
Transformer
0.458
0.379
0.369
0.356
0.404
0.300
Exchange
Linear
0.051
0.119
0.048
0.050
Now
0.045
0,042
Trunsformer
0.101
0.511
0.100
0.099
0.253
0.097
0.091
Weather
Transformer
Lincar
0.142
0.251
0.169
0.168
0.131
0.180
0.133
0.178
0,198
None
0.130
0.164
0.128
0.161
Electrictty
Transfomer
0.195
0,250
0.185
0.196
0.196
0.185
0.188
0.196
0.215
0.195
0.183
0.181
0.192
5.2 SHORT-TERM FORECASTING PERFORMANCE
We evaluate xCPD on univariate short-term forecasting using the M4 and Stock datasets. As shown in Table 4, xCPD consistently outperforms baselines through principled trequencydoomal modeling via shared Fourier bases, spectral grouping, and adaptive filtering. Performance gains exhibit clear horizon dependency: long-term forecasting benefits substantially from frequency decomposition of seasonal patterns, while short-term predictions show modest improvements due to limited spectral diversity in local temporal dynamics. Despite this inherent constraint, xCPD maintains consistent gains through patch-level denoising, demonstrating its generalizability across temporal scales.
5.3 ZERO-SHOT FORECASTING PERFORMANCE
Most MTSF methods are coupled to specific datasets, limiting generalization to unseen scenarios. In contrast, xCPD leverages learned frequency-awar filters capture universal spectral structures,
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Table 4: Short-term forecasting results on M4 dataset in terms of SMAPE, MASE, and OWA, and Stock dataset in terms of MSE and MAE. The forecasting horizon is (7,24) for the Stock.
Modd
LIFT
.CCM
1ACCD
.LIPT
-CCM
+.000
PACETST
-CCM
$.590
.CCM
SMAPE
11:20
14.38
14.67
14.56
5.96
15.05
14.33
14.13
13.45
13.55
13.30
13.33
15.37
15.13
14:42
14.36
MI Misrls
M14SE
33343
3.341
3,500
3.338
4.285
41008
3.141
3.131
3.010
3.000
3.907
2.009
3.554
3.513
3.448
3.434
0.738
SMAPE
11.18
11/0
10.35
10.35
12.14
$1.45
10.31
10.25
10.35
$2.36
10.35
18.35
10.45
10.25
10.32
10.10
MirQueeyy
OWA
0.988
0.92
0784
0.5%
1.00
1.88
0031
0.338
0.921
0.91
0915
0.912
02223
0293
0.857
0.589
M4 (Nhonda5)
ASASL:
3122
1225
1101
19935
11977
0.85
0961
03537
3.330
1.022
11942
0.554
OWA
054
0365
0344
0.335
0.556
0.65
0935
0.931
0.905
0.500
0.855
0.892
0.55
0.31
0.851
0.583
SMAPE
1/67
1095
7.124
7.845
6.55
5.100
6.385
4.55
18855
1,63
6431
0.013
0.500
5216
5.118
M44 (Okann)
MASE
3,877
3,422
5,302
5.208
4.55
4.09)
4.713
4.825
3.347
3523
3.125
3.132
4.527
4.344
3.82
3.86
OWA
1,642
1.335
1.407
1.45
1.39
1.278
1.590
1162
0097
0:385
1,35
1.39
1.217
1.316
SMAPE
13.96
12.03
12.00
12.79
13.55
13.42
12.54
12.43
12.05
1:50
1185
11.11
12:88
12.55
11.11
11:73
MI (Agg
ASASE
1.887
1372
1.54
1.60
1.60
1230
12333
1823
1.6
1587
1.573
1.873
1734
1829
1.992
1951
Sock 71
MRR
0.900
0.98
0.31
0.32
0963
0.83
0.879
0.806
0.733
0.981
0,97
0.89
0.33
0837
0915
0.912
Stock (Hootoon
MSE
1007
1,055
0291
0.386
0.596
0.905
0017
0.912
0.530
0.912
0389
0.878
0.99
0353
0
0.933
enabling effective knowledge transfer for zero-shot forecasting without retraining, Following prior work (Jin et al., 2024), we evaluate zero-shot performance on the ETT benchmark comprising four datasets with varying regional and temporal resolutions. As shown in Table 5 and Table 13, XCPD consistently improves zero-shot performance across all 48 evaluation settings. Notably, (i) performance gains increase with horizon length, demonstrating xCPD's ability to transfer frequency knowledge to more challenging long-range dependencies; and (ii) Cl-based models benefit more substantially, with 12.0% and 15.2% average improvements for DLinear and PStchTST versus 6.7% and 11.1% for TSMixer and TimesNet. These results indicate promising transferability under frequency-aligned settings. Broader cross-domain transfer is left for future work.
Table 5: Zero-shot forecasting MSE results on ETT datasets. The forecasting horizon is (96, 720).
Model
TSMixer
+CCM
+XCPD
DLinerr
+CC
+CCPD
PaichTST
+CCM
xCPD
TimesNet
CCM
cCPD
IMP(%)
hi1-h2
0.288
0,283
0.290
0.358
0.335
0.442
0.418
0.388
2.318
0.374
0.370
0.366
0.855
0.843
0.822
0.582
0.571
0.544
0.540
0.516
0,510
2,366
h1 sm1
0.763
1,252
0.710
1.215
0.702
1.198
0.782
1,245
0.723
1.224
0.702
1.212
0.754
1,892
0.092
1.135
0.678
1.128
0.887
1.633
0.827
1.601
1.589
0.801
2.300
0.937
-m2
0,959
0.937
0.928
1,213
0.912
0,890
1,132
0.903
1,892
1.199
1.122
1,0066
1,950
1,765
1,758
1.712
2.132
1.732
1,680
2,015
1.712
1.874
2,214
0.443
0.689
0.632
0.512
h2--h1
0.695
0.510
0.528
0.533
0.195
0.492
1,032
0.986
0.944
1.274
0.845
0.834
2,098
h2-m2
0.913
1.472
0.876
1.464
0,865
1.422
0.758
1,892
0.732
1.422
0,708
1.274
0.855
1,899
0.781
1.644
0,772
1.542
1,250
1.361
1.671
1.061
1.033
1.6.3
5,439
2,064
h2 mm1
1,254
2.275
1.073
1.254
1.721
1.022
2,012
1.243
0.939
1,850
0.898
1.792
2.732
1.012
0,809
1.832
0.794
1.732
2.183
1.049
0.904
1.742
0.794
3.054
5.4 EFFICIENCY EVALUATION
Traffle 96.96
Figure 2 evaluates the computational ef- ficiency on Traffic under the input-96- predict-96 setting. The overhead of LIFT and CCM is highly model-dependent, in- creasing substantially with large forecast- ing models. While CCM achieves strong performance, its quadratic cost with re-
0.70
TSMD
0.65-
Performancs (MSE)
0.60
TSMIurri CCM
0.55
TSMiuer LIIT 31395 MIL, 45 melter
2200 MI. 385 rulline
0.50
TaNAIST-<CTO
TaLn MB. ailler
15054 MB. 161 unllley
0.45
PudF 35351 MIl molter
spect to the number of clusters limits scalability. In contrast, xCPD maintains poodel-ggosttc efficiency through linear- complexity spectral operations, achieving comparable performance with significantly lower overhead. Table 14 in Appendix C.9 further shows that the time overhead introduced by xCPD accounts for only 9%-11% of the backbone model's training cost. These efficiency advantages make xCPD particularly suitable for large-scale scale and real-time forecasting applications.
