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AsyTO: Asymmetric Temporal Operator for Parameter-Efficient Multivariate Time Series Forecasting
Xiachong Lin, Du Yin, Hao Xue, Wen Hu, Imran Razzak, Arian Prabowo, Matthew Amos, Flora D. Salim
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Summary
The paper introduces AsyTO, an Asymmetric Temporal Operator for parameter-efficient multivariate time series forecasting. It addresses the trade-off between parameter efficiency and flexibility by factorizing the forecasting operator into shared history-reading and future-writing temporal modes, combined with a low-rank periodic prototype. AsyTO achieves state-of-the-art lightweight error rates across eleven benchmarks, locating at the accuracy-compute Pareto frontier.
Entities (12)
Relation Signals (11)
AsyTO → evaluatedon → Electricity
confidence 95% · We evaluate AsyTO on datasets including: ... Electricity
AsyTO → evaluatedon → PEMS
confidence 95% · We evaluate AsyTO on datasets including: ... 4 PEMS datasets
AsyTO → evaluatedon → Weather
confidence 95% · We evaluate AsyTO on datasets including: ... Weather
AsyTO → evaluatedon → Traffic
confidence 95% · We evaluate AsyTO on datasets including: ... Traffic
AsyTO → evaluatedon → ETT
confidence 95% · We evaluate AsyTO on datasets including: the ETT family
AsyTO → solves → Multivariate Time Series Forecasting
confidence 95% · we propose AsyTO, an Asymmetric Temporal Operator that factorizes the tensor of per-variable operators
AsyTO → outperforms → PatchTST
confidence 90% · AsyTO attains the best lightweight error in 30 of 44 dataset-horizon settings
AsyTO → outperforms → CycleNet
confidence 90% · AsyTO attains the best lightweight error in 30 of 44 dataset-horizon settings
AsyTO → outperforms →
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Abstract
Abstract:Multivariate time-series forecasting faces a structural dilemma: sharing one temporal predictor across variables is parameter-efficient but forces heterogeneous variables through an identical history-to-future map, whereas learning an independent predictor per variable restores flexibility at a cost that grows with the product of variable count, context length, and horizon. We argue that this dilemma dissolves once the object being compressed is the forecasting operator rather than the observed series. Auditing per-variable linear history-to-future maps across standard benchmarks, we find that a phase-locked seasonal component paired with a compact residual operator outperforms a dense phase-blind reference in most audited settings. The residual transport is also directional: lag-invariant alternatives consistently underperform asymmetric history-to-future maps. Guided by this structure, we propose AsyTO, an Asymmetric Temporal Operator that factorizes the tensor of per-variable operators into shared but distinct history-reading and future-writing temporal modes with per-variable mode-wise gains, complemented by a low-rank periodic prototype and a cycle-separable factorization of the temporal modes. Each forecast reads only its own variable's history, so parameters and compute grow linearly in the number of variables. Across eleven benchmarks and multiple forecast horizons, AsyTO attains the best lightweight error in 30 of 44 dataset-horizon settings, locating at the accuracy-compute Pareto frontier.
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- Source: https://arxiv.org/abs/2608.16098v1
- Canonical: https://arxiv.org/abs/2608.16098v1
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AsyTO: Asymmetric Temporal Operator for Parameter-Efficient Multivariate Time Series Forecasting Xiachong Lin Du Yin Hao Xue Wen Hu Imran Razzak Arian Prabowo Matthew Amos Flora D. Salim Abstract Multivariate time-series forecasting faces a structural dilemma: sharing one temporal predictor across variables is parameter-efficient but forces heterogeneous variables through an identical history-to-future map, whereas learning an independent predictor per variable restores flexibility at a cost that grows with the product of variable count, context length, and horizon. We argue that this dilemma dissolves once the object being compressed is the forecasting operator rather than the observed series. Auditing per-variable linear history-to-future maps across standard benchmarks, we find that a phase-locked seasonal component paired with a compact residual operator outperforms a dense phase-blind reference in most audited settings. The residual transport is also directional: lag-invariant alternatives consistently underperform asymmetric history-to-future maps. Guided by this structure, we propose AsyTO, an Asymmetric Temporal Operator that factorizes the tensor of per-variable operators into shared but distinct history-reading and future-writing temporal modes with per-variable mode-wise gains, complemented by a low-rank periodic prototype and a