Nonparametric Regression on PₘC, Gp,p+qC , 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 Gp,p+qC,(ii) Schubert varieties Ω_λ ⊂ Gp,p+qC, which aresingular stratified spaces, and(iii) complex flag manifolds Fl(d₁,,dᵣ;n)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.