Multivariate Time Series Forecasting with Hybrid Euclidean-SPD Manifold Graph Neural Networks

2025-12-15

AI Summary

Existing multivariate time series forecasting methods typically rely solely on Euclidean or Riemannian space modeling, failing to capture heterogeneous geometric structures and complex spatiotemporal dependencies in real-world data. To address this limitation, we propose the first dual-geometric graph neural network framework that jointly leverages Euclidean space and the Symmetric Positive Definite (SPD) manifold. Our key contributions are: (1) a hybrid Euclidean–Riemannian geometric representation; (2) Submanifold Cross-Segment embedding (SCS), which explicitly models long-range temporal dependencies via SPD submanifold alignment; and (3) an Adaptive Distance Bank (ADB) coupled with Fusion Graph Convolution (FGCN), enabling learnable, co-optimized message passing across both geometries. Extensive experiments on three benchmark datasets demonstrate state-of-the-art performance, achieving up to 13.8% reduction in prediction error compared to the best existing methods.

Technology Category

• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Machine Learning: Learning with Manifolds
• Data Mining & Knowledge Management: Mining of Spatial, Temporal or Spatio-Temporal Data

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Search and Retrieval-Augmented AI: Multilingual and cross-lingual Web search

Abstract

Multivariate Time Series (MTS) forecasting plays a vital role in various real-world applications, such as traffic management and predictive maintenance. Existing approaches typically model MTS data in either Euclidean or Riemannian space, limiting their ability to capture the diverse geometric structures and complex spatio-temporal dependencies inherent in real-world data. To overcome this limitation, we propose the Hybrid Symmetric Positive-Definite Manifold Graph Neural Network (HSMGNN), a novel graph neural network-based model that captures data geometry within a hybrid Euclidean-Riemannian framework. To the best of our knowledge, this is the first work to leverage hybrid geometric representations for MTS forecasting, enabling expressive and comprehensive modeling of geometric properties. Specifically, we introduce a Submanifold-Cross-Segment (SCS) embedding to project input MTS into both Euclidean and Riemannian spaces, thereby capturing spatio-temporal variations across distinct geometric domains. To alleviate the high computational cost of Riemannian distance, we further design an Adaptive-Distance-Bank (ADB) layer with a trainable memory mechanism. Finally, a Fusion Graph Convolutional Network (FGCN) is devised to integrate features from the dual spaces via a learnable fusion operator for accurate prediction. Experiments on three benchmark datasets demonstrate that HSMGNN achieves up to a 13.8 percent improvement over state-of-the-art baselines in forecasting accuracy.

Problem

• Capturing diverse geometric structures in multivariate time series
• Modeling complex spatio-temporal dependencies across Euclidean and Riemannian spaces
• Reducing computational cost of Riemannian distance in hybrid frameworks

Innovation

• Hybrid Euclidean-Riemannian framework for MTS forecasting
• Submanifold-Cross-Segment embedding to capture spatio-temporal variations
• Adaptive-Distance-Bank layer to reduce Riemannian computational cost