Riemannian Geometry

From News to Returns: A Granger-Causal Hypergraph Transformer on the Sphere ArXiv ID: 2510.04357 “View on arXiv” Authors: Anoushka Harit, Zhongtian Sun, Jongmin Yu Abstract We propose the Causal Sphere Hypergraph Transformer (CSHT), a novel architecture for interpretable financial time-series forecasting that unifies \emph{“Granger-causal hypergraph structure”}, \emph{“Riemannian geometry”}, and \emph{“causally masked Transformer attention”}. CSHT models the directional influence of financial news and sentiment on asset returns by extracting multivariate Granger-causal dependencies, which are encoded as directional hyperedges on the surface of a hypersphere. Attention is constrained via angular masks that preserve both temporal directionality and geometric consistency. Evaluated on S&P 500 data from 2018 to 2023, including the 2020 COVID-19 shock, CSHT consistently outperforms baselines across return prediction, regime classification, and top-asset ranking tasks. By enforcing predictive causal structure and embedding variables in a Riemannian manifold, CSHT delivers both \emph{“robust generalisation across market regimes”} and \emph{“transparent attribution pathways”} from macroeconomic events to stock-level responses. These results suggest that CSHT is a principled and practical solution for trustworthy financial forecasting under uncertainty. ...

Geometric Deep Learning for Realized Covariance Matrix Forecasting ArXiv ID: 2412.09517 “View on arXiv” Authors: Unknown Abstract Traditional methods employed in matrix volatility forecasting often overlook the inherent Riemannian manifold structure of symmetric positive definite matrices, treating them as elements of Euclidean space, which can lead to suboptimal predictive performance. Moreover, they often struggle to handle high-dimensional matrices. In this paper, we propose a novel approach for forecasting realized covariance matrices of asset returns using a Riemannian-geometry-aware deep learning framework. In this way, we account for the geometric properties of the covariance matrices, including possible non-linear dynamics and efficient handling of high-dimensionality. Moreover, building upon a Fréchet sample mean of realized covariance matrices, we are able to extend the HAR model to the matrix-variate. We demonstrate the efficacy of our approach using daily realized covariance matrices for the 50 most capitalized companies in the S&P 500 index, showing that our method outperforms traditional approaches in terms of predictive accuracy. ...