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 ofAdouani 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 aresingular 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-spaceprojection operators, and a convergent optimization algorithm.The complex structure forces Hermitian rather than symmetric tangent-spaceconditions; the Schubert setting forces a cell-wise stratified algorithmbecause the variety is not a manifold; the flag setting couples multipleGrassmannian factors through block-skew-Hermitian constraints.We provide a comparative table situating the method against existingnonparametric regression techniques on Grassmannians and flag manifolds,and we discuss the originality and limitations of each extension.