cycle-separable factorization of the temporal modes. Each forecast reads only its own variable’s history, so parameters and compute grow linearly in the number of variables. Across eleven benchmarks and multiple forecast horizons, AsyTO attains the best lightweight error in 3030 of 4444 dataset-horizon settings, locating at the accuracy-compute Pareto frontier. Introduction Long-horizon multivariate time series forecasting has recently been reshaped by lightweight models. Linear and frequency-domain predictors with modest parameter counts (17; 14; 5) match or outperform Transformer-based forecasters (20; 13; 10) on standard benchmarks. The central question has therefore shifted from whether a forecaster can be small to how its limited parameter budget should be allocated. Compact forecasters spend this budget along both variable and temporal dimensions. Sharing one map across variables is efficient but imposes a common temporal response, whereas learning a dense map for each variable preserves heterogeneity but scales poorly with the panel size. Temporal compression reduces this cost, yet common constructions describe both ends of the forecasting map with the same object, such as basis, kernel, or pattern set. Their efficiency is therefore obtained by tying how historical evidence is read to how the future trajectory is written. We argue that these roles should remain distinct. Reading the history is an evidence-extraction problem that may concentrate on a few informative lags, whereas writing the forecast requires coordinating the entire horizon. We call this functional distinction history–future asymmetry. Preserving it ordinarily requires separate temporal parameters, creating a tension between role specialization and model size. The key question is whether asymmetry can be retained without surrendering compactness. Our model resolves this tension through periodic structure. A low-rank periodic prototype first estimates the recurring component and produces a residual series. The Asymmetric Temporal Operator (AsyTO) acts exclusively on this residual, using separate history-reading and future-writing factors shared across variables. Variable-specific gains are generated from coordinates learned by the periodic prototype, while cycle-separable factorization compresses both temporal factors. The final forecast combines the periodic component with the predicted residual, and each variable is predicted solely from its own history. Main contributions of this work are organized as below: • We formulate parameter-efficient multivariate time series forecasting as structured compression of the pervariable history-to-future operator tensor, and formalize history–future asymmetry as a distinct budget-aware inductive bias. • We propose AsyTO, which factorizes each variable-specific operator and combines it with a low-rank, phase-aligned periodic prototype and cycle-separable temporal factors. Prototype-conditioned gains encode heterogeneous temporal responses in a shared low-dimensional space. • Extensive structural controls and benchmark evaluations validate the proposed design across several real-world time-series datasets. Our AsyTO improves forecasting accuracy across most datasets while requiring substantially fewer parameters. The proposed modules further improve several popular lightweight backbones in most cases, respectively. Related Works Compact Forecasting and Temporal Compression. Lightweight temporal models range from linear predictors such as DLinear (17), MixLinear (8), and LightTS (19) to sparse or spectral designs including FITS (14), SparseTSF (5), FreTS (16), and FilterNet (15). An orthogonal distinction concerns variable interaction: iTransformer (7) and query-based models (3) mix information across the panel, whereas PatchTST (10) enforces channel independence; this trade-off is studied in (1). Existing approaches also compress the history-to-forecast map through fixed-basis spectral interpolation (14), relative-lag convolutions (6), or learned temporal modes (9). AsyTO follows channel-independent data flow while sharing its temporal factors across variables, but differs from these compression schemes by assigning separate learned factors to history reading and forecast generation under the same parameter budget. Explicit Periodic Modeling. CycleNet (4) removes a learned cycle before forecasting the residual, while SparseTSF (5), PhaseFormer (11), and FreqCycle (18) exploit periodicity through downsampling, phase tokenization, and cycle-aligned spectra. AsyTO shares CycleNet’s decomposition step and makes no novelty claim for periodic removal. Its distinction is to reuse the periodic prototype to parameterize residual forecasting: prototype coordinates generate variable responses, while cycle–phase structure compresses the asymmetric temporal factors. Methodology Figure 1: AsyTO. A low-rank periodic prototype =⊤Q=UV is removed from the history