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.

翻译：将实值数据逼近或分解方法推广至黎曼流形上的数据时，基于切空间的方法因其线性空间特性而极具吸引力。一个尚未解决的挑战在于：如何在保持广义方法适用于一般黎曼流形的同时，兼顾全局几何感知与计算可行性。现有方法始终无法同时满足这三个关键要素。本文采用系统化方法构建能兼顾三要素的理论框架。首先，我们将研究对象限定为——仍具一般性的——对称黎曼流形，并揭示曲率如何影响一般流形值张量逼近方案。继而展示如何利用上述发现构建具有全局几何感知能力的逼近策略。最后，在确保普适性与全局几何感知的前提下，我们以低秩逼近为案例进行专项研究，阐明计算可行性实现路径，并提出曲率修正截断高阶奇异值分解（CC-tHOSVD）。该方法在正负曲率对称黎曼流形上的合成数据与真实数据数值实验中均得到性能验证。