仿射子空间的格拉斯曼
The Grassmannian of affine subspaces
摘要：
仿射子空间的格拉斯曼是欧几里得空间的自然推广，点是零维仿射子空间，而通常的格拉斯曼，线性子空间是仿射子空间的特殊情况。我们表明，像格拉斯曼，仿射格拉斯曼有丰富的几何和拓扑性质：它有一个齐性空间，一个微分流形，代数簇，向量丛，分类空间，在许多更多的结构，此外，它提供了一个类似的舒伯特演算和它的（上）同调和同伦群可以很容易地确定。另一方面，像欧几里得空间一样，仿射格拉斯曼作为一个具体的计算平台，在其上可以通过数值线性代数明确定义和计算各种距离、度量、概率密度。此外，机器学习和统计学中的许多标准问题-线性回归、变量误差回归、主成分分析、支持向量机，或者更一般地，任何寻求变量之间的线性关系的问题，这些变量要么最好地表示它们，要么将它们分成分量-可以自然地被公式化为仿射格拉斯曼问题。
英文摘要：
The Grassmannian of affine subspaces is a natural generalization of both the Euclidean space, points being zero-dimensional affine subspaces, and the usual Grassmannian, linear subspaces being special cases of affine subspaces. We show that, like the Grassmannian, the affine Grassmannian has rich geometrical and topological properties: It has the structure of a homogeneous space, a differential manifold, an algebraic variety, a vector bundle, a classifying space, among many more structures; furthermore; it affords an analogue of Schubert calculus and its (co)homology and homotopy groups may be readily determined. On the other hand, like the Euclidean space, the affine Grassmannian serves as a concrete computational platform on which various distances, metrics, probability densities may be explicitly defined and computed via numerical linear algebra. Moreover, many standard problems in machine learning and statistics --- linear regression, errors-in-variables regression, principal components analysis, support vector machines, or more generally any problem that seeks linear relations among variables that either best represent them or separate them into components --- may be naturally formulated as problems on the affine Grassmannian.
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