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 \mathbb{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}= {\rm {ran}}(X) to the subspace \mathcal {S}_{new} ={\rm {ran}}(X_{new}) ={\rm {ran}}(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.