Lin molter
TINNNeCCCM 20014 MIL 172 mallee
0.30
25
150
Running Time (ms/iter)
Figure 2: Model efficiency comparison on Traffic.
5.5 ABLATION STUDY
We conduct ablation studies to evaluate each component's contribution in xCPD, shown in Table 6. Component Removal: (1) w/o Shared Basis removes the shared Fourier basis, using separate bases per batch. (2) w/o Freq. Partition removes learnable frequency boundaries, using fixed equal splits.
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Table 6: Ablation study of xCPD, Look-back horizon 96 and results are averaged over forecasting horizons {96, 192, 336, 720). Best and second-best results are marked in bold and underline.
ETThl
ETTml
Exchange
Solar-Eng.
Weather
Electricity
Traffic
Variants
MSE
MAE|
MSE
MAB
MSE
MAB
MSE
MAB|
MSE
MAE
MSE
MAE
MSE
MAE
DI DDD
0..45
0.435
0.389
0.395
0.312
0.380
0.322
0.393
0.253
0.301
0.197
0.287
0.6066
0.372
w/o Shured Basis
0.462
0.456
0.404
0.415
0.331
0.395
0.341
0.399
0.267
0.310
0.205
0.295
0.618
0.379
Remove
w/o Preq. Parrition
0.448
0.394
0.399
0.403
0.318
0.388
0.329
0.398
0.264
0.307
0.203
0.292
0.610
0.376
w/o Node Group
0.461
0,455
0.406
0,415
0.329
0.394
0.344
0,402
0.266
0.310
0.204
0.293
0.616
0.377
wo Filters
0.465
0.456
0.408
0.416
0.333
0.397
0.344
0.402
0.272
0.313
0.209
0.298
0.623
0.383
w/o Layn & Live
0.401
0.325
0.390
0.396
0.302
0.201
0.609
Top-K
0.464
0.456
0.410
0.420
0.333
0.396
0.343
0,420
0.269
0.314
0.208
0.298
0.621
0.380
Replace
Random-K
0.465
0.456
0.407
0.418
0.332
0.395
0.343
0,419
0.269
0.313
0.209
0.299
0.620
0.380
RegionTop-4
0.459
0.453
0.405
0.415
0.331
0.392
0.335
0,398
0.265
0.308
0.206
0.295
0.617
0.378
TimeFilter
0.457
0.452
0.406
0.328
0.390
0.336
0.402
0.305
0.201
0.615
(3) w/o Node Group replaces energy-based node grouping with random assignment. (4) w/o Filter omits the filtering step and uses full ego-graphs. (5) w/o & removes regularization on expert diversity and balance. Strategy Replacement: (6) Top-K selects top-K edges by weight in each ego-graph. (7) Random-K randomly samples K edges. (8) RegionnTop- selects top-K edges within each frequency band. (9) TimeFilter applies a spatial-temporal filter (Hu et al., 2025b). Please note that, to ensure a fair mechanism-level comparison within our ablation framework, we adopt only the core filtering module of TimeFilter and integrate as a plugin into the DLinear backbone (denoted as Fimell rather than using the full TimeFilter architecture. All replacement strategies lead to performance drops, highlighting the effectiveness of xCPD's spectral filters. Appendix C.7 provides a more detailed analysis.
5.6 QUALITATIVE STUDY
Figure illustrates the correlation between spectral energy and temporal patterns in channel-patch nodes. The red node ex- hibits smooth temporal trends with grad- ual variations, concentrating energy in low- frequency bands and primarily aggregat- ing dependencies from other low-frequency nodes. In contrast, the green node displays abrupt fluctuations and rapid state changes, resulting in dominant high-frequency spec- tral energy that drives its connections to other figh-freeuency nodes. Importantly, these frequency assignments are not manu- ally predetermined. They emerge naturally from the learned spectral energy response, which allows the model to discover frequency structure inherent in the data rather than relying on heuristic thresh- olds. This interpretability difficult to obtain from temporal-domain attention mechanisms, where trend, fluctuation, and noise components remain mixed and cannot be disentangled. Theorem 4.2 further guarantees that spectral energy preserves all information in the original patch embedding. As result, the alignment between temporal behaviors and spectral energy distributions observed in Figure 3 represents a faithful and principled consequence of spectral decomposition, providing intuitive and robust insight into the learned frequency-base -bsed grouping.
Timestens (8-
Timestens [30-32
Value
QT
Low Fre
01:73
0.32
MUFI
Mid
0.01
3.4
MULL
LURL
High Hre.
0.09
0.67
LULL
24
26
20
Timeslop
Figure 3: Filters visualization of HULL on ETThI.
6 CONCLUSION
In this paper, we propose XCPD, a lightweight plugin module that enhances multivariate time series forecasting by modeling fine-grained, frequency-aware channel-patch dependencies. Extensive experiments demonstrate that xCPD consistently improves forecasting accuracy across diverse datasets and backbones while maintaining linear computational complexity. Its model-agnostic design enables seamless integration into existing forecasting models, establishing spectral decomposition and adaptive routing as powerful tools for capturing complex temporal dependencies in MTSF.
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ETHICS STATEMENT
This work presents technical contribution to multivariate time series forecasting and raises no ethical concerns. Our research does not involve human subjects, uses only publicly available benchmark datasets with proper citations, and contains no personally identifiable information. The proposed method, xCPD, is a general purpose forecasting enhancement technique with no inherent potential for harm or misuse beyond standard time series prediction applications. We have conflicts of interest to declare. The computational requirements of xCPD are modest and environmentally responsible compared to cxisting models. We confirm that this research complies with the ICLR Code of Ethics.
REPRODUCIBILITY STATEMENT
To ensure the reproducibility of our work, we provide comprehensive implementation details and experimental protocols throughout the paper and the supplementary materials. The complete ar- chitecture of the model and the algorithmic details are presented in Section 4, with mathematical formulations for all key components. Section 5 describes the experimental setup, including the 9 benchmark datasets used, baseline methods, and implementation details such as optimizer settings, learning rates, and random seed protocols. Appendix B provides detailed statistics and pre-processing steps of the dataset. Appendix C contains extended experimental results and additional implementa- tion specifics, including batch sizes, training epochs, and early stopping criteria. Appendix D lists all evaluation metrics and the complete hyperparameter search spaces with selected values for each dataset. All experiments were carried using PyTorch on NVIDIA RTX A100 GPUs, with average results of random seeds to ensure statistical reliability. The anonymous repository provided contains the full source code, data reepprocessing scripts, and trained model checkpoints facilitate the exact reproduction of all reported results.
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APPENDIX CONTENTS
A The Use of Large Language Models (LLMs)
17
Datasets
17
B.1 Long-term Time Series Forecasting Datasets
17
B.2 Short-term Time Series Forecasting Datasets
17
Additional Experiments
17
C.1 Additional Details of Adjacency Matrix Construction
17
C.2 Comparison with PCD and PRReg
18
C.3 Performance on Additional Backbones
18
C.4 Impact of Input Length on Performance
19
C.5 Dataset-dependent Performance Improvements
20
C.6 Robustness to Non-stationary Dynamics
20
C.7 Analysis of Ablation Study
C.8 Zero-shot Forecasting Performance
22
C,9 Computational Efficiency Analysis
22
D Experiment Details
23
D.1 Metrics
23
D.2 Hyperparameters
24
D.2.1 Sensitivity to k-NN Sparsification Ratio.
