Title: Curvature corrected tangent space-based approximation of manifold-valued data Show Chinese title

Title: 曲率修正的流形数据切空间近似

Abstract: When generalizing schemes for real-valued data approximation or decomposition to data living in Riemannian manifolds, tangent space-based schemes are very attractive for the simple reason that these spaces are linear. An open challenge is to do this in such a way that the generalized scheme is applicable to general Riemannian manifolds, is global-geometry aware and is computationally feasible. Existing schemes have been unable to account for all three of these key factors at the same time. In this work, we take a systematic approach to developing a framework that is able to account for all three factors. First, we will restrict ourselves to the -- still general -- class of symmetric Riemannian manifolds and show how curvature affects general manifold-valued tensor approximation schemes. Next, we show how the latter observations can be used in a general strategy for developing approximation schemes that are also global-geometry aware. Finally, having general applicability and global-geometry awareness taken into account we restrict ourselves once more in a case study on low-rank approximation. Here we show how computational feasibility can be achieved and propose the curvature-corrected truncated higher-order singular value decomposition (CC-tHOSVD), whose performance is subsequently tested in numerical experiments with both synthetic and real data living in symmetric Riemannian manifolds with both positive and negative curvature. Show Chinese abstract

Abstract: 当将适用于实值数据逼近或分解的方案推广到生活在黎曼流形上的数据时，基于切空间的方案非常有吸引力，因为这些空间是线性的。 一个开放的挑战是以这样一种方式做到这一点，即广义方案适用于一般的黎曼流形，具有全局几何意识，并且计算上可行。 现有的方案无法同时考虑这三个关键因素。 在本工作中，我们采取系统的方法来开发一个能够考虑所有三个因素的框架。 首先，我们将自己限制在仍然通用的对称黎曼流形类中，并展示曲率如何影响一般的流形值张量逼近方案。 接下来，我们展示如何将后者的观察结果用于开发也具有全局几何意识的逼近方案的一般策略。 最后，在考虑到普遍适用性和全局几何意识之后，我们再次通过一个案例研究来限制自己在低秩逼近上。 在这里，我们展示如何实现计算可行性，并提出校正曲率的截断高阶奇异值分解（CC-tHOSVD），其性能随后在具有正曲率和负曲率的对称黎曼流形上的合成数据和真实数据的数值实验中进行了测试。