A geometric approach to subspace updates and orthogonal matrix decompositions under rank-one modifications

For a matrix $X\in \mathbb{R}^{n\times p}$, we provide an analytic formula that keeps track of an orthonormal basis for the range of $X$ under rank-one modifications. More precisely, we consider rank-one adaptations $X_{new} = X+ab^T$ of a given $X$ with known matrix factorization $X = UW$, where $U\in\mathbb{R}^{n\times p}$ is column-orthogonal and $W\in \R^{p\times p}$ is invertible. Arguably, the most important methods that produce such factorizations are the singular value decomposition (SVD), where $X=UW=U(\Sigma V^T)$, and the QR-decomposition, where $X = UW = QR$.
We give a geometric description of rank-one adaptations and derive a closed-form expression for the geodesic line that travels from the subspace $\mathcal{S}= \colspan(X)$ to the subspace $\mathcal{S}_{new} =\colspan(X_{new}) =\colspan(U_{new}W_{new})$. This leads to update formulas for orthogonal matrix decompositions, where both $U_{new}$ and $W_{new}$ are obtained via elementary rank-one matrix updates in $\mathcal{O}(np)$ time for $n\gg p$.
Moreover, this allows us to determine the subspace distance and the Riemannian midpoint between the subspaces $\mathcal{S}$ and $\mathcal{S}_{new}$ without additional computational effort.