Orthonormality of Y of Grassmann manifold

[Hiroyuki Kasai]

Dear Nicolas and Bamdev,

How are you both?

I would like to ask you about the orthonormality of Y of Grassmann manifold.

In the some papers including the book written by Prof.Absil, Y is nonorthonormal. 

However, in my understanding, Manopt assumes that "Y'*Y = eye(p) if k = 1" 

from the comment in the source code as well as tutorial pages.

Is this correct? Does this depend on the structure definition of Grass(p,n), which 

are O_n/(O_p x O_(n-p) or ST(p,n)/GL_p ? The former appears in Eldelman's paper.

In this correct?

If it is yes, are there some difference to be taken care of when they are used?

In addition, how to select them in a practical application?

Many thanks in advance, 

Hiro

[BM]

Hello Hiro, 

Thank you for the questions. Below is my take. 

The answer to your first questions is "yes" and Manopt uses orthonormal matrices. However, for the second question, the answer is "no", i.e., both the mentioned representations are equivalent on the Grassmann manifold. The differences are in the implementation only. Orthonormal matrices should be preferred from a practical viewpoint. 

Regards,

Bamdev

[Hiroyuki Kasai]

Hi Bamdev,

I really thank you for your quick response and clear answer. 

Then, I will use the orthonormal none.

Regards,

Hiro

[Nicolas Boumal]

Hello Hiro!

I think, one of the advantages of using the orthonormal representation is that it naturally takes care of some of the invariance and, more importantly, it avoids ill-conditioning. Orthonormal matrices are perfectly conditioned, and that's a nice property to have.

The price you pay for this is that you need to orthonormalize the matrices at each iteration. That's quite okay for most applications, even more so considering that in the non-ortohonormal version you have to invert small matrices anyway, but for some applications, it might be a reason to not work with orthonormal matrices and try something else instead.

I hope this made sense. Two further comments:

1) regarding the Grassmann geometry in Edelman et al., be careful with the metric: I remember that at least one geometry proposed in their paper uses a different metric that the one used in the Manopt implementation.

2) in general, Manopt will work with any proper representation of the Grassmannian, with any proper metric; Manopt ships with one, ready-to-use factory, but if you feel comfortable with that, you could write a different factory for a different representation: all the solvers will still work, and it would be easy to compare the merits of different factories.

Cheers,

Nicolas

[Hiroyuki Kasai]

Hi Nicolas,

I hope you are very well in the new place.

I really thank you for the detailed explanation. Sometimes, I am confused multiple definitions which 

seem to be quite different for me.

Recently, I strongly confirmed the extensibility and flexibility of Manopt. It is great framework. 

Thank you!

Regards,

Hiro

[Nicolas Boumal]

Glad to hear it! :)