Complex unit circle manifold

[khalhuj...@gmail.com]

Hi everyone,

I need to understand the mathematical theory in optimization over complex unit circle, i.e., the tangent space and retractions.

I am looking for a reference about this manifold, like the reference that is used to develop the complexcirclefactory manifold in Manopt. Is it possible to get such a reference?

Many thanks'
Khaled

[Nicolas Boumal]

Hello Khaled,

Perhaps it is easiest to first understand the circle S = {x \in R^2 : x_1^2 + x_2^2 = 1}: this is just a unit circle in the plane R^2, as usual. You will find all the details for getting a tangent space, a retraction and more in Appendix A of https://arxiv.org/pdf/1605.08101v1.pdf.

Then, to understand the complex circle, it is just a matter of thinking of the complex circle as the circle above, S, but you represent a point x = (x_1, x_2) as the complex number z = x_1 + i x_2. From there, you will easily figure out that, for example, ||x|| = |z|, and hence the retraction (x+v)/||x+v|| becomes (z+u)/|z+u|, where |.| is the complex modulus (abs in Matlab), and u is a tangent vector. The inner product between two complex numbers becomes real(u'*v): just work out this formula and compare to what happens in the real case: it's the same thing.

I hope this helps.

Best,

Nicolas

On Thursday, November 16, 2017 at 4:14:13 PM UTC-5,

[khalhuj...@gmail.com]

Dear Nicolas,

Many thanks for your very helpful response.