and restored on the future axis by phase lookup; the periodicity-exclusive residual is transported by an asymmetric operator whose history and future factors (,)( , ) are separate and shared across variables, with a per-variable modal response G. Both temporal factors are further compressed into a cycle-index and a within-cycle phase factor (CeST), and G reuses the prototype’s variable coordinates, so each additional variable costs only RpR_p parameters. Problem Formulation Denote a batch of historical time-series observations ∈ℝB×L×CX ^B× L× C, the objective is to predict the subsequent H observations ∈ℝB×H×CY ^B× H× C, where B, L, and H denote the batch size, context length, and forecast horizon. For sample b and variable c, b,:,c∈ℝLX_b,:,c ^L and b,:,c∈ℝHY_b,:,c ^H are its historical context and future trajectory. The forecasting problem for the given variable is defined as ^b,:,c=fθ(b,:,c;c),c∈1,…,C. Y_b,:,c=f_θ\! (X_b,:,c;c ), c∈\1,…,C\. (1) where fθ∈ℝL×Hf_θ ^L× H is the forecaster operator bridging the information transport from history to future. Low-Rank Periodic Prototype Let P be the dataset period and sb∈0,…,P−1s_b∈\0,…,P-1\ the phase of the first historical observation of sample b, so position τ carries phase πb(τ)=(sb+τ−1)modP _b(τ)=(s_b+τ-1) P. Instead of a dense phase–variable table in ℝP×CR^P× C, we learn a low-rank prototype =⊤,∈ℝP×Rp,∈ℝC×Rp, =UV , ^P× R_p,\;V ^C× R_p, (2) where U holds within-cycle patterns shared by all variables and c,:V_c,: places variable c in that subspace, reducing the cost from PCPC to Rp(P+C)R_p(P+C). The assumption is low rank of the periodic template, not of the observations themselves. History and future components are read off the same prototype by phase, b,t,chist=πb(t),cQ^hist_b,t,c=Q_ _b(t),c and b,h,cfuture=πb(L+h),cQ^future_b,h,c=Q_ _b(L+h),c. Subtracting the first leaves the periodicity-exclusive residual ℛbhist=~b−bhist∈ℝL×C, ^hist_b= X_b-Q^hist_b ^L× C, (3) which the operator transports, while futureQ^future supplies the phase-aligned future component and will be added back eventually. We initialize U orthogonally and V to zero, so training starts from a neutral template while V still receives gradient. Asymmetric Temporal Operator The residual ℛhistR^hist must now be transported to the future axis. Giving every variable its own dense map means an operator ∈ℝC×H×LW ^C× H× L with CHLCHL parameters; sharing one map across variables is far smaller but forces identical temporal responses. We interpolate between the two with a CP factorization into three semantically distinct factors: history-side factors ∈ℝL×Rm ^L× R_m, future-side factors ∈ℝH×Rm ^H× R_m, and a variable–mode response ∈ℝC×RmG ^C× R_m whose c-th row is GcG_c. For variable c, c=diag(Gc)⊤=∑m=1RmGc,mϕmψm⊤, _c= diag(G_c) = _m=1^R_mG_c,m _m _m , (4) and ℛfutureb,:,c=cℛhistb,:,cR^future_b,:,c=W_c\,R^hist_b,:,c. Here ψm=:,m _m= _:,m reads a pattern out of the historical residual and ϕm=:,m _m= _:,m writes the corresponding future trajectory; both are shared across variables, and Gc,mG_c,m sets how strongly variable c responds to mode m. The operator never has to be materialized. Contracting in mode space, Zb,m,c=∑t=1LΨt,mℛb,t,chist,ℛb,h,cfuture=∑m=1RmΦh,mGc,mZb,m,c, Z_b,m,c= _t=1^L _t,mR^hist_b,t,c, ^future_b,h,c= _m=1^R_m _h,mG_c,mZ_b,m,c, (5) costs Rm(L+H+C)R_m(L+H+C) parameters and (BC(L+H)Rm)O(BC(L+H)R_m) arithmetic. It also makes the target-only property explicit: the forecast for variable c reads ℛhistb,:,cR^hist_b,:,c and never another variable’s observations. Crucially, and are parameterized separately. Even when L=HL=H, reading evidence from a past position and generating a future one are different operations, and tying Ψ=Φ = would force a symmetric temporal map. The ordered pair (Ψ,Φ)( , ) is what makes the transport directional, and this asymmetry is the property the operator cannot give up (Figure 3). Cycle-Separable Temporal Factorization Equation (5) already shares Ψ and Φ across variables, so they now dominate the budget: Rm(L+H)R_m(L+H) against only RmCR_mC for the response. Both factors are indexed by absolute time, yet the signal is periodic, so a position matters mainly through the cycle it falls in and its phase within that cycle. Writing t−1=(k−1)P+pt-1=(k-1)P+p for cycle index k and phase p, we give each temporal factor a rank-RkR_k Cycle-Separable Temporal (CeST) form, Ψt,m=∑r=1RkAk,r,mΨBp,r,mΨ, _t,m= _r=1^R_kA _k,r,mB _p,r,m, (6) a Kronecker product of a cycle-index factor AΨ∈ℝK×Rk×RmA ^K× R_k× R_m and a within-cycle phase factor BΨ∈ℝP×Rk×RmB ^P× R_k× R_m, with K=⌈L/P⌉K= L/P . The cost falls from LRmLR_m to RkRm(K+P)R_kR_m(K+P), and the two factors separate how far back evidence lies from where in the cycle it sits. History-End CeST. Applying Eq. (6) to Ψ gives the mode coefficients Zb,m,cZ_b,m,c of Eq. (5) without materializing Ψ . A strictly periodic basis cannot express a transient in the last few observations, so