Asymmetric Metrics on the Full Grassmannian of Subspaces of Different Dimensions

Metrics on Grassmannians have a wide array of applications: machine learning, wireless communication, computer vision, etc. But the available distances between subspaces of distinct dimensions present problems, and the dimensional asymmetry of the subspaces calls for the use of asymmetric metrics. We extend the Fubini-Study metric as an asymmetric angle with useful properties, and whose relations to products of Grassmann and Clifford geometric algebras make it easy to compute. We also describe related angles that provide extra information, and a method to extend other Grassmannian metrics to asymmetric metrics on the full Grassmannian.