Nonparametric Regression on $\mathbb{P}_m\mathbb{C}$, $G_{p,p+q}\mathbb{C}$ , Schubert Varieties, and Complex Flag Manifolds

We extend the optimization-based nonparametric regression framework of
Adouani and Samir (2021), originally developed on the real Stiefel manifold,
to three progressively richer complex geometric settings:
(i) the complex Grassmann manifold $G_{p,p+q}^{\mathbb{C}}$,
(ii) Schubert varieties $\Omega_\lambda \subset G_{p,p+q}^{\mathbb{C}}$, which are
singular stratified spaces, and
(iii) complex flag manifolds $Fl(d_1,\dots,d_r;n)^{\mathbb{C}}$.
For each setting we derive explicit coordinate formulas, the Riemannian
(or stratified) gradient of the regularized energy, the tangent-space
projection operators, and a convergent optimization algorithm.
The complex structure forces Hermitian rather than symmetric tangent-space
conditions; the Schubert setting forces a cell-wise stratified algorithm
because the variety is not a manifold; the flag setting couples multiple
Grassmannian factors through block-skew-Hermitian constraints.
We provide a comparative table situating the method against existing
nonparametric regression techniques on Grassmannians and flag manifolds,
and we discuss the originality and limitations of each extension.