A flag is a sequence of nested subspaces. Flags are ubiquitous in numerical analysis, arising in finite elements, multigrid, spectral, and pseudospectral methods for numerical pde; they arise in the form of Krylov subspaces in matrix computations, and as multiresolution analysis in wavelets constructions. They are common in statistics too—principal component, canonical correlation, and correspondence analyses may all be viewed as methods for extracting flags from a data set. The main goal of this article is to develop the tools needed for optimizing over a set of flags, which is a smooth manifold called the flag manifold, and it contains the Grassmannian as the simplest special case. We will derive closed-form analytic expressions for various differential geometric objects required for Riemannian optimization algorithms on the flag manifold; introducing various systems of extrinsic coordinates that allow us to parameterize points, metrics, tangent spaces, geodesics, distances, parallel transports, gradients, Hessians in terms of matrices and matrix operations; and thereby permitting us to formulate steepest descent, conjugate gradient, and Newton algorithms on the flag manifold using only standard numerical linear algebra.

标志是嵌套子空间的序列。标志在数值分析中无处不在，出现在数值pde 的有限元、多重网格、谱和伪谱方法中; 它们在矩阵计算中以 Krylov 子空间的形式出现，在小波构造中以多分辨率分析的形式出现。它们在统计学中也很常见——主成分、典型相关和对应分析都可以被视为从数据集中提取标志的方法。本文的主要目标是开发优化一组标志所需的工具，这是一个称为标志流形的平滑流形，它包含 Grassmannian 作为最简单的特例。我们将推导出标志流形上黎曼优化算法所需的各种微分几何对象的闭式解析表达式；引入各种外部坐标系统，使我们能够参数化点、度量、切线空间、测地线、距离、平行传输、梯度、Hessians 在矩阵和矩阵运算方面；从而允许我们仅使用标准数值线性代数在标志流形上制定最速下降、共轭梯度和牛顿算法。