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# Updating the singular value decomposition

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with small off-diagonal and distinct diagonal elements can be approximated using a direct scheme of R.

by using $\mathbf U$, $\mathbf S$, and $\mathbf V$), without recalculating SVD from scratch? Fast online SVD revisions for lightweight recommender systems by Brand is an accessible first paper.

I have not seen something for SVD already implemented in R unfortunately.

Let $A$ be a symmetric matrix with eigenvalue decomposition $UDU^T$. Has anything similar been done for the case where the update is of the form $A B$, where $B=uv^t vu^t$ is a rank-two symmetric matrix (note we can't just do two rank-one symmetric updates)?

Just to clarify what rank-one and rank-two mean so you do not get confused, if your new $A^*$ is such that: \begin A^* = A - uv^T \end Where $u$ and $v$ are vectors then you refer to this as a rank-one update (or ).

Note also that this means you are only given values for one in eighty five of the cells. Netflix has then posed a "quiz" which consists of a bunch of question marks plopped into previously blank slots, and your job is to fill in best-guess ratings in their place.

They have chosen mean squared error as the measure of accuracy, which means if you guess 1.5 and the actual rating was 2, you get docked for (2-1.5)^2 points, or 0.25.

What if we relaxed the insistence that $B$ be symmetric and asked instead for an efficient computation of the SVD of the update $A B$?

References or thoughts would be greatly appreciated.