Skip to first unread message
Message has been deleted
Florian
Sep 21, 2014, 11:33:03 PM9/21/14
to manopt...@googlegroups.com, ijingji...@gmail.com
Dear Jing,
Your idea seems to be very interesting, but I am not quite sure about the choice of manifold for optimizing the function F(Y).
I see perfectly how it could be optimized on the Grassman manifold but not on the manifold of rank-k symmetric positive semidefinite matrices.
Could you show us how you expressed your function F(Y) in terms of X ?
In the code you showed, it seems that you are optimizing F(X) and not F(Y) and since your function is based on logarithm, F(Y) is defined but not F(X).
About the computation of the euclidean, I obtain the following (but I did not find time to check it numerically) :
Gradient F(Y) = 4 Y'A Dlog (Y'AY) [log(Y'AY)-log(Y'BY)] + 4 Y'B Dlog (Y'BY) [log(Y'BY)-log(Y'AY)]
For obtaining this, I used the directionnal derivative and Dlog is the directionnal derivative of the log.
We discussed some of this issues in a previous post :
Differentiation a cost function based on a matrix logarithm
About the Hessian, I have no idea how to compute it, but it may not be needed, since the trustregion can approximate it.
Cheers
Florian
Your idea seems to be very interesting, but I am not quite sure about the choice of manifold for optimizing the function F(Y).
I see perfectly how it could be optimized on the Grassman manifold but not on the manifold of rank-k symmetric positive semidefinite matrices.
Could you show us how you expressed your function F(Y) in terms of X ?
In the code you showed, it seems that you are optimizing F(X) and not F(Y) and since your function is based on logarithm, F(Y) is defined but not F(X).
About the computation of the euclidean, I obtain the following (but I did not find time to check it numerically) :
Gradient F(Y) = 4 Y'A Dlog (Y'AY) [log(Y'AY)-log(Y'BY)] + 4 Y'B Dlog (Y'BY) [log(Y'BY)-log(Y'AY)]
For obtaining this, I used the directionnal derivative and Dlog is the directionnal derivative of the log.
We discussed some of this issues in a previous post :
Differentiation a cost function based on a matrix logarithm
About the Hessian, I have no idea how to compute it, but it may not be needed, since the trustregion can approximate it.
Cheers
Florian
Message has been deleted
BM
Sep 22, 2014, 9:05:46 AM9/22/14
to
(The attachments have been modified)
Hello all,
Thank you Jing for the question, and thank you Florian for such a prompt and nice answer. Florian, your calculations are indeed correct. \m/
Thank you Jing for the question, and thank you Florian for such a prompt and nice answer. Florian, your calculations are indeed correct. \m/
The code is attached below. Download the files and click on the .html file to see the code.
The dlogm code is the copy-paste of Nicola's code here, https://groups.google.com/d/msg/manopttoolbox/nM2QpdsTGWU/FZP9IZyvsUMJ.
Follow the entire thread for an interesting discussion on the topic.
Cheers,
BM
P.S. I am working on the Hessian and will let you know at the earliest.
Florian
Sep 23, 2014, 9:17:53 PM9/23/14
to manopt...@googlegroups.com
Hi Bamdev,
Thanks for checking my formula :-D
If ever you have some update about the hessian, I would be really glad to hear about it.
Indeed, computing the second order derivative through the matrix logarithm is far from obvious to me.
Cheers
Florian
Thanks for checking my formula :-D
If ever you have some update about the hessian, I would be really glad to hear about it.
Indeed, computing the second order derivative through the matrix logarithm is far from obvious to me.
Cheers
Florian
BM
Sep 26, 2014, 10:18:22 AM9/26/14
to manopt...@googlegroups.com
Hello Florian,
I will surely let you know of the developments.
Cheers,
Bamdev
Reply all
Reply to author
Forward
0 new messages