Riemannian Geometry of Grassmann Manifolds with a View on Algorithmic Computation

Abstract

Abstract. We give simple formulas for the canonical metric, gradient, Lie derivative, Riemannian connection, parallel translation, geodesics and distance on the Grassmann manifold of p-planes in R n . In these formulas, p-planes are represented as the column space of n × p matrices. The Newton method on abstract Riemannian manifolds proposed by Smith is made explicit on the Grassmann manifold. Two applications -computing an invariant subspace of a matrix and the mean of subspaces -are worked out. (2000): 65J05, 53C05, 14M15. Mathematics Subject Classifications

Kernel Learning for Extrinsic Classification of Manifold Features

“…We refer the interested readers to [7,1] for further discussions on the Grassmann manifold. Grassmann kernels: Grassmann manifold can be mapped to Euclidean spaces by using Mercer kernels [8].…”

Optimization On Manifolds: Methods and Applications

“…4 The Stiefel manifold is named in honor of Eduard L. Stiefel who studied its topology in [55]. Stiefel is perhaps better known for proposing with M. R. Hestenes the conjugate gradient method [25].…”

“…Most applications bear some relation with dimensionality reduction: [24,18,40,53,4,39,54,12,23,52,59,15,27].…”

Flag Manifolds for Subspace ICA Problems

“…Fl(n, d ; R) is locally isomorphic 1 to St(n, p ; R) as a homogeneous space when all d i (1 ≤ i ≤ r) = 1, and it reduces to a Grassmann manifold if r = 1. It may well be said that the ag manifold is a generalization of the Stiefel and Grassmann manifolds in this sense.…”

“…, w p ) ∈ R n×p |W W = I p , n ≥ p , and the Grassmann manifold Gr(n, p ; R) of unoriented pplanes, corresponding to the subspaces spanned by n × p full rank matrices. Stiefel manifolds have been used in ICA and PCA in the case where the number of the extracted components is less than the number of the mixed signals [8], while Grassmann manifolds have been utilized for invariant subspace computation and subspace tracking [1]. One-unit ICA extracts one independent component at a time, while ordinary ICA extracts several components simultaneously by optimization over the Stiefel manifold.…”