Efficient Continuous Manifold Learning for Time Series Modeling

Modeling non-Euclidean data is drawing attention along with the unprecedented successes of deep neural networks in diverse fields. In particular, symmetric positive definite (SPD) matrix is being actively studied in computer vision, signal processing, and medical image analysis, thanks to its ability to learn appropriate statistical representations. However, due to its strong constraints, it remains challenging for optimization problems or inefficient computation costs, especially, within a deep learning framework. In this paper, we propose 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 reduce computation costs greatly. Further, in order for dynamics modeling in time series data, we devise a continuous manifold learning method by integrating a manifold ordinary differential equation and a gated recurrent neural network in a systematic manner. It is noteworthy that because of the nice parameterization of matrices in a Cholesky space, it is straightforward to train our proposed network with Riemannian geometric metrics equipped. We demonstrate through experiments that the proposed model can be efficiently and reliably trained as well as outperform existing manifold methods and state-of-the-art methods in two classification tasks: action recognition and sleep staging classification.