论文标题
在对称阳性矩阵的流形上，具有微分方程的几何形状谎言组集成符
Geometry-preserving Lie Group Integrators For Differential Equations On The Manifold Of Symmetric Positive Definite Matrices
论文作者
论文摘要
在许多应用中，一个人会遇到位于流形而不是欧几里得空间上的信号。特别是，协方差矩阵是具有非欧几里得结构的普遍数学对象的示例。欧几里得方法在集成在此类对象上的微分方程的应用不尊重歧管的几何形状，这可能会导致许多数值问题。在本文中，我们建议使用谎言组方法在对称阳性确定矩阵的歧管上定义具有几何形状的数值积分方案。这些可以应用于有关实际利益的协方差矩阵的许多微分方程。我们表明，与其他示例相比，它们比其他经典或天真的集成方案更稳定，更健壮。
In many applications, one encounters signals that lie on manifolds rather than a Euclidean space. In particular, covariance matrices are examples of ubiquitous mathematical objects that have a non Euclidean structure. The application of Euclidean methods to integrate differential equations lying on such objects does not respect the geometry of the manifold, which can cause many numerical issues. In this paper, we propose to use Lie group methods to define geometry-preserving numerical integration schemes on the manifold of symmetric positive definite matrices. These can be applied to a number of differential equations on covariance matrices of practical interest. We show that they are more stable and robust than other classical or naive integration schemes on an example.
