A geometric approach to subspace updates and orthogonal matrix decompositions under rank-one modifications,Mathematics of Computation

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A geometric approach to subspace updates and orthogonal matrix decompositions under rank-one modifications
Mathematics of Computation ( IF 2 ) Pub Date : 2020-10-23 , DOI: 10.1090/mcom/3574
Ralf Zimmermann

Abstract: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

under rank-one modifications. More precisely, we consider rank-one adaptations $X_{new} = X+ab^T$ of a given

with known matrix factorization

, 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

. 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.

中文翻译：

秩修正下子空间更新和正交矩阵分解的一种几何方法

摘要：对于矩阵，我们提供了一个解析公式，该公式跟踪不足一级修改的范围的正交基础。更确切地说，我们考虑已知矩阵分解的给定的秩适应，其中列正交且可逆。可以说，产生这种分解的最重要的方法是奇异值分解（SVD），其中，以及QR分解，其中。我们给秩一适配的几何描述和导出用于从所述子空间行进的大地线的闭合形式的表达到子空间。这导致了正交矩阵分解的更新公式，其中 $X \ in \ mathbb {R} ^ {n \ times p}$

$X_ {new} = X + ab ^ T$

$U \ in \ mathbb {R} ^ {n \ times p}$ $W \ in \ mathbb {R} ^ {p \ times p}$ $X = UW = U（\ Sigma V ^ T）$

$\ mathcal {S} = {\ rm {ran}}（X）$ $\ mathcal {S} _ {new} = {\ rm {ran}}（X_ {new}）= {\ rm {ran}}（U_ {new} W_ {new}）$ $U_ {new}$ 并且 $W_ {new}$ 通过及时更新基本秩一矩阵获得。此外，这使我们能够确定的子空间距离和子空间之间的中点黎曼并没有额外的计算工作。 $\ mathcal {O}（np）$ $n \ gg p$ $\数学{S}$ $\ mathcal {S} _ {new}$

更新日期：2020-10-23

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