COLLAR: combating low-rank temporal latent representation for high-dimensional multivariate time series prediction using dynamic Koopman regularization

Recent advancements in high-dimensional multivariate time series forecasting have yielded promising outcomes by effectively capturing temporal dynamics within a low-dimensional latent subspace. Recurrent Neural Networks (RNNs), particularly Gated Recurrent Units (GRUs), excel at modeling complex sequential dependencies within these subspaces. However, their simplistic decoder structures often fail to fully represent the diversity inherent in observed data, while overly complex encoders risk overfitting. To address these challenges, we introduce COLLAR, a novel architecture that integrates dynamic linearized regularization to balance nonlinear temporal structures with linear evolution in multivariate time series. COLLAR employs an Auto-Encoder (AE) framework to factorize high-dimensional time series into a low-dimensional latent subspace, where GRUs construct nonlinear predictions for latent sequences. Simultaneously, we incorporate a data-driven Koopman operator into a modified AE architecture to generate linear dynamics regularization terms, mitigating uncertainty in nonlinear predictions. The eigenfunctions of the Koopman operator provide intrinsic coordinates that globally linearize the dynamics, enabling state predictions based solely on current states. Evaluated on five publicly available datasets, COLLAR consistently outperforms state-of-the-art baselines, demonstrating superior forecasting accuracy and robustness. The code and models are available at https://github.com/Nsyyyy/COLLAR .