GatedLinear: Adaptive Routing of Complementary Linear Bases for Time Series Forecasting

Abstract: Time series forecasting requires models to capture diverse, often mutually exclusive, temporal dynamics, from smooth trend continuation to nonstationary drift and strict phase-aligned recurrence. While recent deep learning models have improved accuracy, they typically force these diverse patterns through a single computational backbone governed by fixed algorithmic inductive biases (e.g., self-attention or spectral filtering). This single-mechanism approach often struggles with the profound heterogeneity of real-world series, where different variables and forecast horizons necessitate fundamentally different predictive treatments. To address this, we propose GatedLinear: a lightweight framework that frames forecasting as the adaptive routing of complementary linear bases. GatedLinear leverages a pool of three specialized mechanisms: a global trend-seasonal basis for smooth projection, a difference-based incremental basis for nonstationary drift, and a phase-aligned recurrence basis for explicit cyclic reuse. To dynamically orchestrate these distinct behaviors, we introduce a Tri-Factorized Fusion Gate that disentangles routing decisions into channel-specific preferences, horizon-aware offsets, and phase-indexed biases derived from known future time marks. This design allows the model to perform highly granular, point-wise soft routing across different predictive regimes without stacking computationally heavy neural modules. Experiments on standard benchmarks show that our method achieves state-of-the-art or highly competitive accuracy against recent complex foundational models, while offering explicitly interpretable routing patterns and operating with a substantially smaller parameter footprint.

• The paper introduces GatedLinear, which combines trend-seasonal, difference-based, and phase-aligned linear predictors through a horizon- and channel-specific Tri-Factorized Fusion Gate.
• The model achieves the best results on 11 of 16 dataset-level metrics across eight benchmarks, with average MSE of 0.260 and MAE of 0.298 while avoiding the quadratic cost of attention.
• Ablations show that structured gating and phase-aware routing are central to performance, although the channel-independent design, periodicity assumptions, and limited linear basis pool constrain results on multivariate or nonlinear series.

GatedLinear addresses a recurring weakness in modern time series forecasting: most deep architectures impose a single algorithmic inductive bias—self-attention, spectral filtering, or multi-scale mixing—on all inputs, even though real-world series exhibit heterogeneous, sometimes mutually exclusive temporal regimes. The paper's central claim is that forecasting accuracy can be improved not by deepening a backbone but by adaptively routing each forecast point among a small pool of complementary linear mechanisms. The proposed model achieves state-of-the-art or highly competitive results on standard benchmarks while remaining lightweight and interpretable (2607.09537).

Motivation: mechanism heterogeneity

The authors first establish empirically that temporal traits (trend strength, exact periodicity, phase consistency, mean/scale shift, spectral entropy) vary substantially across benchmark datasets and across channels within a single dataset. They cite SynTSBench as evidence that few deep forecasters perform uniformly well across programmable pattern types. A motivating comparison shows PatchTST underfitting strict trends, over-smoothing cyclic spikes, and lagging nonstationary drift on synthetic data. This motivates a routing paradigm rather than a single-mechanism backbone.

Tri-basis linear architecture

GatedLinear instantiates three candidate predictors, each built from elementary operations plus one-layer temporal affine maps:

• Global trend–seasonal basis: moving-average decomposition into seasonal and trend components, each projected directly from length $L$ to horizon $H$ via channel-wise linear layers.
• Difference-based incremental basis: first-order differencing of the input, linear projection of future increments, and cumulative integration anchored strictly at the last observed value $X_{L,c}$.
• Phase-aligned recurrence basis: averaging the last $K$ complete cycles of period $P$ into a template $\mathbf{T} \in \mathbb{R}^{P \times C}$, tiling it forward, and adding a learned linear residual.

The final prediction is a convex combination of the three branches with weights $\mathbf{W} \in \mathbb{R}^{H \times C \times 3}$.

Tri-Factorized Fusion Gate

The gating logits are factorized into three disjoint learnable components:

$z_{h,c,k} = a_{c,k} + u_{h,c,k} + v_{\phi_h, c, k}$

where $a_{c,k}$ is a persistent channel-specific base tendency, $u_{h,c,k}$ is a horizon-dependent offset, and $H$0 is a phase-indexed bias selected by the future phase index $H$1, derived from decoder-side time marks when available or from horizon modulo $H$2 otherwise. A temperature-controlled softmax ($H$3) normalizes the weights. This design enables point-wise soft routing that distinguishes, for example, a noon forecast from a midnight forecast at identical horizon offsets. Total parameter count scales as $H$4 and inference complexity as $H$5, avoiding the $H$6 cost of attention-based models.

