Matrix optimization problem for sparseness and closeness.
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If I have a matrix $$bf M= left[beginarraylbf M_k+epsilon\bf M_uendarrayright], casesbf M_k text known\bf M_u text unknown$$
How can I then find $bf epsilon$ (matrix) which is minimal (or regularized) in some norm and that $bf M_u$ and $bf M_k+epsilon$ is sparse, $bf M^-1$ exists and is also sparse?
linear-algebra reference-request soft-question nonlinear-optimization signal-processing
asked Sep 4 at 11:46
mathreadler
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up vote
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down vote
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If I have a matrix $$bf M= left[beginarraylbf M_k+epsilon\bf M_uendarrayright], casesbf M_k text known\bf M_u text unknown$$
How can I then find $bf epsilon$ (matrix) which is minimal (or regularized) in some norm and that $bf M_u$ and $bf M_k+epsilon$ is sparse, $bf M^-1$ exists and is also sparse?
linear-algebra reference-request soft-question nonlinear-optimization signal-processing
asked Sep 4 at 11:46
mathreadler
13.9k72058
Just to be clear, $M_k$ is fixed and $epsilon$ and $M_u$ are variables to be optimized over, right?
– Brian Borchers
Sep 4 at 14:29
@BrianBorchers Yes.
– mathreadler
Sep 4 at 14:47
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
If I have a matrix $$bf M= left[beginarraylbf M_k+epsilon\bf M_uendarrayright], casesbf M_k text known\bf M_u text unknown$$
How can I then find $bf epsilon$ (matrix) which is minimal (or regularized) in some norm and that $bf M_u$ and $bf M_k+epsilon$ is sparse, $bf M^-1$ exists and is also sparse?
linear-algebra reference-request soft-question nonlinear-optimization signal-processing
asked Sep 4 at 11:46
mathreadler
13.9k72058
If I have a matrix $$bf M= left[beginarraylbf M_k+epsilon\bf M_uendarrayright], casesbf M_k text known\bf M_u text unknown$$
How can I then find $bf epsilon$ (matrix) which is minimal (or regularized) in some norm and that $bf M_u$ and $bf M_k+epsilon$ is sparse, $bf M^-1$ exists and is also sparse?
linear-algebra reference-request soft-question nonlinear-optimization signal-processing
linear-algebra reference-request soft-question nonlinear-optimization signal-processing
asked Sep 4 at 11:46
mathreadler
13.9k72058
asked Sep 4 at 11:46
mathreadler
13.9k72058
edited Sep 8 at 16:03
asked Sep 4 at 11:46
mathreadler
13.9k72058
asked Sep 4 at 11:46
mathreadler
13.9k72058
asked Sep 4 at 11:46
mathreadler
13.9k72058
13.9k72058
Just to be clear, $M_k$ is fixed and $epsilon$ and $M_u$ are variables to be optimized over, right?
– Brian Borchers
Sep 4 at 14:29
@BrianBorchers Yes.
– mathreadler
Sep 4 at 14:47
add a comment |Â
Just to be clear, $M_k$ is fixed and $epsilon$ and $M_u$ are variables to be optimized over, right?
– Brian Borchers
Sep 4 at 14:29
@BrianBorchers Yes.
– mathreadler
Sep 4 at 14:47
Just to be clear, $M_k$ is fixed and $epsilon$ and $M_u$ are variables to be optimized over, right?
– Brian Borchers
Sep 4 at 14:29
Just to be clear, $M_k$ is fixed and $epsilon$ and $M_u$ are variables to be optimized over, right?
– Brian Borchers
Sep 4 at 14:29
@BrianBorchers Yes.
– mathreadler
Sep 4 at 14:47
@BrianBorchers Yes.
– mathreadler
Sep 4 at 14:47
add a comment |Â
1 Answer
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1
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There's a body of research on finding a sparse matrix that is an approximate inverse for a given matrix. This is used to find an approximate inverse for a sample covariance matrix which typically isn't of full rank and thus doesn't have an exact inverse.
