Question about complex sphere

[Fengcheng-Pei]

Hello,

In my problem, the variable is a complex vector w and its norm ||w||=5, so I think it is a complex sphere manifold. But how could I define this complex sphere manifold? I only find the real sphere manifold.

Best regards,
Fengcheng Pei

[antoinecollas]

Hello, indeed, the set of ${w \in \mathbb{C}^n: || w || = 1}$ is a manifold. Unfortunately, it is only available as a real-valued manifold in Pymanopt.

If you implement a complex-valued version, you can open a PR. It consists of copy-paste the class Sphereas well as its unit tests and adapting real-valued operators (such as the transpose) to the complex-valued counterparts (such as the conjugate transpose).

[kellertuer]

Hi,
I was not sure whether it was available, so thanks Antoine for checking that; you can “steal” quite a lot of ideas and information from the Julia implementation: Manifolds.jl does cover the complex case as well, see https://juliamanifolds.github.io/Manifolds.jl/stable/manifolds/sphere.html#Manifolds.Sphere

[antoinecollas]

By the way, if the goal is only to do optimization on the complex sphere manifold, I think it is better to directly implement the complex Stiefel manifold (which reduces to the complex sphere for $p=1$).

[NicolasBoumal]

Also, as a proxy for now, you can work with the sphere in R^2n instead of the sphere in C^n, and use the map g(x) = x(1:n) + i * x((n+1):2n) to transform a vector from R^2n to C^n. It's a Riemannian isometry (with the usual metrics), so there's no mathematical difference at all. Of course, eventually, it'd be nicer to just have the complex sphere, complex oblique and complex Stiefel implemented directly.