Theoretical justification for the difference basis

The appendix provides an analysis under an integrated random walk with drift ($H$7). The MMSE predictor is $H$8. A global linear map over raw values requires $H$9 to be shift-invariant; unconstrained SGD with weight decay cannot guarantee this, so any unseen level shift induces bias of order $X_{L,c}$0. The difference basis, by contrast, operates on stationary increments and reconstructs levels by cumulative summation from $X_{L,c}$1, structurally matching the MMSE predictor with error variance exactly $X_{L,c}$2. The authors present this as a theoretically necessary inductive bias rather than an engineering choice, though the argument applies specifically to this data-generating process.

Empirical results

With input length $X_{L,c}$3 and horizons $X_{L,c}$4, GatedLinear attains the best result on 11 of 16 dataset-level metrics across eight benchmarks and the lowest average MSE (0.260) and MAE (0.298), compared against DLinear, TimeMixer, MixLinear, PaiFilter, PatchTST, PhaseFormer, TQNet, TimeAlign, and iTransformer. Representative averages:

Dataset	GatedLinear MSE	Best baseline MSE
ECL	0.138	0.141 (PaiFilter/TQNet/TimeAlign)
ETTm1	0.305	0.307 (TimeAlign)
Weather	0.173	0.174 (TimeAlign)
ETTh1	0.404	0.383 (MixLinear)
Traffic	0.405	0.367 (iTransformer)

The model is strongest on ECL, ETTm1, ETTm2, ETTh2, Exchange, and Weather, but concedes ETTh1 to MixLinear and Traffic to variate-aware methods such as iTransformer and TQNet—an outcome consistent with its channel-independent design.

Ablations confirm both sources of gain. Removing any single branch degrades performance (average relative increases of 1.60–2.07%), removing phase-indexed gate logits costs 2.11%, and replacing the factorized gate with a plain shared softmax costs 2.52%—the largest single ablation effect, indicating that structured routing contributes more than any individual basis.

Interpretability

Because the gate parameters are structured lookups, learned routing is directly inspectable. Across datasets, the difference branch receives larger weights at short horizons and declines with horizon length, while the phase-aligned branch grows in prominence at longer horizons—consistent with error accumulation in increment prediction versus stable cyclic reuse. Channel-level scatter plots show Traffic variables clustering tightly while Weather variables disperse broadly, mirroring measured trait heterogeneity. Spearman correlations between channel traits and aggregated branch preferences align semantically: the difference branch correlates with mean/scale shift, and the phase branch correlates with seasonality strength and phase consistency. These correlations support the claim that the gate learns meaningful preferences rather than arbitrary weights.

Efficiency measurements on Electricity place GatedLinear in the low-error, low-training-time, low-memory region compared with heavier baselines. Sensitivity studies show stable performance for fusion temperature $X_{L,c}$5 and best results with $X_{L,c}$6–$X_{L,c}$7 phase cycles; larger $X_{L,c}$8 degrades accuracy because older cycles dilute the template under level or amplitude changes.

Limitations and open questions

The authors identify three constraints. First, the channel-independent design does not model cross-variable dependencies, which explains the weaker Traffic results where variate-level attention helps. Second, the phase basis and phase-indexed gate assume a usable base period and meaningful future phase indices; weakly periodic series, drifting periods, holiday effects, and irregular sampling weaken this assumption. Third, restricting the predictor pool to three linear bases bounds expressiveness, so highly nonlinear dynamics or abrupt regime shifts may not be captured—the qualitative visualizations indeed show remaining errors concentrated around sudden regime changes. Additionally, reported numbers are averaged over three seeds without confidence intervals or significance tests, so fine-grained differences between top methods should be interpreted cautiously. An open question left by the paper is whether the tri-basis pool can be extended with cross-variate or nonlinear bases while preserving the interpretability and efficiency of the factorized gate.

Conclusion

GatedLinear demonstrates that adaptive, point-wise routing among three simple complementary linear bases—decomposed direct projection, difference-based incremental evolution, and phase-aligned recurrence—can match or exceed recent complex forecasters on standard benchmarks, with the lowest average MSE and MAE among nine baselines. The Tri-Factorized Fusion Gate provides interpretable, semantically aligned routing at negligible parameter and compute cost. The evidence supports the paper's position that structured mechanism selection is a viable alternative to architectural escalation, subject to the stated assumptions of channel independence, usable periodicity, and a deliberately limited basis pool.