A Geometric Deep Learning Approach to Forecasting the Time Series of Covariance Matrices

The forecasting approaches of time-varying covariance matrices often overlook the geometric properties of symmetric positive definite matrices, ignoring the fact that these are points on a Riemannian manifold. This may lead to suboptimal forecast accuracy and might result in an over-parameterized model, making it infeasible to work with high-dimensional matrices. This paper introduces an innovative approach to forecasting time series of covariance matrices using a deep learning method based on Riemannian optimization. In an application with simulated data, we show that when geometric properties of the predicted object are taken into account, the prediction accuracy improves significantly.