Numerical algorithms on the affine Grassmannian

TL;DR

This paper extends Riemannian optimization algorithms to the affine Grassmannian by realizing it as a matrix manifold, enabling efficient computation of algorithms like steepest descent, Newton, and conjugate gradient.

Contribution

It introduces a matrix manifold realization of the affine Grassmannian and adapts classical Riemannian optimization algorithms for this setting.

Findings

Algorithms are based on standard linear algebra operations.

Methods are computationally efficient and readily implementable.

Extends optimization techniques to a broader class of geometric structures.

Abstract

The affine Grassmannian is a noncompact smooth manifold that parameterizes all affine subspaces of a fixed dimension. It is a natural generalization of Euclidean space, points being zero-dimensional affine subspaces. We will realize the affine Grassmannian as a matrix manifold and extend Riemannian optimization algorithms including steepest descent, Newton method, and conjugate gradient, to real-valued functions on the affine Grassmannian. Like their counterparts for the Grassmannian, these algorithms are in the style of Edelman--Arias--Smith --- they rely only on standard numerical linear algebra and are readily computable.