24
D.2.2 Selection of Patch Size and Overlap
24
E Theory Assumptions and Proofs
25
E.1
Theorem 4.1 Proofs
25
E.2
Theorem 4.2 Proofs
25
E.3
Limitations
26
E,4
Future Works
26
E.5
Social Impacts
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A THE USE OF LARGE LANGUAGE MODELS (LLMS)
We acknowledge the use of large language models to assist with manuscript preparation. Specifically, LLMs were employed solely for language refinement tasks, including grammatical corrections, vocabulary enhancement, and sentence structure improvements. All scientific content, including research methodology, experimental design, data analysis, theoretical developments, and technical conclusions, represents the original intellectual contribution of the authors. The LLMs did not generate any substantive scientific content, interpretations, or novel insights presented in this work.
B DATASETS
B. LONG-TERM TIME SERIES FORECASTING DATASETS
In Table 7, we summarize the detailed statistics of all datasets used our experiments. We conduct extensive evaluations on 9 widely used multivariate time series datasets for long-ten seng-term forecasting and two datasets for short-term forecasting. The ETT dataset (Zhou et al., 2021) contains temperature and power load data collected at hourly (ETTh) and 15-minute (ETTm) intervals. Exchange (Lai et al., 2018) compiles daily exchange rate information from 1990 to 2016. Solar-Energy (Lai et al., 2018) provides 10-minute solar power readings from 137 PV plants. The Weather (Wu et al., 2023), Electricity (Wu et al., 2023), and Traffic (Wu et al., 2023) datasets include high-resolution environmental and usage signals, recorded respectively 10-minute, hourly, and hourly intervals.
B.2 SHORT-TERM TIME SERIES FORECASTING DATASETS
For short-term univariate forecasting, we use the M4 dataset (Chen et al., 2024), comprising 100,000 time series across multiple domains and six frequeecy- based subgroups. Additionally, we stock forecasting benchmark (Chen et al., 2024) based on 10 years of 5-minute price data from 1,390 stocks. The data span from November 25, 2013 September 1, 2023, with missing timestamps interpolated to ensure sequence alignment. This stock dataset emphasizes short-term dynamics and market fluctuations, with forecasting horizons set to and 24 time steps.
Table 7: Statistics of all datasets used in our experiments. "Frequency" denotes the sampling interval of time points.
Dataset
Type
Lookback Length
Prediction Length
Total Length
Frequency
Domain
Long-term Jerm Forecasting Darasets
ETTh1 ETTh2
Multivariate
96
(96, 192, 336, 720)
17,420
Hourly
Electricity
Multivariate
96
196, 192, 336, 720
17,420
Hourly
Electricity
Multivariate
336,
15
Electricity
ETTm2
Multivariate
96
(96, 192, 336, 720
69,680
15-min
Electricity
336, 720)
Meteorology
Ellccticity
Multivariate
321 96
196, 192, 336, 720)
26,304
Hourly
Energy
Traffic
Multivariate
96,, 336, 720)
Hourly
Transportation
Lixchunge
Multivariate
8
96
96, 192, 336, 720)
7,588
Daily
Finance
Solar-Energy
Multivariate
96. 336, 720)
$2,560
10-min
Energy
Short temm Foreasting Dataxets
M4-Yeal Yearly
Univariate
[12, 18, 24, 30, 36, 42)
6
Various
Yearly
Mixed
M4. Mua-rurryryyy
Univariate
16, 24, 32, 40, 48, 56]
8
Varinus
Quarterly
Mixed
Momthly
Univariate
(36. 54, 39,
126)
18
Monbly
M4- Wee-le
Univariate
39, 52, 65, 78, 91}
13
Various
Weekly
Mixed
Daily
Univariate
42, 70,
Daily
Mixed
M4-Hourly
Univariate
(96. 144, 192, 240, 288, 336)
48
Various
Hourly
Mixed
Stock
Univariate
(24.48)
(7.24)
2,026,990
5-min
Finance
C ADDITIONAL EXPERIMENTS
C.1 ADDITIONAL DETAILS OF ADJACENCY MATRIX CONSTRUCTION
This subsection provides additional details on the construction of the adjacency matrix A' mentioned in Section 4.1, focusing three key aspects: (i) non-triviality of dense similarity graphs, (ii)
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numerical stability through Laplacian normalization, and (iii) the operational sparsity enforeed during routing.
The adjacency matrix is constructed using cosine similarity cos (X; xgmbb.), which guarantees that all affinity values lie within the bounded interval [-1, 1]. This bounded- ness ensures numerical stability and prevents degenerate scaling of affinities. Moreover, cosine similarity preserves angular variation between embeddings, yielding a highly non-uniform structure even when the matrix is dense. The diagonal elements naturally satisfy A = 1, climinating the need for explicit self-loop regularization. Dense posine-similarte graphs of this form are standard in spectral graph methods, where global affinity patterns are crucial for identifying meaningful frequency components.
The shared Fourier basis is derived from the normalized Laplacian L' I- whose eigenvalues are guaranteed lie within [0. 2]. This normalization controls the operator scale, stabilizes the eigendecomposition and promotes clear separation between low- and high-frequency components regardless of the density of A'. These properties ensure that spectral representations are well-conditioned across timesteps, which is essential for learning the shared Fourier basis in Theorem 4.1.
Although A' is dense, downstream routing does not operate on the full dense graph. Instead, xCPD applies k-NN sparsification procedure to each node, retaining only the most relevant neighbors based on similarity. Furthermore, the frequency-aware MoE router activates only a subset of spectral components for each sample, resulting in selective and structured dependency pathways. Consequently, the effective computation graph is sparse and informative, avoiding the trivial connectivity patterns typically associated with dense graphs.
C.2 COMPARISON WITH PCD AND PRREG
PCD (Lee et al., 2025) is not included in our main experimental tables due to its architectural con- straints. As detailed in their original work, PCD's core mechanisms require self-attention operations, limiting its applicability to Transformer-bast based models. This restriction prevents fair evaluation across the diverse model families considered in study, including linear models (DLinear) and CNN-based architectures (TimesNet). To provide comprehensive analysis, we conduct additional experiments comparing xCPD and PCD PatchTST, Transformer backbone where both plugins applicable. As shown in Table 8, xCPD consistently outperforms PCD across all eight benchmark datasets, demonstrating superior performance even within PCD's intended architectural domain.
Table 8: Long-term forecasting results with lookback window 96 and prediction horizons {96, 192, 336, 720). Results show MSE/MAE averaged over three runs with best in bold.
Method
ETTh1
ETTh2
ETTml
ETTm2
Exchange
Weather
Electricity
Traflic
PaC
0.468/0.456
0,379/0.399
0.382/0.393
0.272/0.325
0.366/0.403
0.255/0.281
0. 0.198/0.286
0.482/0.300
PaCHTST+<<PDD
0.455/0.443
0.367M0.393
0.371/0.388
0.265/0.317
0.358/03397
0.248/0.272
0.191/0.280
0.466/0.292
PRReg (Han et al., 2024) differs fundamentally from plugin modules in its design philosophy. While we compare against PRReg in Table 3 using their proposed regularization framework, PRReg operates through loss regularization terms requiring joint optimization and careful hyperparameter tuning for each backbone-dated combination. This contrasts with xCPD's plug-and-play design, which can be applied post-hoc without modifying the training process. Furthermore, PRReg's 's performance varies significantly across architectures due its dependence on model-specific gradient dynamics, whereas xCPD maintains consistent improvements through its architecture-agnoste spectral operations. Despite these differences, xCPD outperforms PRReg even when evaluated within PRReg's 's own optimization framework (Table 3), further validating our approach.
