Graph diffusion in riemannian and euclidean spaces for multidimensional time-series decoding

Abstract

The problem of decoding multidimensional time series with high variance and strong covariance between components is considered. Previously proposed prediction methods do not account for the spatial structure of time series. It is proposed to model this structure using graph-based methods, such as graph neural diffusion — a generalization of graph neural networks obtained by solving an anisotropic diffusion equation discretized on a graph. In this work, a model for multidimensional time-series prediction based on the combination of graph neural diffusion and Riemannian geometry of covariance representations is proposed. In the Euclidean domain, diffusion on a graph of signal components is used to analyze correlation dependencies between the components of the series. Simultaneously, on the Riemannian manifold of positive semidefinite matrices, a graph of covariance matrices corresponding to different temporal windows is constructed. Representations on a Riemannian manifold are invariant to linear transformations of time series components and are robust to noise; these properties enable the use of the Riemannian manifold to derive informative features for time series forecasting. Diffusion on this graph is performed in the tangent space of the manifold using logarithmic and exponential mappings. Predictions obtained in both domains are summed in the original space, and model training is performed using a consistency-based loss function. The proposed approach simultaneously captures local temporal dynamics and global inter-component dependencies. Computational experiments on multidimensional EEG data demonstrate improved prediction accuracy compared to autoregressive methods.