Abstract:Flag manifolds are generalizations of projective spaces and other Grassmannians: they parametrize flags, which are nested sequences of subspaces in a given vector space. These are important objects in algebraic and differential geometry, but are also increasingly being used in data science, where many types of data are properly understood as subspaces rather than vectors. In this paper we discuss partially oriented flag manifolds, which parametrize flags in which some of the subspaces may be endowed with an orientation. We compute the expected distance between random points on some low-dimensional examples, which we view as a statistical baseline against which to compare the distances between particular partially oriented flags coming from geometry or data.

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中文翻译：

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部分定向标志的流形上的预期距离

摘要：标志流形是射影空间和其他 Grassmannians 的推广：它们参数化标志，标志是给定向量空间中子空间的嵌套序列。这些是代数和微分几何中的重要对象，但也越来越多地用于数据科学，其中许多类型的数据被正确理解为子空间而不是向量。在本文中，我们讨论部分定向标志流形，其参数化标志，其中一些子空间可能被赋予一个方向。我们计算一些低维示例上随机点之间的预期距离，我们将其视为统计基线，用于比较来自几何或数据的特定部分定向标志之间的距离。

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