complexcirclefactory

[Hamide Zebardast]

Hi,

My question is how the projection and retraction defined in complexcirclefactory manifold. 

Wold you please introduce me a good reference ?

Best, 

Hamideh

 

[Nicolas Boumal]

Hello,

A complex circle is really just a circle in R^2. Retraction to the circle is implemented in the usual way for spheres: (x+v)/norm(x+v). Only,  complexcirclefactory allows you to have a matrix where each entry is on a circle (that is, we have a product manifold), so the retraction is computed for each entry individually (each entry for its own circle). Likewise for projections to tangent spaces: you just project to the tangent space space of each circle.

Best,

Nicolas

[Ronny Bergmann]

HI,

the manual I am linking to is from another programming language (Manifolds.jl in Julia) – not to convince you to switch, just because I drew some illustrations for the case of projections and retractions therein:

Projections of a point (q in the complex plane) can be done in two interpretations: project onto the manifold (called projection/project in Manopt) – which is s_1 in the image

https://juliamanifolds.github.io/Manifolds.jl/latest/interface.html#Projections

and onto tangent spaces – which is the point s_2 in the image.

For a retraction, imagine you have a point p on the circle and a tangent vector X at that point, then p+X is just a point in the complex plane – and the retraction is given by projecting this back onto the manifold, see s1 case above, or

https://juliamanifolds.github.io/Manifolds.jl/latest/interface.html#Retractions-and-inverse-Retractions

for an image in comparison to the exponential map.

This is done element wise for the case Nicolas mentioned, i.e. where you have a product manifold.

Best,

Ronny

[Hamide Zebardast]

Thanks, I got it. I have read your book including the general concept of retraction and projection. Now I want to cite a reference  in my paper including projection and retraction specifically for complex circle case. Do you have any relevant suggestion? 

Best, 

Hamideh

[Ronny Bergmann]

I would either use the book by Absil, Mahony, Sepulchre as a reference / citation – and/or Nicolas book.