A geometric note on subspace updates and orthogonal matrix decompositions under rank-one modifications

In this work, we consider rank-one adaptations $X_{new} = X+ab^T$ of a given matrix $X\in \mathbb{R}^{n\times p}$ with known matrix factorization $X = UW$, where $U\in\mathbb{R}^{n\times p}$ is column-orthogonal, i.e. $U^TU=I$. 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 $X = UW = QR$. An elementary approach to produce a column-orthogonal matrix $U_{new}$, whose columns span the same subspace as the columns of the rank-one modified $X_{new} = X +ab^T$ is via applying a suitable coordinate change such that in the new coordinates, the update affects a single column and subsequently performing a Gram-Schmidt step for reorthogonalization. This may be interpreted as a rank-one adaptation of the $U$-factor in the SVD or a rank-one adaptation of the $Q$-factor in the QR-decomposition, respectively, and leads to a decomposition for the adapted matrix $X_{new} = U_{new}W_{new}$. By using a geometric approach, we show that this operation is equivalent to traveling from the subspace $\mathcal{S}= \text{ran}(X)$ to the subspace $\mathcal{S}_{new} =\text{ran}(X_{new})$ on a geodesic line on the Grassmann manifold and we derive a closed-form expression for this geodesic. In addition, this allows us to determine the subspace distance between the subspaces $\mathcal{S}$ and $\mathcal{S}_{new}$ without additional computational effort. Both $U_{new}$ and $W_{new}$ are obtained via elementary rank-one matrix updates in $\mathcal{O}(np)$ time for $n\gg p$.