we add a short dense correction over the final M steps, Zb,m,c=∑t=1LΨt,mℛb,t,chist+∑j=1MEj,mℛb,L−M+j,chist, Z_b,m,c= _t=1^L _t,mR^hist_b,t,c+ _j=1^ME_j,mR^hist_b,L-M+j,c, (7) with E∈ℝM×RmE ^M× R_m initialized to zero, so the model starts from the purely periodic basis and departs from it only if the data demand it. History-side cost is RkRm(K+P)+MRmR_kR_m(K+P)+MR_m. Future-End CeST. The same factorization applies to Φ with K′=⌈H/P⌉K = H/P output cycles. It is only used when the horizon spans at least one full cycle: if H≤PH≤ P there is no cycle index to separate, and Φ stays dense. This condition, rather than any property of the data, is what makes the future-side term inactive on the short-horizon PEMS settings. Cycle-Shared Variable Response. The response ∈ℝC×RmG ^C× R_m is the last term that still grows with C. The prototype already assigns every variable a coordinate c,:∈ℝRpV_c,: ^R_p, so we reuse it, =+γtanh(G),G∈ℝRp×Rm, =1+γ (VM_G), _G ^R_p× R_m, (8) where γ bounds the modulation. This replaces CRmCR_m parameters by RpRmR_pR_m. Since Rp≤CR_p≤ C by construction, the factorization is never more expensive, and each additional variable then costs only the RpR_p entries of its prototype coordinate. The phase coordinates U supply an analogous modulation of Φ across future phases. Residual Echo. The observation at the same phase one cycle earlier is a direct local cue that the mode basis need not carry. For future step h we read s(h)=L−P+((h−1)modP)+1s(h)=L-P+((h-1) P)+1 and add a phase-conditioned, cycle-decayed copy of ℛb,s(h),chistR^hist_b,s(h),c, gated to zero whenever s(h)<1s(h)<1. The term is therefore identically zero when the window is shorter than one cycle. Experimental Results Datasets We evaluate AsyTO on datasets including: the ETT family (20), Electricity and Traffic (2), Weather (13), and 4 PEMS datasets (6). The appendix summarizes the statistics, including the period P each dataset is assumed to have. For the 7 non-PEMS datasets, we use a context length of L=720L=720 and forecasting horizons of H∈96,192,336,720H∈\96,192,336,720\. ETT follows the standard calendar split, while Electricity, Traffic, and Weather use chronological 70%/10%/20% splits. Following the standard short-horizon traffic forecasting protocol, we evaluate the PEMS datasets using L=96L=96, H∈12,24,48,96H∈\12,24,48,96\, and chronological splits of 60%/20%/20%. Baselines and Metrics We compare against 12 published forecasters, including lightweight group: SparseTSF (5), MixLinear (8), PhaseFormer (11), FITS (14), CycleNet (4), DLinear (17) and FreqCycle (18). The conventional group: FilterNet (15), PatchTST (10), TQNet (3), iTransformer (7) and FreTS (16). Mean squared error (MSE) and mean absolute error (MAE) on the standardized series are reported for evaluation, averaged over seeds 2024,2025,2026\2024,2025,2026\. Implementation Details All models in this work are implemented with PyTorch (12), training uses the Adam optimizer with MSE loss. All the experiments are conducted on NVIDIA H100 GPUs. Main Results Table 1: The main forecasting results, averaged over 3 seeds. Left: lightweight models; right: conventional models. The best and second-best results within each model group are shown in bold red and underlined blue, respectively. H Lightweight Models Conventional Models Ours SparseTSF MixLinear PhaseFormer FITS CycleNet DLinear FreqCycle FilterNet PatchTST TQNet iTrans. FreTS MSE MAE MSE MAE MSE MAE MSE MAE MSE MAE MSE MAE MSE MAE MSE MAE MSE MAE MSE MAE MSE MAE MSE MAE MSE MAE ETTh1 96 0.353 0.393 0.374 0.405 0.367 0.401 0.374 0.395 0.379 0.402 0.379 0.403 0.395 0.419 0.368 0.401 0.407 0.435 0.379 0.410 0.382 0.410 0.395 0.426 0.452 0.462 192 0.389 0.415 0.404 0.412 0.396 0.419 0.407 0.414 0.413 0.423 0.418 0.427 0.439 0.450 0.404 0.427 0.443 0.453 0.417 0.435 0.426 0.437 0.429 0.449 0.500 0.495 336 0.418 0.434 0.432 0.427 0.412 0.436 0.439 0.435 0.433 0.439 0.449 0.447 0.467 0.470 0.428 0.441 0.454 0.466 0.444 0.455 0.457 0.457 0.493 0.493 0.527 0.510 720 0.449 0.469 0.427 0.450 0.423 0.453 0.425 0.444 0.431 0.457 0.475 0.482 0.512 0.524 0.484 0.493 0.500 0.502 0.468 0.484 0.503 0.507 0.674 0.598 0.606 0.562 ETTh2 96 0.273 0.335 0.286 0.344 0.287 0.345 0.281 0.343 0.273 0.338 0.272 0.337 0.323 0.382 0.282 0.345 0.327 0.380 0.278 0.341 0.284 0.349 0.306 0.362 0.366 0.408 192 0.340 0.380 0.343 0.379 0.340 0.379 0.343 0.381 0.332 0.376 0.333 0.380 0.408 0.432 0.345 0.391 0.381 0.410 0.345 0.383 0.359 0.396 0.403 0.419 0.442 0.454 336 0.375 0.409 0.358 0.397 0.359 0.397 0.375 0.408 0.355 0.397 0.365 0.410 0.495 0.488 0.387 0.427 0.400 0.432 0.370 0.407 0.395 0.422 0.460 0.458 0.608 0.538 720 0.399 0.437 0.381 0.423 0.382 0.424 0.421 0.453 0.378 0.424 0.416 0.450 0.833 0.646 0.441 0.468 0.422 0.451 0.403 0.440 0.420 0.452 0.433 0.463 1.202 0.765 ETTm1 96 0.287 0.340 0.343 0.374 0.333 0.374 0.298 0.346 0.319 0.360 0.319 0.362 0.315 0.359 0.296 0.351 0.323 0.372 0.297 