Asymmetric Metrics on the Full Grassmannian of Subspaces of Different Dimensions

TL;DR

This paper introduces asymmetric metrics on the full Grassmannian of subspaces with different dimensions, extending existing metrics to handle dimensional asymmetry and providing computationally efficient methods.

Contribution

It extends the Fubini-Study metric into an asymmetric angle and develops a framework to adapt other Grassmannian metrics for subspaces of different dimensions.

Findings

Introduces an asymmetric angle based on the Fubini-Study metric.

Provides a method to extend symmetric Grassmannian metrics asymmetrically.

Establishes relations to Grassmann and Clifford geometric algebras for computation.

Abstract

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.