Research map / Time series / Long-term forecasting

Multivariate Time Series Forecasting needs Cross Variable Loss

NeurIPS 2026 · Time series › Long-term forecasting · approach: Graph-based multivariate forecasting · 0 citations (OpenAlex)

Read the paperCheck your idea against 108,967 papers

AI-extracted summary. Written by AI from the paper's full text. Numbers and claims can be wrong or missing context, so check the paper before you cite or build on it.

Proposes CvLoss, a plug-in graph total-variation regularizer on forecast residuals across variables and patches, added to point-wise MSE to improve multivariate time series forecasting backbones.

Problem

Direct forecasting models trained with point-wise MSE ignore cross-variable and lagged dependencies among future values, so residual structure is unconstrained and the objective mismatches the joint spatiotemporal error distribution.

Approach

Adds an l1 graph total-variation penalty on edge-wise differences of forecast residuals over a cross-variable graph of patch nodes (fully connected by default, with random edge sampling), fused with MSE via a fixed weight alpha; no change to architecture or inference.

Key results

• CvLoss on TimeFilter improves all 24 dataset-level MSE/MAE entries over the same backbone trained with standard MSE (mean reduction 3.20% MSE, 3.12% MAE).
• Across 114 fixed-backbone comparisons over seven backbones, CvLoss wins 111, ties 2, loses 1, with mean relative error reduction of 3.39% MSE and 3.61% MAE.
• On ECL, CvLoss with iTransformer reduces MSE by 8.6% and MAE by 7.9%; on Traffic with iTransformer, MSE reduction reported as 17.7%.

Surprising findings

• The l1 penalty used in practice is not the Gaussian (squared) form that the theory derives; the theory only motivates a relation-aware constraint.
• Setting alpha=1 (pure structural penalty) is not always optimal; best ECL results occur near alpha=0.5.

Limitations

• Uses a predefined uniform fully connected cross-variable graph that may include uninformative interactions, especially for high-dimensional data.
• Theory does not literally derive the l1 objective; the norm choice is empirical.
• Evaluated only on deterministic forecasting on regular benchmark datasets.

Experimental setup

Datasets: ETTh1 ETTh2 ETTm1 ETTm2 Weather ECL Traffic Solar-Energy PEMS03 PEMS04 PEMS07 PEMS08

Benchmarks: ETTh1 ETTh2 ETTm1 ETTm2 Weather ECL Traffic Solar-Energy PEMS03 PEMS04 PEMS07 PEMS08

Metrics: MSE MAE

Baselines: TimeFilter TQNet iTransformer Leddam SOFTS PatchTST Crossformer TimesNet DLinear FEDformer QDF Time-o1

Builds on: TimeFilter TQNet iTransformer FreDF

Closest papers by meaning

• Revitalizing Multivariate Time Series Forecasting: Learnable Decomposition with Inter-Series Dependencies and Intra-Series Variations Modeling (ICML 2024 · similarity 0.88)
• Patch-wise Structural Loss for Time Series Forecasting (ICML 2025 · similarity 0.87)
• Structured Matrix Basis for Multivariate Time Series Forecasting with Interpretable Dynamics (NeurIPS 2024 · similarity 0.87)
• ARM: Refining Multivariate Forecasting with Adaptive Temporal-Contextual Learning (ICLR 2024 · similarity 0.86)
• Unlocking the Power of Patch: Patch-Based MLP for Long-Term Time Series Forecasting (AAAI 2025 · similarity 0.86)
• Loss Shaping Constraints for Long-Term Time Series Forecasting (ICML 2024 · similarity 0.86)