C.3 PERFORMANCE ON ADDITIONAL BACKBONES
To evaluate xCPD's generalizability beyond our primary baselines, conduct experiments two recent forecasting architectures: iTransformer (Liu et al., 2024a) and TimeMixer (Wang et al., 2024a). These models represent state-of-the-art advances in attention-based and ML.P- MLP-mixing architectures,
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respectively. Table 9 presents results across six benchmark datasets. xCPD achieves consistent improvements, with average MSE reductions of 3.7% for iTransformer and 2.8% for TimeMixer. The improvements are particularly pronounced on Solar-Energy (3.0% and 2.8% respectively), where complex multi-scale temporal patterns benefit from our frequency-aware modeling. These results confirm xCPD'ss compatibility with diverse architectural paradigms beyond the linear, transformer, and CNN-based models evaluated in our main experiments.
Table 9: Performance on iTransformer and TimeMixer. Results averaged over forecasting horizons (96, 192, 336, 720} with lookback window 96. Values show MSE/MAE with best results in bold.
Model
ETThl
ETTh2
ETTml
ETTm2
Solar-Energy
Weather
iTransformer
0.455/0.447
0.383/0.400
0.407/0.411
0.288/0.332
0.258/0.27 0.233/0.262
ITransformer-xCPD
0.438/0.440
0.375/0.400
0.394/0.401
0,277/0.325
0.226/0.256
0.250/0.269
TimeMixer
0.447/0.444
0.365/0.396
0.381/0.395
0.275/0.323
0.216/0.280
0.240/0.27
TimeMixe+xCCPD
0.435/0.432
0.358/0.389
0.372/0.388
0.268/0.317
0.210/0.275
0.233/0.266
DUET
0.443/0.437
0.373/0.398
0.390/0.394
0.280/0.325
0.238/0.235
0.252/0.277
DUET +XCPD
0.436/0.433
0.364/0.392
0.381/0.390 0.275/0.318
0.231/0.234
0.246/0.277
Beyond the CI/CD backbones evaluated in this paper, we were also interested in examining whether xCPD remains effective when paired with CP-based backbone. To this end, we additionally include DUET Qiu et al. (2024), which performs channel-level dependency modeling through global channel clustering. Following the same experimental setup as in our main experiments, we freeze all backbone parameters and fine-tune only the parameters introduced by our xCPD plugin across six datasets. As shown in Table 9, observe two interesting findings: (1) xCPD continues to yield performance improvements even on a CP backbone. This is because DUET models channel level dependencies, whereas xCPD captures channel-patch level dependencies that provide finer-grained relational information; (2) The improvement achieved by DUET is less pronounced than that of CI/CD backbones such as TSMixer and iTransformer. We hypothesize that DUET's hard channel clustering partially disrupts the original dependency structure in the raw signal, making subsequent patch-level CPD more difficult to learn.
C.4 IMPACT OF INPUT LENGTH ON PERFORMANCE
The performance of DLinear (Zeng et al., 2023) on the Traffic dataset differs between our Table and the original paper. This discrepancy arises from different lookback window configurations: DLinear's original experiments use a lookback window of 336, while we adopt the standardized setting of 96 following recent work (Liu et al., 2024a) ensure fair comparison across all baselines. To investigate the impact of input length and validate xCPD's 's robustness, we conduct experiments with both configurations. Table 10 and Table 11 present results under lookback horizons of 96 and 336, respectively.
Table 10: Performance on Traffic dataset with look-back horizon 96.
Forecasting Horizon
Method
96 MSE/MAE
192
336
720
MSE/MAE
MSE/MAE
MSE/MAE
DLinear
0.650/0.396
0.598/0.370
0.605/0.373
0,645/0.394
DLiDear*xCPD
0.636/0.386
0.574/0.356
0.586/0.362
0.630/0.387
The results demonstrate two key findings: (1) DLinear's performance with lookback window 336
aligns with the original paper, confirming our implementation correctness; (2) xCPD consistently improves performance across both configurations, achieving average MSE reductions of 3.2% (look- back=96) and 2.3% (lookback=336), Notably, the longer input window provides better baseline performance (0.435 vs 0.625 average MSE), yet xCPD maintains its effectiveness regardless of input length. This confirms that performance differences stem from experimental settings rather than implementation issues, while validating xCPD's robustness across varying input contexts.
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Table 11: Performance on Traffic dataset with look-back horizon 336.
Prediction Horizon
Method
96 MSE/MAE
192 MSE/MAB
336
720
MSE/MAE
MSE/MAE
DLinear
0.412/0.282 0.406/0.288
0,423//.287 0.411/0.283
0,437/0.296 0.428/0.295
0.468/0.316
DLinear+XCPD
0.457/0.311
C.5 DATASET-DEPENDENT PERFORMANCE IMPROVEMENTS
We observe that the performance gains brought by xCPD are not uniform across datasets, and these differences are closely tied the spectral and multivariate characteristics of the underlying signals. Since xCPD explicitly models fine-grained, frequency-awas channel-patch dependencies, datasets with richer spectral variability naturally benefit more from our plugin. We summarize the dataset-dependent behavior as follows.
Small improvements (1 (1%)))) ETT and SolarEnergy. They exhibit highly regular, low- frequency periodicity dominated by daily or hourly seasonal cycles. Their cross-channel relationships relatively stable and already well captured by CI/CD backbones such as TSMixer and iTransformer. As a result, xCPD provides moderate but consistent improvements by modeling subtle patch-level variations that are not fully exploited by these backbones.
Medium improvements (3 (3%-4% ): Exchange and Weather. They contain mixed spectral structures, combining smooth seasonal trends with occasional short-term fluctuations. These datasets feature moderate multivariate interactions that are partially captured by the backbone but still benefit from xCPD's s ability to extract localized, frequency- channel-pateh dependencies. Consequently, xCPD yields steady improvements in this mid-range category.
Large improvements (4%-7%): Electricity and Traffic. They display the richest multi- scale frequency patterns among all datasets, including local bursts, irregular short-term spikes, asynchronous channel shifts, and strong high-frequency components. Such volatile and rapidly changing cross-channel relationships challenge backbones that rely on coarse channel mixing. In these settings, xCPD's 's fine-grat fine-grained frequency- modeling becomes particularly effective, consistently producing the largest performance gains across all benchmarks.
In summary, the improvement pattern across datasets aligns strongly with their spectral complexity: the richer and more volatile the frequency structure, the greater the benefit from explicit frequency- aware channel-patch dependency modeling. This analysis provides practical guidance on when xCPD yields the most substantial improvements and highlights the importance of spectral diversity in multivariate time-series forecasting.
C.6 ROBUSTNESS TO NON-STATIONARY DYNAMICS
Real-world multivariate time series often exhibit non-stationary behavior with evolving inter-channel dependencies. To evaluate xCPD's robustness under such conditions, we conduct controlled experi- ments with synthetically induced ssaaiioattiit..
Experimental Setup. We perturb the ETTml dataset to simulate temporal misalignment and amplitude variations commonly observed in real-world scenarios. Specifically, we apply two types of perturbations to each channel: (i) random temporal shifts within 1 time steps to disrupt cross- channel synchronization, and (ii) random amplitude scaling factors sampled from 24(0.8, 1.2) to introduce magnitude variations. These perturbations directly challenge the stationarity assumption underlying our shared Fourier basis (Theorem 4.),, providing a rigorous test of xCPD's robustness.
