Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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 $n\gg p$.

28
The categories of this paper are: alg-geom
15
The categories of this paper are: nucl-ex
4
The categories of this paper are: cond-mat
36
The categories of this paper are: cmp-lg
11
The categories of this paper are: hep-lat
37
The categories of this paper are: supr-con
3
The categories of this paper are: physics
10
The categories of this paper are: q-bio
17
The categories of this paper are: q-fin
22
The categories of this paper are: chao-dyn
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The categories of this paper are: ao-sci
16
The categories of this paper are: stat
5
The categories of this paper are: gr-qc
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The categories of this paper are: bayes-an
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The categories of this paper are: mtrl-th
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The categories of this paper are: funct-an