Exponential and Logarithm Mapping on Stiefel Manifold

Question

The Stiefel Manifold is defined as

$$ \mathrm{St}(p,n):= \{ X\in \mathbb{R}^{n\times p} :\ X^T X = I_p \}. $$

Recall that the tangent space at a point $X\in \mathrm{St}(p,n)$ is given by $$ T_X{\mathrm{St}(p,n)} = \{\xi\in \mathbb{R}^{n\times p}:\ X^T\xi + \xi^T X = 0 \}. $$

Given a point $X\in \mathrm{St}(p,n)$ and a tangent vector $\xi \in T_X{\mathrm{St}(p,n)}$, it is possible to express the exponential map $\exp_X(\xi)$ using the matrix exponential function. A formula is given in the paper www.mit.edu/~wingated/introductions/stiefel-mfld.pdf .

My question is whether the inverse $\log_X(\cdot): \mathrm{St}(p,n) \to T_X\mathrm{St}(p,n)$ can also be expressed using the matrix logarithm.

A related question is the following: Instead of the exponential function one may define other retractions such as $$ R_X(\xi) = (X+\xi)(I_p + \xi^T\xi)^{-1/2} $$ or the closest point projection $$ R_X(\xi) = \pi(X+\xi), $$ where $\pi$ maps a matrix $A$ to the closest element in the Stiefel manifold. This projection can be easily computed using SVD.

Again, the question is wheter one can find simple formulas for the inverses of these retractions.

Any helpful comments would be greatly appreciated.

edit: at this point I do not really care which metric is used (the Euclidean metric or the one inherited from the orthogonal group).

Answer 1

If one uses the Hilbert-Schmidt inner product then Section 3.1.3 of

"A new geometric metric in the space of curves, and applications to tracking deforming objects by prediction and filtering" by Sundaramoorthi, Mennuci, Soatto, and Yezzi in SIAM Journal on Imaging Sciences (2010)

gives a numerical method for computing the log. So I guess that implies that as of 2010 the authors were not able to find any closed form expression in the literature.

Answer 2

There are some potentially useful formulae in Section 7 of this document:

http://neil-strickland.staff.shef.ac.uk/research/universes.pdf

I have not seen them in the literature, though I may not have looked in the right place. They are directly applicable to complex Grassmannians, but it may be possible to adapt them.

Answer 3

If the question is still relevant to you, have a look at the paper:

R. Zimmermann: A Matrix-Algebraic Algorithm for the Riemannian Logarithm on the Stiefel Manifold under the Canonical Metric, SIAM. J. Matrix Anal. & Appl. 38-2 (2017), pp. 322-342, arXiv:1604.05054, doi: 10.1137/16M1074485

Answer 4

Peter, I'm a non-geometer but I'm a bit curious about a possible subtlety related to your answer. On page 316 of

www.mit.edu/~wingated/introductions/stiefel-mfld.pdf

the authors make a careful distinction between the Euclidean metric $$g_{E}(Δ,Δ)=Tr(Δ^{∗}Δ)$$ on the tangent space and the so-called "canonical metric" $$g_{C}(Δ,Δ)=Tr(Δ^{∗}(I-(1/2)UU^{∗})Δ)$$ on the tangent space at U. The latter metric is called "canonical" because it comes from that comes from viewing the manifold $V_{n,p}$ of isometries $U:C^{p}→C^n$ as the quotient $$V_{n,p}=O_{n}/O_{n-p}.$$

Am I correct in assuming that your answer applies to the canonical metric but not to the euclidean one?

Answer 5

A list of possible retractions for the compact Stiefel manifold (including the one mentioned in the question) and their inverses is available in:

T. Kaneko, S. Fiori and T. Tanaka, "Empirical Arithmetic Averaging over the Compact Stiefel Manifold," IEEE Transactions on Signal Processing, Vol. 61, No. 4, pp. 883 - 894, February 2013

Answer 6

Have a look at the paper:

• Y. A. Neretin: On Jordan angles and the triangle inequality in Grassmann manifold, Geometriae Dedicata, 86 (2001), arXiv:math/0005059, doi: 10.1023/A:1011974705094.

There are explicit formulas for geodesics and even for the geodesic distance on real Grassmannians, and the Riemannian logarithm in terms of $\arccos$. Geodesics on the Grassmannian correspond to horizontal geodesics on the Stiefel manifold, and I am sure that one adapt Neretins formulas to Stiefel manifolds.