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.

翻译：对非欧几里得数据的建模，随着深度神经网络在多个领域取得前所未有的成功，正受到广泛关注。特别是，由于对称正定矩阵能够学习有益的统计表示，它在计算机视觉、信号处理和医学图像分析中被积极研究。然而，由于其严格的限制，它在优化问题和计算成本方面仍面临挑战，尤其在与深度学习框架结合时。本文提出一种框架，利用黎曼流形与乔列斯基空间之间的微分同胚映射，这不仅使得高效解决优化问题成为可能，还大幅降低了计算成本。此外，针对时间序列数据的动态建模，我们设计了一种连续流形学习方法，通过系统集成流形常微分方程和门控循环神经网络来实现。值得注意的是，由于乔列斯基空间中矩阵的优良参数化特性，训练配备黎曼几何度量的所提网络是直接可行的。通过在规则和不规则时间序列数据集上的实验，我们证明所提模型能够高效且可靠地训练，并在多种时间序列任务中优于现有流形方法和最新方法。