Results and Analysis. Table 12 presents the performance comparison with and without the shared spectral basis under synthetic non-stationarity. Despite induced perturbations. xCPD maintains strong performance with only marginal degradation compared to the stationary setting (Table 1). Notably, removing the shared basis results in significant performance deterioration (8.3% MSE increase), confirming its critical role in maintaining spectral alignment under non-stationary conditions.
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Table 6 compares the proposed xCPD with several alternative dependency modeling mechanisms using a unified backbone (DLinear). We note that the MSE/MAE values of TimeFilter Table differ from those reported in the original TimeFilter paper (Hu et al., 2025b). We provide below an explanation to clarify this discrepancy and avoid potential misunderstanding. The original TimeFilter is introduced as complete forecasting architecture built upon temporal graph neural networks, equipped with its own encoder, normalization layers, and additional architectural components. These design choices contribute substantially to its reported performance. In contrast, the purpose of Table is evaluate dependency mechanisms controlled and unified setting. Therefore, we re- re-implemented only the core ppatio-temporal filtering module of TimeFilter and integrated it as a lightweight plugin into the DLinear backbone. This variant denoted as DLinear+ Tomeerr+Timmeeee Time Filter. logic, without incorporating the Published as a conference paper at ICLR 2026
Table 12: Ablation study under synthetic non-stationarity on ETTml (lookback window: 96, forecast- ing horizon: 96). The shared spectral basis maintains performance despite temporal perturbations.
Model
Configuration
MSE
MAE
w/ Shared Basis
0.389
0.395
DLincar+XCPD
w/o Shared Basis
0.413
0.425
w/ Shared Basis
0.373
0.384
TimesNet+xCPD
w/o Shared Basis
0.395
0.403
This robustness stems from complementary mechanisms. First, formalized in Theorem 4. the shared basis U approximates time-varying bases U(()) up orthogonal rotation, preserving spectral relationships despite temporal variations. The bound || - UR S. C Lavg ensures graceful degradation rather than catastrophic failure when stationarity is violated. Second, the dynamic MoE routing provides local adaptability by selecting frequency-specife filters based on instantaneous node characteristics, effectively compensating for global approximation errors introduced by the shared basis. These results, combined with strong performance inherently non-stationary datasets, validate that xCPD effectively handles time-varying dependencies through the synergy of global spectral alignment and local adaptive routing. This dual mechanism enables robust performance across both controlled perturbations and real-world non-stationary scenarios.
C.7 ANALYSIS ABLATION STUDY
The ablation w/o Freq. Partition replaces learnable boundaries with fixed frequency band allocation (33%/3496/33)) Despite limiting adaptation to datasct-specific spectral patterns, this variant shows minimal performance degradation compared to other ablations. This robustness comes from three complementary factors:
First, the spectral grouping mechanism remains functional. Despite fixed boundaries, nodes compute spectral energy responses across three bands, enabling firqueecy- graph construction that captures coarse but discriminative patterns (e.g.. wwwffreqenncy dominance for trends).
Second, the shared Fourier basis U maintains consistent spectral representation across batches, thereby ensuring stable energy computation that remains independent of boundary adaptation.
Third, the MoE routing module retains full adaptability by dynamically adjusting filter selection based on node-specific energy profiles, which partially compensates for the limitations imposed by fixed partitioning.
These results suggest that while learnable partitioning enhances fine- ime-ggained spectral alignment, the three-band decomposition itself provides strong inductive bias for frequency-aware modeling. In contrast, ablations that disrupt core mechanisms, such as spectral grouping (w/o Node Group) or adaptive filtering (w/o Fillers), yield larger performance drops, confirming their critical roles.
The plugin implementation isolates solely the dependency-filet architectural advantages of the full TimeFilter model.
Furthermore, Table is designed to answer the following question: Which dependency mechanism is more effective when integrated into a generic backbone? Under this controlled setup, DLnD consistently outperforms DL inear+ mhearTTmmefi TiteFilter across all datasets. This demonstrates that the pro-
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posed spectral DyMoE mechanism provides effective, generalizable, and backbone- dependency modeling capability than the camporal-domate filtering strategy used in TimeFilter.
C.8 ZERO-SHIOT FORECASTING PERFORMANCE
Our zero-shot evaluation assesses the transferability of plugin modules across domains without re- training. This setting requires plugins to be architecure--amostic and maining-decoupp Under these criteria, compare xCPD with CCM, the only existing baseline that satisfies these requirements. LIFT (Zhao & Shen, 2024) is excluded from this evaluation due to fundamental incompatiiilities; (i) modifies backbone architectures through additional layers and loss terms, requiring joint optimiza- tion rather than modular integration: (ii) it learns datasct-specific transformations without storing transferable patterns or prototypes; (iii) it lacks decoupled components necessary for cross-domain deployment. In contrast, both CCM and xCPD maintain explicit, reusable representations, i.e., CCM through frequency prototypes and xCPD through spectral bases, enabling zero-shot transfer.
Table 13 presents the zero-shot forecasting results, where the plugins trained on the source datasets are directly applied to the target datasets without adaptation. xCPD demonstrates superior transferability, achieving consistent improvements across diverse domain pairs. This validates our design principle of learning universal spectral representations rather than dptasct-specific transformations.
Table 13: Zero-shot forecasting results on ETT datasets. Look-back horizon 96 and the forecasting horizon is (96, 720}. Best performance on each setting is in bold.
Makel
TSMi
CTM
-TTM
+.C MSE/MAA
ParSTST MAE
CTM
MSE/M4A
Timeshor
MAR
CCM
0.28//0.35
ELSO6/0 /O 035
0.564/0033 (1.992
HL362
DU-38 (0L000
0.501 F1412
0:58 FILAEO
8.564/ BuaNi
0094//0.444
0.06 01.96
100S FULS22
10543 (087)
8323 (UA64
TURS 00.555
BL211 H0L92
6:44 (ES5)
0515//0/91
8:508 6.426
1.252/0.13
1215/0093
1.99/00222
1243/00.33
/0.11
1.313/0.76 (0.76
10552
1.23 06221
1.59 R.992
11-122
0009/03954 594
0007 10.99
12255 996
1130 7U.700
EANO FOUSE)
FUATI
100
(0/251
32-111
0.96/72.622
0195/3554
0,85 (0.26
0,63/ (6A51
0.516 FLLES
0,45 10,022
0/00
0.80
DASS (0,777
0/00 (URDA
0:733 (0,99)
0.455 (B16
8:735
13-02
1.722/0.92
1,866 996
1443 (6864
1995//0.53
1,455 10.575
1.21 (1.46)
1,999/1 190
1664/0.99 (0.95)
1.543 (6373
//50//1900 1116
1.61 FBNNT
1.33 F8903
/0/791
2.37511313
/1
2003/1 135
1,20/1100 1855
(1/62
2/55/11 199
1553
3/167/1
C.9 COMPUTATIONAL EFFICIENCY ANALYSIS
We evaluate the computational overhead of different plugin methods on the Traffic dataset, measuring both memory consumption and training time per iteration. Table 14 presents a comparative analysis across three backbone models with and without plugins.
Table 14: Computational efficiency comparison on Traffic dataset. Memory usage and training time measured on single NVIDIA A100 GPU.
