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

• URL: http://arxiv.org/abs/2606.09917v1
• Date: Sat, 06 Jun 2026 17:04:57 GMT
• Title: SPDM: Geometry-Modulated State Space Modeling with Manifold Constraints for Time Series Forecasting
• Abstract summary: Multivariable time series forecasting requires capturing the continuously evolving correlation structure among interacting variables.<n>We address this limitation by introducing manifold constraints into state-space modeling.<n>We propose SPDM, a geometry-aware SSM architecture that realizes this principle through two cooperating mechanisms.
• Score: 15.612678669559516
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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