  Modeling non-Euclidean data is drawing extensive attention along with the
unprecedented successes of deep neural networks in diverse fields.
Particularly, a symmetric positive definite matrix is being actively studied in
computer vision, signal processing, and medical image analysis, due to its
ability to learn beneficial statistical representations. However, owing to its
rigid constraints, it remains challenging to optimization problems and
inefficient computational costs, especially, when incorporating it with a deep
learning framework. In this paper, we propose a framework to exploit a
diffeomorphism mapping between Riemannian manifolds and a Cholesky space, by
which it becomes feasible not only to efficiently solve optimization problems
but also to greatly reduce computation costs. Further, for dynamic modeling of
time-series data, we devise a continuous manifold learning method by
systematically integrating a manifold ordinary differential equation and a
gated recurrent neural network. It is worth noting that due to the nice
parameterization of matrices in a Cholesky space, training our proposed network
equipped with Riemannian geometric metrics is straightforward. We demonstrate
through experiments over regular and irregular time-series datasets that our
proposed model can be efficiently and reliably trained and outperforms existing
manifold methods and state-of-the-art methods in various time-series tasks.
