About the manifold with Hermitian positive semidefinite matrices and fixed rank constraints

[Zhao Xingyu]

Could you provide more materials about the manifold with Hermitian positive semidefinite matrices and fixed rank constraints?

For example:

1.What is the name of this manifold with Hermitian positive semidefinite matrices and fixed rank constraints?

2.What are the definitions of the tangent space, Riemannian gradient, and retraction?

And other similar fundamental definations.

Thank you very much!

[Nicolas Boumal]

Hi,

The relevant factory is  symfixedrankYYcomplexfactory. 

The code for it is here: https://github.com/NicolasBoumal/manopt/blob/master/manopt/manifolds/symfixedrank/symfixedrankYYcomplexfactory.m

The comments at the top of the code (displayed from the command line with help symfixedrankYYcomplexfactory) provide some basic background for that manifold, including references to relevant papers with more background.