The Grassmannian of affine subspaces is a natural generalization of both the Euclidean space, points being 0-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.

仿射子空间的Grassmannian是欧氏空间（点是0维仿射子空间）和通常的Grassmannian线性子空间是仿射子空间的特殊情况的自然概括。我们证明，仿射格拉斯曼像格拉斯曼一样具有丰富的几何和拓扑特性：它具有均匀空间，微分流形，代数变体，向量束，分类空间等结构；此外，它提供了舒伯特演算的类似物，其（共）同构和同型基团很容易确定。另一方面，像欧几里德空间一样，仿射格拉斯曼像作为具体的计算平台，可以在其上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。