Derivation of ehess2rhess for Stiefel and Grassmann manifolds.

[Jamie Townsend]

I'm trying to get my head around ehess2rhess in the Stiefel and Grassmann manifolds. I wondered if you might have a derivation written down for these?

[Nicolas Boumal]

Hello Jamie,

The (orthogonal) Stiefel manifold, that is, the set of all nxp matrices with orthogonal columns, is a submanifold of R^{nxp}. If we use the standard metric <A, B> = Trace(A^T B), it is a Riemannian submanifold of the Euclidean space R^{nxp}. This means the formulas in Absil's book apply:

equations 3.37 and 5.15

Section 4.1 in this paper by Absil gives the details:

An extrinsic look at the Riemannian Hessian

http://sites.uclouvain.be/absil/2013-01/Weingarten_05PA_techrep.pdf

See section 4.1.

For the Grassmann manifold, it's a bit more tricky, because it's a quotient manifold, and things depend a bit of how you look at them. There are also formulas in Absil's book (close by the ones I referenced). You can also take a look at the explicit (but somewhat expedited) deriviation in this paper:

Low-rank matrix completion via preconditioned optimization on the Grassmann manifold

http://www.sciencedirect.com/science/article/pii/S0024379515001342

See section 2.

I hope this helps. Don't hesitate if this leaves unanswered questions.

Best,

Nicolas

[Jamie Townsend]

Very helpful, thanks Nicolas

[pierr...@gmail.com]

In order to see how section 4.1 in the report http://sites.uclouvain.be/absil/2013-01/Weingarten_07PA_techrep.pdf yields the implementation of ehess2rhess in Manopt 2.0's stiefelfactory.m, you should look at the line

P_X D_Z P V = −P_X Z S

in the report (along with equation (7) which shows that why this "PDP" term is worth considering), and not at the equivalent formulation obtained a few lines below. It seems that both formulations have the same dominant flop count, and the former is slightly more compact to implement than the latter.

PA.

[Jamie Townsend]

Thanks, that's also very helpful. I'm writing code to automatically generate a function which calculates equation (7) in that paper, based on the projection and cost.

[Jamie Townsend]

https://github.com/pymanopt/pymanopt/blob/master/pymanopt/tools/testing.py