Simple matrix models for the flag, Grassmann, and Stiefel manifolds

TL;DR

This paper introduces three comprehensive families of orthogonally-equivariant matrix models for the Grassmann, flag, and Stiefel manifolds, enabling efficient and accurate geometric computations.

Contribution

It provides an exhaustive classification of minimal-dimension matrix models for these manifolds with desirable computational properties.

Findings

Models are orthogonally-equivariant and computationally efficient.

Exhaustive classification of minimal-dimension models.

Identification of the Stiefel model family with the Cartan manifold.

Abstract

We derive three families of orthogonally-equivariant matrix submanifold models for the Grassmann, flag, and Stiefel manifolds respectively. These families are exhaustive -- every orthogonally-equivariant submanifold model of the lowest dimension for any of these manifolds is necessarily a member of the respective family, with a small number of exceptions. They have several computationally desirable features. The orthogonal equivariance allows one to obtain, for various differential geometric objects and operations, closed-form analytic expressions that are readily computable with standard numerical linear algebra. The minimal dimension aspect translates directly to a speed advantage in computations. And having an exhaustive list of all possible matrix models permits one to identify the model with the lowest matrix condition number, which translates to an accuracy advantage in computations. As an interesting aside, we will see that the family of models for the Stiefel manifold is naturally parameterized by the Cartan manifold, i.e., the positive definite cone equipped with its natural Riemannian metric.