Complex Circle

[Hamide Zebardast]

Hi,

Hope everything goes well,

I am using "Complex Circle" Factory to solve my optimization problem in the continuous phase domain. Now I want to set my vector Z in the discrete phase domain. Would you please let me know how to choose this domain? 

Best

[Hamideh Zebardast]

Hi again

I will be so grateful if you help me about the mentioned problem,

Best.

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[Nicolas Boumal]

Hello,

Happy to help, but I need more information. What do you call the discrete phase domain?

Best,

Nicolas

[Robert Qiu]

Hello,

This is also a problem I encountered. For the complex circle manifold, the phase of each element in the variable can be continuously taken from 0 to 2pi. If we limit the value of the phase to a finite set of discrete values with equal deviations, can the complex circle manifold still be applicable?

[Nicolas Boumal]

I see: you would like to optimize over sets such as S = {e^(2 pi i k / K) : k = 0,..., K-1}? 

If so, that is a discrete set: it is a manifold, sure, but it is disconnected with many connected components. Therefore, Riemannian optimization  is not the right tool. You can, however, optimize over continuous phases and round. Sometimes that works well.

Best, 

Nicolas 

[Robert Qiu]

Hello,

Thanks for your help. It is the problem I need to solve.

Now, I need to solve a quartic optimization problem under complex circle manifold. 

Several algorithms in the toolbox "monopt" works well. Thank you very much!

The manopt toolbox directly gives the expressions  of the  notions of complex circle manifold, like tangent space, projrction operator and retraction. But I want to know how to derive the specific expressions of these notions. 

I found the relevant conclusions of Obique Manifold in your book in chapter 7 in , and realized that  complex circle manifold  is a special kind of  Obique Manifold,  but the book does not give a detailed process. Can you give me some help or provide some relevant materials.