SPDM: Geometry-Modulated State Space Modeling with Manifold Constraints for Time Series Forecasting

2026-06-06

AI Summary

This work addresses the limitation of existing state space models in multivariate time series forecasting, which often overlook the dynamically evolving geometric relationships among variables. To remedy this, the paper introduces, for the first time, a symmetric positive definite (SPD) manifold constraint into state space modeling, leveraging Riemannian geometric features as structural regularizers to enable geometry-aware temporal modeling. The proposed method integrates projection from the SPD manifold to its tangent space, a geometry-guided gating mechanism based on manifold-valued signals, and Mamba’s linear-complexity parallel scan architecture. Extensive experiments on eleven real-world benchmark datasets demonstrate state-of-the-art performance, underscoring the critical role of geometric constraints in enhancing forecasting accuracy.

Abstract

Multivariate time series forecasting requires capturing the continuously evolving correlation structure among interacting variables. Existing state-space models process time series by scanning tokenized temporal or spatial sequences, discarding the evolutionary geometric structure. We address this limitation by introducing manifold constraints into state-space modeling: treating the cross-variable correlation structure as a continuous trajectory on the symmetric positive definite manifold, whose Riemannian geometric features, tangent space linearity, and Frechet mean centrality act as a principled geometric regularizer that guides and stabilizes the selective scanning dynamics of SSMs. We propose SPDM, a geometry-aware SSM architecture that realizes this principle through two cooperating mechanisms: a manifold trajectory path that projects dynamically evolving covariance matrices from the SPD manifold to a Euclidean tangent space, and a geometric gating scheme that directly modulates SSM's internal selective parameters based on geometric signals derived from the manifold trajectory. The parameterization preserves the linear-time complexity of the Mamba parallel scan while embedding rich structural constraints, making the architecture preserve prediction accuracy and computational efficiency simultaneously. Extensive experiments on eleven real-world benchmark datasets establish state-of-the-art forecasting performance, and further studies confirm that geometrically constrained state-space dynamics are the dominant architectural factor behind its performance gains.

Problem

• multivariate time series forecasting
• state-space models
• geometric structure
• manifold constraints
• correlation dynamics

Innovation

• manifold constraints
• state space modeling
• symmetric positive definite manifold
• geometric regularization
• time series forecasting