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

arXiv: https://arxiv.org/abs/2606.09917 · HTML: https://mlsanity.com/paper/2606.09917/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 22 references that are in the map.
- First posted on arXiv 6 Jun 2026.

## Builds on (most cited first, 9 of 9)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Autoformer: Decomposition Transformers with Auto-Correlation for Long-Term Series Forecasting](https://mlsanity.com/paper/2106.13008/) (2021)
- [TimesNet: Temporal 2D-Variation Modeling for General Time Series Analysis](https://mlsanity.com/paper/2210.02186/) (2022)
- [iTransformer: Inverted Transformers Are Effective for Time Series Forecasting](https://mlsanity.com/paper/2310.06625/) (2023)
- [Long-term Forecasting with TiDE: Time-series Dense Encoder](https://mlsanity.com/paper/2304.08424/) (2023)
- [A Riemannian Network for SPD Matrix Learning](https://mlsanity.com/paper/1608.04233/) (2016)
- [ManifoldFormer: Geometric Deep Learning for Neural Dynamics on Riemannian Manifolds](https://mlsanity.com/paper/2511.16828/) (2025)
- [Time Series Forecasting via Direct Per-Step Probability Distribution Modeling](https://mlsanity.com/paper/2511.23260/) (2025)
- [Ridge Regression on Riemannian Manifolds for Time-Series Prediction](https://mlsanity.com/paper/2411.18339/) (2024)

## Built on by (most cited first, 1 of 1)

- [Structured Latent Space Modeling over Multi-Scale Temporal Patches for Multivariate Time Series Forecasting](https://mlsanity.com/paper/2607.19404/) (2026)

## Abstract (arXiv)

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.