Complex finite-basis PC-VAR optimization on Grassmann

If $X^{\ast }X=I_r$, then $P=X{X}^{\ast }$ is Hermitian and $P^2=X(X^{\ast }X)X^{\ast }=P$, with range equal to the column space of $X$ and therefore rank $r$. For unitary $Q$, $(XQ)(XQ{)}^{\ast }=XQ{Q}^{\ast }{X}^{\ast }=P$. Thus $E(X{X}^{\ast })$ is unitary-invariant and descends to complex Grassmann by [F1].

For every horizontal $Z$, [F2] and [F4] yield the complex Grassmann gradient $2(I-P)F(P)X$. This formula uses the real derivative of the Hermitian-matrix energy, as required for a real-valued objective on complex coordinates.

Conversely, let $P$ be any rank-$r$ orthogonal projector. Choose an orthonormal basis of its $r$-dimensional range by [F7] and collect its vectors as the columns of $X$. Then $X^{\ast }X=I_r$ and $X{X}^{\ast }=P$, so the parameterization covers exactly the model's projector domain. By [F5], the polar-retracted trial remains a complex orthonormal frame, so its projector remains Hermitian, idempotent, and rank $r$; rejection leaves the old projector unchanged by [F6].

Fix any finite ${\mathrm{\Delta }}_{\max }>0$. The set of pairs $(X,\eta )$ with $X^{\ast }X=I_r$, $X^{\ast }\eta =0$, and $\mathrm{\parallel }\eta {\mathrm{\parallel }}_F\le {\mathrm{\Delta }}_{\max }$ is closed and bounded in a finite-dimensional real coordinate space, hence compact by [F8]. The polar retraction is smooth on it, and [F1]–[F2] make the pullback energy $C^2$ along the horizontal tangent bundle. Its fiber Hessian is continuous on this compact set, so its operator norm has a finite maximum by [F9]. Integrating this Hessian along each fiber segment gives a uniform Lipschitz bound for the pullback gradient. Choose the model operator $B_x=0$, which is self-adjoint and uniformly bounded by $M_B=0$. Hence all regularity and model hypotheses in [F6] hold, and a finite cap at least the theorem's trial bound suffices.