Method
Memory (MB)
Time (ms/iter)
Baseline Models
PatchTST
12,458
29
TimesNet
13,005
70
TSMixer
13,088
89
With CCM Plugin
PatchTST + CCM
25,034
161
TimesNet + CCM
20,614
172
TSMixer + CCM
22,698
183
With xCP Plugin
PatchTST + xCPD xCPD
7,018
TimesNet xCPD
7,018
TSMixer + xCPD
7,018
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The results reveal substantial differences in computational overhead. CCM incurs significant costs, increasing memory usage by 60-100% and training time by 4-5x compared to baseline models. This overhead originates from CCM's iterative clustering mechanism, whose complexity scales linearly with both cluster count and channel dimensionality. In contrast, xCPD achieves remarkable efficiency, maintaining constant memory footprint (Rs 7GB) and minimal iteration time (<100ms regardless of the backbone architecture. Importantly, the memory reported here corresponds to the peak GPU memory required to train the plugin module, the full backbone model. Since xCPD trains only its own lightweight parameters while keeping the backbone frozen, the backbone does not allocate gradient states, optimizer buffers, backward activations. Consequently, the memory consumption during xCPD training is dominated by the plugin itself, leading similar values across different backbones. This efficiency stems from three design principles: (i) spectral decomposition operates on pre-computed Fourier bases, eliminating iterative optimization; (ii) the shared spectral basís amortizes computational costs across batches; and (ii) frequency-awe -ware routing employs only sparse matrix multiplications. These properties make xCPD particularly suitable for large-scal scale deployment and applications.
Table 15: Hyperparameter settings for different datasets, where 7" is the forecasting horizon, e.layers is the number of graph blocks, Ir is the learning rate, and d_ff is the dimension of feed-forward layers.
Tasks
Dataset
Length Patch
e layers
ff
Num. of Epochs
ETTh1
T /16
le-4
256
10
ETTh2
T /16
1
le-4
256
10
ETTml
T' /1/12
2
le-4
256
10
Long-term
ETTm2
T6/6
2
le-4
128
10
Forecasting
Exchange
T /24
2
le-4
256
10
Weather
T
2
1e-4
256
10
Electricity
T
2
le-1
512
15
Traffic
T"
2
le-4
2048
30
Solar- Energy
/2
2
le-4
512
10
M4 Yearly
1
le-3
512
20
M4 Quarterly
1
le-3
512
20
Short-term
M4 Monthly
1e-3
512
20
Forecasting
M4 Weekly
le-3
512
20
M4 Daily
1
1e-3
512
20
M4 Hourly
2
1
1e-3
512
20
Stock
1
1e-3
512
20
D EXPERIMENT DETAILS
D.1 METRICS
We adopt standard evaluation metrics following prior work (Wu et al., 2023; Chen et al., 2024). Specifically, for long-term and stock forecasting, report Mean Absolute Error (MAE) and Mean Squared Error (MSE). For the M4 dataset (Makridakis et al., 2018), we follow its official protocol, using three metries: Symmetric Mean Absolute Percentage Error (SMAPE). Mean Absolute Scaled Error (MASE), and Overall Weighted Average (OWA). MASE is a scale-independent metric that compares predictions against naïve seasonal baseline with periodicity s. SMAPE scales errors by the average of predictions and targets, while OWA combines SMAPE and MASE, normalized by the performance of the Naïve2 baseline (Team et al., 2018).
To formulate these metrics, let X, and X, denote the ground truth and prediction at timestep t, respectively, and let T' be the forecasting horizon. They are formulated as follows:
MAR
X.
Xeb
MSE
Σ X. x.).
SMAPE
200 T
X
(15)
X
1
X,
X,
1
SMAPE
MASE
MASE
OWA
(16)
T
X
X
2
SMAPENaac!
MASE
-
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D.2 HYPERPARAMETERS
D.2.1 SENSITIVITY TO K-NN SPARSIFICATION RATIO
We analyze the sensitivity of the sparsification ratio used in the ego-graph construction. In our implementation, we adopt a constant sparsification ratio by retaining the top-a proportion of edges in each node's ss adjacency row. To evaluate whether adaptive is necessary, we vary from 0.3 to 0.7 and report results across six datasets.
Table 16: Sensitivity to sparsification ratio (top-a edges retained). Reported metrics are averaged MSE/MAE over forecasting horizons {96, 192, 336, 720) with input length 96 on CCD.
Variant
ETTh
ETTb2
ETTml
ETTm2
Exchange
Solar
K-NN (top-30%)
0.449/ 00.438
0.511 A..42
0.399/
0,346 0..394
0.315/0.381
0.326 / 0.396
k-NN (top-40%)
0.447 / 00.43
0.509 0.480
0.391 /0.398
0,342 / 0,388
0.314/0.381
0.325/ 0.3395
k-NN (top-50%)
0.507
0.395
0.340 0.388
0.312/0..8 0.338
0.322 0.393
K-NN (lup-60%)
0.446/ 0.435
0.507 / 0.479
0,389 / 0.396
0.341 / 0.338
0.311/0.379 0
0.321 / 00399
(lop-70%) 70%)
0.450/0.438
0.394/ 0.400
0.346/ 0.393
0.317/0.382
0.325 0.395
As shown in Table 16, the results demonstrate that xCPD highly stable across a wide range of sparsification ratios, with the best performance consistently achieved around 0.5. This indicates that: (i) the model is largely insensitive to the exact choice of k; (ii) there is no evidence suggesting the need for dataset-specific adaptive selection; and (iii) sparsification mainly acts light structural constraint, as frequency-awae modulation. We therefore adopt a 0.5 the default in all experiments. This stability further supports the practical usability of xCPD without extensive hyperparameter tuning.
MoE routing already performs dynamic and selective dependency
Table 17: Running time, forecasting accuracy, and GPU memory usage of DLinear+xCPD the Electricity dataset with forecasting horizon T = 192.
Model
Patch Length
Running Time (ms/iter)
MSE
MAE
Memory (GB)
DI CPD
16
78
0.183
0.270
~7.0
DI CCD
0.181
0.269
55.5
DLinear*XCPD
32
30
0,180
0,268
44.5
DLinear*XCPD
64
6
0.181
0.269
~3.2
DLinear+xCPD
96
5.5
0.184
0.271
~2.8
D.2.2 SELECTION OF PATCH SIZE AND OVERLAP
We further conduct a systematic study of patch configurations to fully address the impact of patch size and patch overlap. The patch sizes used in our experiments are selected by balancing forecasting accuracy against computational efficiency. To illustrate this trade-off more concretely, Table 17 reports the results of DLinear+xCPP We observe that smaller patch sizes slightly improve accuracy but significantly increase computational cost and GPU memory usage, while very large patches degrade accuracy due to oversmoothing. A moderate patch length (e.g.. T²/3 when T" = 192) provides the best efficiency-accuracy trade-off.
the Electricity dataset with forecasting horizon T" = 192.
Table 18: Forecasting accuracy (MSE/MAE) of DLinear*XCPD the Electricity dataset under different patch-overlap settings and forecasting horizons.
Setting
Horizon=192 Horizon=96
Horizon=336
Horizon=720
Overlap 1
0.180 0.267
0.187 / 0.275
0.199 / 0.296
0.245 / 0.328
Overlap = 2
0.178/ 0.267
0.189 / 0.275
0.201 / 0.297
0.248 0.329
No Overlap
0.174/ / 0.262
0.180 / 0.268
0.195 / 0.293
0.240 / 0.326
Furthermore, although our main experiments adopt non-overlapping patches, also examine whether allowing patch overlap could improve performance. Table 18 reports results on the Electricity dataset across four forecasting horizons (96, 192, 336, and 720). We observe that introducing overlap
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consistently degrades accuracy across all settings. Similar trends observed other datasets, where overlap does not yield improvements. This effect arises because overlapping patches blur local temporal boundaries and mix frequency components from adjacent segments. As a result, the patch-wise spectral patterns become less distinctive, weakening the separation among low-, mid- and high-frequency bands and reducing the discriminative power of our frequency-awar MoE. routing. These findings support our choice of using non-overlapping patches as the default configuration.
