There are often scenarios in applications where one needs to extend various distances to objects that have incompatible dimensions, for example, an m-dimensional subspace and an n-dimensional subspace, or an m-by-m covariance matrix and an n-by-n covariance matrix, where m and n are distinct. We will discuss a neat geometric approach for such problems.
The approach is most easily described in the context of subspaces -- the required distance may be interpreted as the distance of a point to a Schubert variety in a Grassmannian. This also provides the impetus for other contexts. This talk is based on joint works with Ke Ye and Rodolphe Sepulchre.