0.352 0.298 0.352 0.319 0.371 0.356 0.392 192 0.324 0.364 0.355 0.379 0.353 0.383 0.331 0.365 0.345 0.374 0.339 0.373 0.347 0.381 0.334 0.372 0.356 0.389 0.337 0.376 0.339 0.377 0.349 0.389 0.388 0.410 336 0.353 0.383 0.381 0.393 0.391 0.408 0.360 0.382 0.372 0.390 0.363 0.386 0.373 0.394 0.364 0.389 0.383 0.400 0.366 0.396 0.373 0.397 0.382 0.410 0.416 0.427 720 0.405 0.411 0.425 0.416 0.432 0.429 0.414 0.412 0.420 0.416 0.412 0.413 0.424 0.425 0.416 0.415 0.432 0.424 0.422 0.422 0.439 0.431 0.448 0.448 0.468 0.460 ETTm2 96 0.161 0.252 0.176 0.264 0.167 0.257 0.174 0.264 0.167 0.258 0.160 0.250 0.178 0.273 0.168 0.260 0.184 0.276 0.165 0.256 0.174 0.262 0.182 0.276 0.180 0.268 192 0.214 0.290 0.226 0.298 0.221 0.295 0.227 0.299 0.221 0.295 0.215 0.289 0.282 0.359 0.222 0.297 0.238 0.309 0.217 0.294 0.225 0.300 0.249 0.320 0.248 0.317 336 0.269 0.328 0.277 0.331 0.271 0.328 0.274 0.329 0.272 0.329 0.267 0.326 0.296 0.359 0.275 0.333 0.286 0.341 0.270 0.329 0.285 0.343 0.303 0.355 0.308 0.354 720 0.353 0.383 0.356 0.381 0.353 0.381 0.353 0.379 0.351 0.380 0.354 0.384 0.414 0.433 0.350 0.384 0.370 0.399 0.352 0.381 0.373 0.395 0.376 0.402 0.380 0.409 Weather 96 0.145 0.198 0.180 0.236 0.175 0.231 0.148 0.193 0.174 0.229 0.165 0.221 0.172 0.234 0.148 0.203 0.157 0.213 0.148 0.199 0.157 0.210 0.178 0.229 0.159 0.223 192 0.193 0.243 0.221 0.269 0.218 0.268 0.194 0.238 0.215 0.263 0.211 0.259 0.215 0.272 0.193 0.245 0.207 0.257 0.192 0.242 0.208 0.257 0.226 0.268 0.201 0.263 336 0.241 0.281 0.263 0.300 0.264 0.303 0.245 0.280 0.260 0.296 0.255 0.292 0.258 0.305 0.247 0.287 0.262 0.297 0.246 0.285 0.258 0.295 0.291 0.313 0.251 0.302 720 0.311 0.332 0.324 0.343 0.327 0.347 0.316 0.331 0.321 0.340 0.321 0.339 0.319 0.357 0.321 0.338 0.314 0.334 0.311 0.332 0.335 0.348 0.358 0.362 0.315 0.347 ECL 96 0.127 0.222 0.139 0.233 0.150 0.245 0.129 0.221 0.142 0.243 0.128 0.223 0.133 0.230 0.129 0.225 0.142 0.240 0.130 0.224 0.136 0.235 0.134 0.229 0.131 0.229 192 0.143 0.237 0.151 0.245 0.163 0.257 0.146 0.236 0.156 0.256 0.143 0.237 0.147 0.244 0.148 0.242 0.170 0.265 0.147 0.241 0.152 0.247 0.153 0.247 0.147 0.243 336 0.159 0.254 0.166 0.260 0.178 0.273 0.167 0.258 0.172 0.271 0.159 0.254 0.162 0.261 0.162 0.259 0.187 0.283 0.162 0.257 0.171 0.267 0.167 0.263 0.164 0.263 720 0.196 0.287 0.205 0.294 0.217 0.305 0.200 0.286 0.210 0.303 0.197 0.287 0.197 0.294 0.198 0.292 0.221 0.311 0.197 0.288 0.198 0.294 0.188 0.284 0.200 0.299 Traffic 96 0.379 0.263 0.389 0.266 0.406 0.281 0.363 0.233 0.393 0.279 0.381 0.265 0.422 0.327 0.393 0.272 0.355 0.260 0.369 0.255 0.397 0.296 0.345 0.250 0.378 0.269 192 0.395 0.270 0.399 0.270 0.419 0.286 0.378 0.242 0.404 0.283 0.394 0.272 0.434 0.332 0.410 0.275 0.371 0.268 0.383 0.261 0.395 0.274 0.360 0.259 0.398 0.279 336 0.409 0.277 0.411 0.276 0.429 0.292 0.397 0.251 0.417 0.288 0.406 0.279 0.449 0.338 0.421 0.281 0.386 0.275 0.397 0.268 0.418 0.295 0.369 0.266 0.417 0.287 720 0.440 0.296 0.448 0.296 0.469 0.313 0.434 0.271 0.454 0.308 0.442 0.301 0.491 0.359 0.458 0.304 0.437 0.298 0.434 0.289 0.440 0.297 0.394 0.280 0.470 0.312 PEMS03 12 0.062 0.167 0.185 0.296 0.197 0.306 0.095 0.205 0.116 0.226 0.079 0.191 0.103 0.218 0.069 0.175 0.068 0.173 0.080 0.190 0.061 0.161 0.067 0.171 0.080 0.190 24 0.081 0.188 0.324 0.395 0.429 0.474 0.156 0.265 0.234 0.323 0.121 0.238 0.180 0.293 0.100 0.212 0.096 0.204 0.131 0.244 0.078 0.183 0.093 0.202 0.123 0.236 48 0.115 0.219 0.650 0.586 0.776 0.661 0.292 0.368 0.542 0.522 0.156 0.257 0.317 0.407 0.167 0.276 0.149 0.258 0.232 0.330 0.107 0.215 0.151 0.260 0.196 0.305 96 0.156 0.249 1.186 0.846 1.352 0.927 0.513 0.502 1.063 0.791 0.199 0.292 0.451 0.507 0.256 0.351 0.228 0.326 0.385 0.444 0.149 0.254 0.319 0.392 0.262 0.362 PEMS04 12 0.075 0.180 0.197 0.312 0.209 0.322 0.113 0.226 0.130 0.241 0.089 0.200 0.114 0.228 0.081 0.188 0.078 0.183 0.102 0.215 0.066 0.165 0.085 0.188 0.096 0.207 24 0.090 0.200 0.336 0.412 0.440 0.492 0.195 0.301 0.249 0.341 0.127 0.244 0.188 0.300 0.113 0.225 0.097 0.209 0.159 0.272 0.077 0.180 0.116 0.223 0.142 0.257 48 0.115 0.227 0.675 0.608 0.808 0.684 0.371 0.423 0.570 0.542 0.187 0.305 0.319 0.405 0.179 0.288 0.135 0.253 0.278 0.370 0.096 0.204 0.177 0.280 0.224 0.331 96 0.143 0.252 1.269 0.887 1.429 0.959 0.659 0.585 1.166 0.838 0.190 0.294 0.424 0.481 0.270 0.367 0.201 0.316 0.446 0.489 0.124 0.232 0.271 0.357 0.288 0.382 PEMS07 12 0.058 0.155 0.179 0.300 0.190 0.308 0.091 0.203 0.109 0.221 0.073 0.180 0.100 0.214 0.064 0.165 0.061 0.160 0.075 0.183 0.056 0.153 0.064 0.161 0.077 0.184 24 0.077 0.177 0.328 0.407 0.437 0.489 0.160 0.270 0.229 0.326 0.102 0.207 0.192 0.304 0.094 0.202 0.085 0.188 0.120 0.231 0.064 0.160 0.090 0.192 0.126 0.238 48 0.119 0.208 0.675 0.606 0.806 0.682 0.329 0.393 0.546 0.529 0.154 0.253 0.381 0.440 0.152 0.263 0.125 0.232 0.212 0.310 0.084 0.184 0.137 0.239 0.224 0.320 96 0.182 0.238 1.252 0.881 1.388 0.943 0.607 0.552 1.094 0.797 