E THEORY ASSUMPTIONS AND PROOFS
E. THEOREM 4.1 PROOFS
Theorem E.1 (Shared Graph Fourier Basis). Let L' be graph Laplacian derived from adjacency trix A at the t-th timestep where E }...., Define the average Laplacian as Lag=+ Et= L', with deeen-eccomooitionn = U denotes the shared Fourier basis that approximates each individual eigenbasis U from the decomposition of L', satisfying | UR'I l C L Laggl where C depends on the eigen-gap of Long and R is an orthogonal rotation matrix.
Proof. Let us define the Laplacian perturbation as:
A : L' Lavg.
We assume that Lavg and L' are symmetric and positive semi-definite matrices, which is always true for graph Laplacians.
Let Lavg UAUT be the eigendecomposition of Lav. where U E Raxa an orthonormal matrix of eigenvectors and A is diagonal matrix of eigenvalues Àι S ... < An- Similarly, let L' UU' be the eigendecomposition of L with eigenvectors Ut and eigenvalues A
We aim to bound the difference between U and U (up to rotation). This can be achieved by applying a classical result from matrix perturbation theory-Davis DavisKaaaaahn Kahan sin θ Theorem. Let U and U be the subspaces spanned by columns of U and U respectively. The Davis-Kahan theorem asserts that the principal angles between these subspaces satisfy:
sin (U, | ||d||
5
where 8 = miniys |^: Ay1 is the eigengap of Luug, assuming is strictly positive.
To obtain a bound on the #iffference between the orthonormal eigenbases, we use the following consequence of the Davis-Kahan theorem Park & Kim (2017): There exists orthogonal matrix R such that:
l0' - -UR - v2I sin @U, U'l
Combining the two inequalities:
I UR' S 12. |
Thus, defining C = 22/8 gives:
IU-- 1F S C
Lavgll
E.2 THEOREM 4.2 PROOPS
Theorem E.2 (Spectral Energy Response), Given spectral node embeddings X and graph Fourier basis U, spectral energy response of node i for frequency j is S = U. x : The orthonormal- ity of U ensures energy preservation between spatial and spectral domains: E SLg = | " | confirming spectral energy response retains all information from original channel-patch embeddings.
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Proof. Let X Ruxd denote the spatial-domain embeddings and X U X the spectral embeddings, where U E R=** is the orthonormal Fourier basis satisfying U -I For node i, its spatial representation can be recovered as:
12
Σ
Un
X
(17)
The energy of node in the spatial domain is:
2
n
Uii X J;
xp
(18)
Since the Fourier basis vectors are orthogonal, the cross terms vanish, yielding:
X
Σ
X 1
Σ Sij-
(19)
where Sij = ||0g) X 11 is the spectral energy response as defined.
Ji
Remark (Role of Energy Preservation xCPD). While energy preservation under orthogonal transformation is well-established mathematical property. Theorem 4.2 serves specific functional purpose in our framework beyond norm conservation. The theorem establishes two critical com- ponents for xCPD: (1) Node-level frequency characterization. The spectral energy response S.j quantifies how strongly node responds to frequency component j. This node- code-freeuency response forms the basis for our frequency-aware grouping, enabling identification of nodes are dominated by low-frequency. mid-frequency, or high-frequency patterns. (2) Foundation for spectral routing, By decomposing each node's total energy across frequency bands, we compute probabilistic assignments that guide our MoE routing mechanism. This frequency- based decomposition allows the model to ap- ply specialized filters tailored to each node. Unlike prior spectral methods in time series analysis that focus on global frequency analysis or channel-wise spectral features, xCPD leverages node-specific frequency responses to construct localized, frequency-alige aligned ego-graphs. This design enables fine- grained modeling of varying frequency-specine dependencies, providing principled approach to adaptive graph construction multivariate time series forecasting.
E.3 LIMITATIONS
Although xCPD is computationally efficient due to its ego-graph design and channel-patch modeling, its reliance on graph spectral decomposition and Trequency-aware grouping introduces moderate overhead, particularly when the number of channels patches becomes large. Although we alleviate this with learnable frequency bands and node-level filtering, the trade-off between fine-grained modeling and runtime efficiency still requires careful balancing. In addition, the assumption that spectral energy can effectively guide dependency selection may not hold in cases where temporal patterns are highly irregular or when the frequency distribution shifts significantly across domains. Lastly, while xCPD operates as general plugin, integrating it with more complex backbones (c.g.. autoregressive or hicrarchical models) may require additional architectural tuning.
E.4 FUTURE WORKS
Future work can extend xCPD in several directions. One promising avenue is incorporating external domain knowledge (c.g., meteorological or physiological priors) guide the spectral grouping and filtering process. Additionally, developing more adaptive and interpretable graph filtering strategies, such as continuous expert blending or attention- bbaed routing, could further improve robustness and transparency. Another important direction is to explore the applicability of xCPD to wider spectrum of structured prediction tasks beyond time series forecasting, including spatiotemporal event forecast- ing and multivariate anomaly detection. Finally, with the advent of time series foundation models, xCPD is positiostt to serve as a lightweight plugin to augment their inductive biases across modalities and domains. Integrating spectral-awar dependency modeling into large-scale pretrained time series backbones may enhance their ability to generalize, adapt, and transfer knowledge to unseen environments with minimal fine-tuning.
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E.5 SOCIAL IMPACTS
xCPD holds significant promise for critical real-world applications by enhancing forecasting accuracy and robustness in dynamic, multi-sensor environments. In healthcare, xCPD could help in early warn- ing systems for patient deterioration by better modeling evolving physiological patterns. Similarly, in domains like smart grids and traffic systems, its frequency-aware filtering could prove instrumental in mitigating noise to extract actionable trends. Ultimately, by providing a modular and interpretable framework, xCPD contributes to the crucial goal of developing scalable and trustworthy forecasting systems for high-stakes decision-making.
