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Research output: Contribution to journal › Journal article › Research › peer-review
For a matrix X ∈ R^{n×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 ∈ R^{n×p} is column-orthogonal and W ∈ R^{p×p} is invertible. Arguably, the most important methods that produce such factorizations are the singular value decomposition (SVD), where X = UW = U(ΣV ^{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 S = ran(X) to the subspace S_{new} = ran(X_{new}) = 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 O(np) time for n » p. Moreover, this allows us to determine the subspace distance and the Riemannian midpoint between the subspaces S and S_{new} without additional computational effort.
Original language | English |
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Journal | Mathematics of Computation |
Volume | 90 |
Issue number | 328 |
Pages (from-to) | 671-688 |
ISSN | 0025-5718 |
DOIs | |
Publication status | Published - 2021 |
Research output: Contribution to journal › Journal article › Research › peer-review