0.204 0.291 0.580 0.542 0.248 0.343 0.180 0.287 0.344 0.403 0.106 0.206 0.279 0.362 0.323 0.392 PEMS08 12 0.080 0.183 0.188 0.304 0.201 0.313 0.105 0.214 0.127 0.241 0.091 0.201 0.113 0.226 0.081 0.187 0.078 0.179 0.099 0.207 0.070 0.169 0.078 0.178 0.095 0.202 24 0.117 0.222 0.323 0.404 0.426 0.485 0.180 0.282 0.252 0.348 0.143 0.253 0.197 0.301 0.119 0.227 0.108 0.213 0.167 0.272 0.094 0.195 0.111 0.211 0.150 0.255 48 0.190 0.286 0.678 0.609 0.817 0.689 0.353 0.404 0.604 0.568 0.264 0.340 0.389 0.431 0.202 0.296 0.170 0.264 0.314 0.384 0.152 0.247 0.179 0.266 0.247 0.334 96 0.336 0.379 1.346 0.900 1.509 0.974 0.703 0.569 1.322 0.893 0.281 0.335 0.667 0.542 0.371 0.394 0.285 0.329 0.561 0.521 0.260 0.310 0.326 0.361 0.356 0.399 1st1^stCnt∗ 30 21 0 3 2 0 4 14 3 2 4 4 0 0 1 0 – – – – – – – – – – 1st1^stCnt† 13 7 0 3 2 0 0 13 3 2 3 3 0 0 1 0 0 0 1 0 16 15 5 1 0 0 Best and Second-best within each model group. For the lightweight group, the original source annotations retain tie-breaking based on unrounded values. 1st1^stCnt∗ within the lightweight group, counted per metric over all settings, ties are broken on unrounded values. 1st1^stCnt† among all compared models. Table 1 reports MSE and MAE across all 11 benchmarks. Within the lightweight group, AsyTO attains the lowest MSE in 3030 of the 4444 cells and the lowest MAE in 2121, and its mean rank among the eight lightweight models is 1.961.96 on the 7 standard benchmarks and 1.061.06 on PEMS. The advantage is concentrated where the periodic structure the model assumes is actually present, and it holds across horizons rather than only short ones. AsyTO is best across all four horizons on ETTm1, Weather, and Electricity, and in 1515 of the 1616 PEMS cells (where the regime with the shortest look-back relative to the period is least identifiable from the data). On ETTh2 the lightweight methods are separated by less than the seed-to-seed spread, and the leader changes with the horizon, so no method can be said to win there. Traffic is the one benchmark where AsyTO loses at every horizon, trailing PhaseFormer by 4.5%4.5\%, 4.5%4.5\%, 2.9%2.9\% and 1.3%1.3\% as the horizon grows. This is the expected cost of the target-only design: Traffic has 862862 sensors whose predictive information is largely shared, and a model that forecasts each variable from its own history alone cannot recover it. The gap narrows as the horizon lengthens, consistent with cross-sensor information being most useful at short range. We treat this as a scope boundary rather than a tuning failure, and return to it in the discussion. Against the conventional group AsyTO remains competitive without matching their budget: it attains the lowest MSE of all thirteen models in 1313 of the 4444 cells while using two to three orders of magnitude fewer parameters, which is the trade-off Figure 2 makes explicit. Efficiency Analysis Figure 2 places AsyTO and baselines on the accuracy-compute plane at L=720→H=96L=720→ H=96, using multiply accumulate operations (MACs) per forward sample as a hardware-agnostic measure of compute and trainable parameters as bubble area. AsyTO sits on the Pareto frontier, and it does so at a compute scale set by the cheap end of the field rather than the expensive one: on Electricity, it uses 47.5×47.5×, 62.2×62.2×, and 590.4×590.4× fewer MACs than TQNet, iTransformer, and PatchTST, and on Weather the gap to PatchTST reaches 2527×2527×. The parameter gap is wider: 92.5×92.5× against TQNet and 155×155× against iTransformer on Electricity, indicating the conventional models carry per-variable or per-token machinery that AsyTO replaces with shared factors. The frontier is crowded at its cheap end, and AsyTO does not dominate on every axis. PhaseFormer matches our compute on Electricity (1.0×1.0× MACs) with a tenth of the parameters, and CycleNet is marginally cheaper (0.9×0.9× MACs); both trail in error by 1.5%1.5\% and 1.0%1.0\%. The claim supported by the figure is therefore specific: among models at this compute scale AsyTO is the most accurate, not the smallest. What does scale favourably is the cost of an additional variable. Instantiating the shipped configuration at C and C+1C+1 variables shows the parameter count growing by exactly RpR_p per variable: 22 on Weather, 44 on ETTm1, 88 on Traffic and 1616 on Electricity. This is because the temporal factors are shared and the variable response is expressed in the prototype’s coordinates rather than stored per variable. A dense per-variable temporal map would instead add HLHL coefficients per variable, or 69,12069,120 at →96720\!