aiqbench analysis
SPDM introduces manifold constraints into state-space models to preserve the evolving geometric structure of multivariate time series, addressing a key limitation in existing architectures that discard such features. By projecting covariance matrices onto a Euclidean tangent space and using geometric signals to modulate internal parameters, the method maintains linear-time complexity while achieving state-of-the-art forecasting accuracy on real-world benchmarks. This work demonstrates that embedding rich structural constraints directly into the dynamics of selective scanning is the primary driver of performance gains over unconstrained baselines.
SPDM: Geometry-Modulated State Space Modeling with Manifold Constraints for Time Series Forecasting
arXiv:2606.09917v1 Announce Type: new 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.
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.
Machine Learning (cs.LG)
arXiv:2606.09917 [cs.LG]
(or arXiv:2606.09917v1 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2606.09917 Focus to learn more arXiv-issued DOI via DataCite
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Full text under the publisher's open licence · 287 words