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

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Abstrakt

For a matrix X ∈ Rn×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 Xnew = X + abT of a given X with known matrix factorization X = UW, where U ∈ Rn×p is column-orthogonal and W ∈ Rp×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 Snew = ran(Xnew) = ran(UnewWnew). This leads to update formulas for orthogonal matrix decompositions, where both Unew and Wnew 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 Snew without additional computational effort.

OriginalsprogEngelsk
TidsskriftMathematics of Computation
Vol/bind90
Udgave nummer328
Sider (fra-til)671-688
ISSN0025-5718
DOI
StatusUdgivet - 2021

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