→\!96. Figure 2: Visualization of accuracy vs. compute Pareto on ETTm1, Weather, Electricity under the setting of L=720→H=96L=720→ H=96. The bubble size indicates the scale of the trainable parameters, where the dashed step in grey indicates the Pareto frontier. Validating the Structural Priors Table 2: Module ablation for structure prior validation. Dataset w/o Q w/o ATO w/o Rct. AsyTO ETTh1 0.406 0.468 0.407 0.402 ETTh2 0.348 0.397 0.354 0.346 ETTm1 0.347 0.352 0.344 0.342 ETTm2 0.251 0.253 0.249 0.249 Weather 0.221 0.218 0.223 0.223 ECL 0.159 0.161 0.157 0.156 Traffic 0.406 0.428 0.410 0.406 PEMS03 0.130 0.134 0.105 0.104 PEMS04 0.142 0.133 0.107 0.106 PEMS07 0.131 0.128 0.109 0.109 PEMS08 0.219 0.232 0.182 0.181 To validate the sources of AsyTO’s performance gain, we design two complementary experiments: module ablations in Table 2 and controlled operator constraints in Figure 3. Table 2 replaces each stage with a simpler alternative while retaining the training protocol. w/o Q applies the operator to the raw series instead of the periodicity-exclusive residual, w/o ATO replaces the factorized operator by an independent dense H×LH× L map per variable, w/o G forces every variable to share one response, and w/o recent removes the dense correction over the latest observations. Entries are raw test MSE, 3-seed means over 4 horizons. The three interventions reveal different roles. Periodic removal changes MSE only marginally on the seven long-context datasets, but improves it by 2020–34%34\% on PEMS, where the look-back is shorter than the dominant cycle. The periodic prototype is therefore a regime-specific necessity rather than a universal source of improvement. In contrast, the dense replacement is worse on ten of eleven datasets despite using 2626–19,509×19,509× more parameters, showing that unstructured per-variable capacity cannot replace shared residual structure. Recent correction has improved the displayed averages in most cases, but its effect is at most 2.3%2.3\%, identifying it as a secondary refinement. Figure 3: Directionality control. Each cell is the 3-seed averaged relative MSE reduction of Asym ( ≠ ≠ ) in percentage, against 3 constrained alternatives: Sym sets = = with a parameter aligned version Sym-PA, and shift-invariant Toeplitz. Blue means Asym is better. The dense ablation replaces the entire residual operator and therefore cannot isolate history–future asymmetry. Figure 3 provides this control using the same operator family at L=HL=H. Asym learns separate history and future factors. Sym ties them by setting = = , testing the common-basis constraint but using fewer parameters. Sym-PA reallocates the saved capacity to match the parameter count of Asym, separating directionality from model size. Toeplitz instead makes the map depend only on relative lag, testing whether a shift-invariant rule is sufficient. Every displayed comparison favors separate factors, by 1.11.1–25.5%25.5\%. After averaging the tested window lengths, Asym wins 2929 of 3333 controlled setting comparisons across all datasets (see Appendix for more details), the 4 reversals occur only on ETTh1/2 and are within 1.4%1.4\%. The persistence of the gap against Sym-PA rules out parameter count as its explanation, while the Toeplitz comparison shows that relative lag alone cannot replace distinct history-reading and future-writing structure. Figure 4: The learned factorization on ETTm1 and Weather at →720720\!→\!720. (a, b) The six highest-energy future-writing and history-reading modes, (c) the variable mode responses, (d) the operator reconstructed for variables, with ρasy=∥−⊤∥F/∥F _asy= -W _F/ _F. Figure 4 shows the asymmetry mechanism, where the history modes are localized in time while future modes spread across the horizon, indicating the two temporal bases are specialized rather than converge. Panel (d) puts a number on the gap: for each dataset’s target variable ρasy _asy is 1.371.37 on ETTm1 and 1.391.39 on Weather, placing 46.9%46.9\% and 48.3%48.3\% of the operator’s energy in its antisymmetric part. Panel (c) illustrates the responses grouping thermo-dynamically related variables, showing that shared modes are adapted across the panel rather than duplicated. Cost of Factorization The preceding experiments establish which structures are needed, this section answers whether their compressed parameterizations discard useful capacity. Table 3 changes one storage choice at a time at L=H=720L=H=720. Shared variants remove variable specificity, Independent variants store a separate prototype or response for every variable, and Dense variants replace one cycle-separable temporal factor with its unrestricted counterpart. Variable-side Storage. The benefit of amortization increases with panel width. Relative to AsyTO, storing an independent prototype QcQ_c increases the total parameter count by 1.06×1.06×, 1.45×1.45×, 2.41×2.41×, and 7.32×7.32× on ETTm1, Weather, ECL, and Traffic, respectively, without improving the reported MSE. Independent responses GcG_c follow the same scaling trend (1.011.01–2.19×2.19×) and match or underperform AsyTO except on Weather, where they reduce MSE from .311.311 to .307.307 for a 4.5%4.5\% increase in parameters. Full sharing is not a uniform substitute either: it helps on