  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.
