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Given two side matrices $P$ and $Q$, extract (find) the diagonal scaling matrix $Sigma$ of a singular value decomposition

0

$begingroup$

I have an application where I have already approximated a given matrix $R$ of size $m times n$ by multiplying two matrices $P$ and $Q^mathrm T$: $hat R=PQ^mathrm T$. $P$ is size $m times k$ and $Q$ is size $n times k$ and $Q^mathrm T$ is size $k times n$. I desire now to use these two matrices to efficiently as possible find a proper singular value decomposition, which has three matrices as you know.

What gives me great hope is that Simon Funk said here that "The end result, it's worth noting, is exactly an SVD if the training set perfectly covers the matrix. Call it what you will when it doesn't. [If you're wondering where the diagonal scaling matrix is, it gets arbitrarily rolled in to the two side matrices, but could be trivially extracted if needed.]"

Can someone describe and detail the trivial extraction process he talked about which I can use to find that third matrix $Sigma$ in the famous SVD equation $hat R = U Sigma V^mathrm T$?

Never mind FunkSVD, as I am not using that algorithm currently, but I do have a pretty well-estimated pair of matrices $P$ and $Q$ as my starting point. I used a gradient descent and machine learning to get $P$ and $Q$ already.

I am required to NOT run SVD from scratch -- instead I must do something very efficient to "trivially extract" the sigma matrix, when given "two side matrices", which dear Mr. Funk said is possible.

Thanks for contributions if any!

matrix-decomposition

share|cite|improve this question

edited Mar 23 at 17:26

Rócherz

3,0263823

asked Aug 2 '18 at 19:30

Geoffrey AndersonGeoffrey Anderson

1012

$endgroup$

add a comment | 

0

$begingroup$

I have an application where I have already approximated a given matrix $R$ of size $m times n$ by multiplying two matrices $P$ and $Q^mathrm T$: $hat R=PQ^mathrm T$. $P$ is size $m times k$ and $Q$ is size $n times k$ and $Q^mathrm T$ is size $k times n$. I desire now to use these two matrices to efficiently as possible find a proper singular value decomposition, which has three matrices as you know.

What gives me great hope is that Simon Funk said here that "The end result, it's worth noting, is exactly an SVD if the training set perfectly covers the matrix. Call it what you will when it doesn't. [If you're wondering where the diagonal scaling matrix is, it gets arbitrarily rolled in to the two side matrices, but could be trivially extracted if needed.]"

Can someone describe and detail the trivial extraction process he talked about which I can use to find that third matrix $Sigma$ in the famous SVD equation $hat R = U Sigma V^mathrm T$?

Never mind FunkSVD, as I am not using that algorithm currently, but I do have a pretty well-estimated pair of matrices $P$ and $Q$ as my starting point. I used a gradient descent and machine learning to get $P$ and $Q$ already.

I am required to NOT run SVD from scratch -- instead I must do something very efficient to "trivially extract" the sigma matrix, when given "two side matrices", which dear Mr. Funk said is possible.

Thanks for contributions if any!

matrix-decomposition

share|cite|improve this question

edited Mar 23 at 17:26

Rócherz

3,0263823

asked Aug 2 '18 at 19:30

Geoffrey AndersonGeoffrey Anderson

1012

$endgroup$

add a comment | 

$begingroup$

I have an application where I have already approximated a given matrix $R$ of size $m times n$ by multiplying two matrices $P$ and $Q^mathrm T$: $hat R=PQ^mathrm T$. $P$ is size $m times k$ and $Q$ is size $n times k$ and $Q^mathrm T$ is size $k times n$. I desire now to use these two matrices to efficiently as possible find a proper singular value decomposition, which has three matrices as you know.

What gives me great hope is that Simon Funk said here that "The end result, it's worth noting, is exactly an SVD if the training set perfectly covers the matrix. Call it what you will when it doesn't. [If you're wondering where the diagonal scaling matrix is, it gets arbitrarily rolled in to the two side matrices, but could be trivially extracted if needed.]"

Can someone describe and detail the trivial extraction process he talked about which I can use to find that third matrix $Sigma$ in the famous SVD equation $hat R = U Sigma V^mathrm T$?

Never mind FunkSVD, as I am not using that algorithm currently, but I do have a pretty well-estimated pair of matrices $P$ and $Q$ as my starting point. I used a gradient descent and machine learning to get $P$ and $Q$ already.

I am required to NOT run SVD from scratch -- instead I must do something very efficient to "trivially extract" the sigma matrix, when given "two side matrices", which dear Mr. Funk said is possible.

Thanks for contributions if any!

matrix-decomposition

share|cite|improve this question

edited Mar 23 at 17:26

Rócherz

3,0263823

asked Aug 2 '18 at 19:30

Geoffrey AndersonGeoffrey Anderson

1012

$endgroup$

share|cite|improve this question

edited Mar 23 at 17:26

Rócherz

3,0263823

asked Aug 2 '18 at 19:30

Geoffrey AndersonGeoffrey Anderson

1012

share|cite|improve this question

edited Mar 23 at 17:26

Rócherz

3,0263823

asked Aug 2 '18 at 19:30

Geoffrey AndersonGeoffrey Anderson

1012

edited Mar 23 at 17:26

Rócherz

3,0263823

edited Mar 23 at 17:26

Rócherz

3,0263823

asked Aug 2 '18 at 19:30

Geoffrey AndersonGeoffrey Anderson

1012

asked Aug 2 '18 at 19:30

Geoffrey AndersonGeoffrey Anderson

1012