Weather but degrades ETTm1 and ECL. The coordinate-generated form therefore provides the useful middle ground between a channel-homogeneous model and costly per-variable storage, with Weather marking a possible capacity boundary when independent responses are inexpensive. Temporal Storage. Replacing either temporal factor with a dense one raises the total model size by 1.631.63–2.90×2.90×, while changing MSE by at most .003.003 and with no consistent direction. Dense factors slightly improve ECL and Traffic, but match or underperform AsyTO on ETTm1 and Weather. The cycle-separable form therefore removes 3939–65%65\% of the parameters required by its dense counterpart while retaining nearly all of its accuracy. The two factorizations address complementary scaling terms: variable-side compression prevents per-variable storage from dominating wide panels, whereas temporal compression controls the shared history–future cost and is most visible on narrow panels. Both are therefore required for efficiency across panel widths. Table 3: Cost of factorization at L=720→H=720L=720→ H=720, the one horizon at which every arm exists. Each row stores one component differently; Par. is its trainable-parameter count. All eleven datasets and horizons are in the appendix. Parameterization ETTm1 Weather Electricity Traffic MSE Par. MSE Par. MSE Par. MSE Par. Low-Rank Periodic Prototype s=⊤Q_s=UV Shared Q 0.411 10,498 0.309 6,846 0.200 38,346 0.440 23,074 Indep. cQ_c 0.410 11,074 0.311 9,726 0.196 92,106 0.446 167,722 Cycle-Shared Variable Response G Shared sG_s 0.406 10,281 0.310 6,691 0.197 37,475 0.440 23,512 Indep. cG_c 0.405 10,498 0.307 7,006 0.196 57,698 0.441 50,234 History-CeST Factorization Dense dense _dense 0.407 27,042 0.312 14,302 0.195 67,042 0.439 37,338 Future-CeST Factorization Dense dense _dense 0.407 30,114 0.311 15,838 0.195 73,186 0.437 40,410 AsyTO 0.405 10,402 0.311 6,702 0.196 38,178 0.440 22,906 Table 4: Transferability analysis on frozen PhaseFormer and MixLinear. T0T0 is the backbone MSE, T1ΔT1_ and T2ΔT2_ are relative MSE changes after adding Q and Q+CeSTQ+CeST. Negative is better, performance gains exceeding 5%5\% are bold. Dataset PhaseFormer MixLinear T0 T1Δ T2Δ T0 T1Δ T2Δ ETTh1 0.419 -0.8% -0.2% 0.400 +3.0% +2.7% ETTh2 0.354 -0.0% -0.2% 0.342 +0.2% +0.1% ETTm1 0.350 -0.2% -0.9% 0.377 -0.4% -8.1% ETTm2 0.257 -0.7% -2.6% 0.253 -1.6% -1.8% Weather 0.225 +0.1% -1.7% 0.249 -0.4% -13.4% ECL 0.160 -1.8% -3.0% 0.174 -4.8% -9.4% Traffic 0.393 -0.6% -0.7% 0.417 -1.7% -2.5% PEMS03 0.176 -12.1% -24.7% 0.587 -43.1% -70.7% PEMS04 0.192 -16.9% -30.0% 0.619 -30.7% -66.2% PEMS07 0.208 -14.9% -31.8% 0.609 -45.1% -72.6% PEMS08 0.363 -16.7% -35.6% 0.634 -30.9% -64.3% Transferability Analysis Ablations establish that the proposed components are useful within AsyTO, but not whether they remain effective outside the architecture for which they were designed. We therefore attach them to two independently trained lightweight forecasters, PhaseFormer (11) and MixLinear (8), while keeping all backbone weights frozen. T0T0 is the frozen backbone, T1T1 trains only the periodic prototype Q, and T2T2 trains Q together with CeST. T2T2 reduces MSE in 3939 of 4444 PhaseFormer cells and 4040 of 4444 MixLinear cells. Across the seven standard benchmarks, the mean reductions are 1.3%1.3\% and 4.6%4.6\%, respectively; on PEMS, where the look-back is shorter than one cycle (L=96<P=288L=96<P=288), they increase to 30.5%30.5\% and 68.4%68.4\%. This contrast is consistent with the periodic path being most useful when the backbone cannot observe a complete cycle, although the ratio L/PL/P is not the only factor that differs between these datasets. CeST also contributes beyond periodic removal: T2T2 outperforms T1T1 on 2121 of the 2222 displayed backbone–dataset pairs. On PEMS, it raises the mean reduction from 15.2%15.2\% to 30.5%30.5\% for PhaseFormer and from 37.5%37.5\% to 68.4%68.4\% for MixLinear. The only material regression relative to T0T0 is MixLinear on ETTh1 (+2.7%+2.7\%), where the long input already spans many cycles. Because these gains are obtained without updating any backbone weight, they demonstrate that the modules provide a transferable correction rather than relying on co-adaptation with the AsyTO architecture. Discussion and Conclusion This work presented AsyTO, a compact forecasting operator that assigns distinct parameterizations to history-side evidence extraction and future-side trajectory generation. After estimating and removing a low-rank periodic component, AsyTO maps the residual through separate history and future factors. Cycle-separable factorization controls the temporal cost, while prototype-derived coordinates generate variable-specific responses. Matched-budget controls substantiate the asymmetric design, and AsyTO achieves the lowest error among lightweight forecasters in 30 of 44 settings. 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