Theory of ehess2rhess

[Hiroyuki Kasai]

Dear all,

I am wondering where the codes for "ehess2rhess" in each manifoldfactory come from. 

Are there in the book "Optimization Algorithms on Matrix Manifolds"? If so, it would be 

appreciated to tell us where they, for example, in Grassmann case or Sphere case, are.

Regards,

Hiro

[Nicolas Boumal]

Hello Hiroyuki,

The formulas for converting the Euclidean Hessian in a Riemannian Hessian are not really explicit in the book by Absil, Mahony and Sepulchre (2008).

The book still gives the essentials in chapter 5 (from memory) abound second order geometry. There, it is stated how you are supposed to compute the Hessian on a submanifold or on a quotient manifold, which covers most things.

There is a nice, short explicit paper by Absil about computing Hessians for some important cases. I don't think that covers Grassmann, but it should cover the sphere:

"An extrinsic look at the Riemannian Hessian"

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

For the Grassmannian specifically, where points are represented as orthonormal matrices (which is the standard thing in Manopt), I believe the formulas should be explained to some extent in this paper of mine (and Absil):

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

http://www.optimization-online.org/DB_HTML/2012/09/3619.html

(skip straight to section 2 --- it's not extremely explicit, and it's not the simplest exposition possible, but it should give good pointers I hope)

Cheers,

Nicolas

[Hiroyuki Kasai]

Dear Nicolas,

Thank you for the pointers for the theory of the conversion into hessian.

Regarding the first paper, I will try to take a look. At a glance, it seems to 

have the direct relationships with the practical implementation in Manopt.

As for the second one, i.e. your paper, which is a very informative and 

great paper, and I had already read last week, I will again take a closer 

look at it.

Once I got another questions, I will post them on this thread.

Have a nice weekend!

Hiro