Grassmann流形手册：基础几何与计算层面
A Grassmann Manifold Handbook: Basic Geometry and Computational Aspects
AI总结：
本文系统阐述Grassmann流形的几何性质，统一正交投影与商空间两种视角，并提出改进的黎曼对数映射算法及平行移动等公式，为低秩优化等问题提供计算支持。
摘要：
线性子空间的Grassmann流形对于众多应用的数学建模至关重要，涵盖机器学习、计算机视觉与图像处理问题，以及低秩矩阵优化问题、动态低秩分解和模型降阶。通过这篇主要为阐述性的工作，我们旨在以适合利用基于矩阵的算法解决上述问题的方式，提供关于Grassmann流形几何的必备事实与公式集合。此外，我们既从用正交投影表示子空间的方法，又从将其视为正交群的商空间的角度来阐述Grassmann几何，在商空间视角下，子空间被识别为（正交）基的等价类。这桥接了相关的研究脉络，并允许在这两种方法之间轻松转换。原创贡献包括一种用于计算Grassmann流形上黎曼对数映射的改进算法，该算法在数值上更具优势，同时也能更基础但更完整地描述切点集与共轭点。我们还推导了在正交投影视角下沿测地线的平行移动公式、指数映射导数的公式，以及在某一点处消失的Jacobi场公式。
英文摘要：
The Grassmann manifold of linear subspaces is important for the mathematical modelling of a multitude of applications, ranging from problems in machine learning, computer vision and image processing to low-rank matrix optimization problems, dynamic low-rank decompositions and model reduction. With this mostly expository work, we aim to provide a collection of the essential facts and formulae on the geometry of the Grassmann manifold in a fashion that is fit for tackling the aforementioned problems with matrix-based algorithms. Moreover, we expose the Grassmann geometry both from the approach of representing subspaces with orthogonal projectors and when viewed as a quotient space of the orthogonal group, where subspaces are identified as equivalence classes of (orthogonal) bases. This bridges the associated research tracks and allows for an easy transition between these two approaches. Original contributions include a modified algorithm for computing the Riemannian logarithm map on the Grassmannian that is advantageous numerically but also allows for a more elementary, yet more complete description of the cut locus and the conjugate points. We also derive a formula for parallel transport along geodesics in the orthogonal projector perspective, formulae for the derivative of the exponential map, as well as a formula for Jacobi fields vanishing at one point.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