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Reviews:
[{"review_id": "ROogtDwkrw", "reviewer": "Reviewer_r5xJ", "summary": "This paper proposes a generic plugin xCPD that can adaptively model the channel-patch dependencies from the perspective of graph spectral decomposition to address the problem in multivariate time series forecasting (TSF). This plugin can be added to an existing TSF model, improving its performance accordingly.", "questions": "1. Why does this paper not include existing CP methods in comparison? \n\n2. In the efficiency evaluation, it does not compare the time difference between the original model and the original+xCPD method.", "limitations": "", "rating": 4, "confidence": 4, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "1. The idea of studying channel relationships is interesting. \n\n2. The experimental settings in this paper are extensive, covering long-term, short-term, and zero-shot forecasting. \n\n3. The writing of this paper is good, which is easy to read.", "weaknesses": "1. This plugin introduces lots of computations, but the improvement in the experiment seems to be trivial. \n\n2. In the experiment, this paper does not include existing CP methods, e.g.. Qiu et al., 2024, Chen et al., 2024,  Hu et al., 2025b, Lee et al., 2025.\n\n3. The module design seems to be straightforward, and the design is not very novel.", "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": "3oeTpXSYil", "reviewer": "Reviewer_noBY", "summary": "This paper introduces xCPD, a generic plugin that adaptively routes channel-patch dependencies in multivariate time series forecasting through a graph spectral decomposition framework. Unlike previous channel-partial or CI/CD approaches, xCPD conducts frequency-domain analysis using a learned, shared graph Fourier basis, classifying channel-patch nodes by spectral energy into low-, mid-, and high-frequency bands. It employs a mixture-of-experts (MoE) routing mechanism at the patch level to enable fine-grained, dynamic selection of inter-channel interactions. The plugin is model-agnostic, fits atop various existing backbones, and demonstrates consistent improvement in forecasting accuracy and generalization across a breadth of benchmarks and domains. Extensive empirical results, ablations, qualitative illustrations, and theoretical justifications are included.", "questions": "1、Can the authors provide more precise formalization for channel-patch edge construction? Are there practical constraints on $k$ in the $k$-NN, and how does patch size/overlap affect results, especially in high-dimensional settings (e.g., Traffic, Electricity datasets)?\n\n2、What is the sensitivity of performance and expert utilization (e.g., routing diversity, collapse) to the threshold $\\tau$ and noise parameter $\\epsilon$? Is there evidence that these parameters are stable across dataset/backbone combinations, or do they require careful tuning?\n\n3、Have the authors considered adaptively selecting the number of frequency bands (rather than fixed to three), e.g., via a learned clustering or continuous assignment, and how would this affect interpretability or accuracy?\n\n4、Can the authors provide statistics or visualizations of the empirical spectral gap across datasets? In scenarios with weak or noisy structure, what is the observed effect on performance and stability of the shared basis?", "limitations": "", "rating": 4, "confidence": 5, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "1、The proposal to model dependencies at the channel-patch level, routed adaptively by frequency band via graph spectral decomposition, provides a fresh and well-argued approach to balancing robustness and expressivity in MTSF, beyond existing CI/CD/CP tactics.\n\n2、The framework is underpinned by rigorous spectral graph theory, with precise mathematical formulations (e.g., Theorems 4.1 and 4.2 on shared basis and energy response, substantiated in the appendix with detailed proofs).\n\n3、The methodology is described with mathematical precision, from input patch embedding (Equation block in 4.1), spectral grouping (Section 4.2, Theorem 4.2), and MoE-based routing (Section 4.3, detailed algorithmic equations). The description in Section 4 rigorously motivates and supports the approach.", "weaknesses": "1、Some equations and algorithm formulations are somewhat terse or overloaded. For instance, the definition of the adjacency matrix in Section 4.1 blends inner products and node similarity without explicit regularization or clarification about self-connections; there is little discussion on normalization/range scaling (see equation defining $\\boldsymbol{A}^{t}$, Page 3).\n\n2、While Figure 1 and the methodological text (Pages 3-4) present the channel-patch notion as a powerful primitive, more care is needed in formally defining the edges: Are patches temporally contiguous? Are there constraints to prevent trivial, densely connected graphs? Is $k$ in $k$-NN constant or adaptive? This lack of detail may hinder reproducibility and may even affect rigor in settings with many variables and timesteps.\n\n3、The MoE routing for patch-level expert filtering, described with stochastic noise and cumulative softmax thresholds (Section 4.3), is algorithmically sound, but its practical behavior is underexplored. For example, the decision threshold $\\tau$ and stochastic term $\\epsilon$ may have non-trivial impact—these parameters are not deeply analyzed for sensitivity beyond a basic experiment. Additionally, while the adaptive routing is well-motivated, more insights or visualization of selected expert patterns across datasets/backbones would strengthen claims.", "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": "ynvGiHKcpD", "reviewer": "Reviewer_7esN", "summary": "This paper proposed a new method facilitating the modeling of cross-channel correlations multivariate time series. The proposed method is built based on the limitation of how existing multivariate time series forecasting methods handle dynamic correlations between channels, and introduce a graph composed of channel-patches to represent correlations between patches. A universal plugin that can be added to existing forecasting methods is introduced, and experiments show that it can bring performance uplift to existing methods.", "questions": "Can the authors further explain the motivation of this paper, so that the novelty of their proposed method is more intuitive?", "limitations": "", "rating": 4, "confidence": 3, "soundness": 2, "presentation": 1, "contribution": 2, "strengths": "1. A universal plugin is introduced based on the proposed novel technique of constructing correlation graphs of patches in multivariate time series. Extensive experiments prove its effectiveness at improving the performance of existing multivariate time series forecasting methods.\n2. The proposed method is described in detail and thus should be highly reproducible.", "weaknesses": "1. The presentation of the background and challenges can be improved. While discussing existing methods, the paper claims that there are limitations in how existing multivariate time series forecasting methods handle cross-channel correlations. However, the paper mainly uses vague descriptions such as \"coarse-grained methods treat each channel as a monolithic unit, missing nuanced interactions between channel segments\", which could use some further explanation. There are also some claims that don't seem to be fully correct. For example, one claim that existing methods \"failing to capture how inter-channel dependencies evolve dynamically\", yet that doesn't seem to be the case for existing methods that dynamically calculate attention weights between channels. In conclusion, it is unintuitive to me why existing methods cannot effectively model correlations in multivariate time series.\n2. Similarly, the explanation of the motivation behind the proposed method can be improved. Right now it is unintuitive why the proposed method can effectively tackle the limitations faced by existing methods, and how it is fundamentally different from existing designs. For example, the fundamental difference between building a graph of patches versus existing methods that calculate attention weights between patches (which essentially is a fully-connected graph).", "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": "KQdj6pwHhl", "reviewer": "Reviewer_KjPE", "summary": "This paper proposes xCPD, a lightweight plugin module designed to enhance multivariate time series forecasting by modeling fine-grained, frequency-aware channel-patch dependencies through graph spectral decomposition and adaptive routing. xCPD introduces spectral channel-patch embedding, frequency-based grouping, and a MoE routing mechanism. Experiments across 9 benchmarks demonstrate consistent improvements when integrating xCPD into both channel-independent and channel-dependent backbones.", "questions": "1. The paper claims that prior methods assume static dependencies, but TimeFilter and CM both model dynamic ones. Could the authors clarify this claim?\n2. Is there any insight into what the learned spectral groups correspond to in temporal terms?\n3. The framework essentially reinterprets channel dependencies in the spectral domain, but without clear evidence that this yields fundamentally different learned relationships among low-, mid-, and high-frequency bands.", "limitations": "", "rating": 4, "confidence": 5, "soundness": 3, "presentation": 4, "contribution": 2, "strengths": "1. xCPD can be plugged into various architectures (linear, transformer, convolutional) with relatively low computational cost.\n2. Evaluations cover both long-term and short-term forecasting, along with ablations and zero-shot tests.\n3. Theorems 4.1 and 4.2 provide formal justification for the spectral construction.", "weaknesses": "1. Questionable novelty claim: The paper claims prior methods like TimeFilter [1] and CM [2] assume static inter-channel dependencies. However, both papers explicitly model **fine-grained**, **adaptive**, and **time-varying** relationships through learned attention or context-dependent graphs. This undermines the motivation that xCPD uniquely addresses “time-varying” dependencies.\n2. The structure, especially the Channel-Patch Filtering with MoE, is highly reminiscent of TimeFilter’s[1] adaptive filtering mechanism. The main distinction—the spectral grouping—feels incremental rather than fundamentally novel.\n3. In Table 6, the MSE/MAE values are worse than those reported in the TimeFilter[1] paper on the same datasets and horizons. This discrepancy needs to be explained.\n\n[1] TimeFilter: Patch-Specific Spatial-Temporal Graph Filtration for Time Series Forecasting\n\n[2] Partial channel dependence with channel masks for time series foundation models", "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)