# Numerical algorithms on the affine Grassmannian

arXiv: https://arxiv.org/abs/1607.01833 · HTML: https://mlsanity.com/paper/1607.01833/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 6 Jul 2016.

## Built on by (most cited first, 3 of 3)

- [Grassmannian diffusion maps based surrogate modeling via geometric harmonics](https://mlsanity.com/paper/2109.13805/) (2021)
- [Nested subspace learning with flags](https://mlsanity.com/paper/2502.06022/) (2025)
- [Riemannian Stochastic Gradient Method for Nested Composition Optimization](https://mlsanity.com/paper/2207.09350/) (2022)

## Abstract (arXiv)

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.