See for example:
Sun, Tingni, and Cun-Hui Zhang. "Sparse matrix inversion with scaled lasso." The Journal of Machine Learning Research 14.1 (2013): 3385-3418.
Let
$U=left[
beginarrayc
M_k \
0
endarray
right]
$
Then find a sparse matrix $V$ such that
$UV approx I$.
Here "approximately" is measured by the spectral norm, $| UV - I |$. The $V$ matrix would then be your $M^-1$.
Note that although $UV$ is approximately $I$, this doesn't mean that $V^-1$ will be very close to $U$ or that $V^-1=M$ would necessarily be sparse.
answered Sep 4 at 16:08
Brian Borchers
5,24111119
Interesting. But this won't let us find $bf M_u$, will it?
– mathreadler
Sep 5 at 9:40
$epsilon$ and $M_u}$ would be the difference between $V^-1$ and $M_k$.
– Brian Borchers
Sep 5 at 11:54
But if I also want to put constraints on them? Is it handled in the paper?
– mathreadler
Sep 5 at 12:00
As I wrote in the answer, this doesn't give you a way to constrain the sparsity of $epsilon$ and $M_u$.
– Brian Borchers
Sep 5 at 12:08
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
There's a body of research on finding a sparse matrix that is an approximate inverse for a given matrix. This is used to find an approximate inverse for a sample covariance matrix which typically isn't of full rank and thus doesn't have an exact inverse.
See for example:
Sun, Tingni, and Cun-Hui Zhang. "Sparse matrix inversion with scaled lasso." The Journal of Machine Learning Research 14.1 (2013): 3385-3418.
Let
$U=left[
beginarrayc
M_k \
0
endarray
right]
$
Then find a sparse matrix $V$ such that
$UV approx I$.
Here "approximately" is measured by the spectral norm, $| UV - I |$. The $V$ matrix would then be your $M^-1$.
Note that although $UV$ is approximately $I$, this doesn't mean that $V^-1$ will be very close to $U$ or that $V^-1=M$ would necessarily be sparse.
answered Sep 4 at 16:08
Brian Borchers
5,24111119
Interesting. But this won't let us find $bf M_u$, will it?
– mathreadler
Sep 5 at 9:40
$epsilon$ and $M_u}$ would be the difference between $V^-1$ and $M_k$.
– Brian Borchers
Sep 5 at 11:54
But if I also want to put constraints on them? Is it handled in the paper?
– mathreadler
Sep 5 at 12:00
As I wrote in the answer, this doesn't give you a way to constrain the sparsity of $epsilon$ and $M_u$.
– Brian Borchers
Sep 5 at 12:08
add a comment |Â
up vote
1
down vote
There's a body of research on finding a sparse matrix that is an approximate inverse for a given matrix. This is used to find an approximate inverse for a sample covariance matrix which typically isn't of full rank and thus doesn't have an exact inverse.
See for example:
Sun, Tingni, and Cun-Hui Zhang. "Sparse matrix inversion with scaled lasso." The Journal of Machine Learning Research 14.1 (2013): 3385-3418.
Let
$U=left[
beginarrayc
M_k \
0
endarray
right]
$
Then find a sparse matrix $V$ such that
$UV approx I$.
Here "approximately" is measured by the spectral norm, $| UV - I |$. The $V$ matrix would then be your $M^-1$.
Note that although $UV$ is approximately $I$, this doesn't mean that $V^-1$ will be very close to $U$ or that $V^-1=M$ would necessarily be sparse.
answered Sep 4 at 16:08
Brian Borchers
5,24111119
Interesting. But this won't let us find $bf M_u$, will it?
– mathreadler
Sep 5 at 9:40
$epsilon$ and $M_u}$ would be the difference between $V^-1$ and $M_k$.
– Brian Borchers
Sep 5 at 11:54
But if I also want to put constraints on them? Is it handled in the paper?
– mathreadler
Sep 5 at 12:00
As I wrote in the answer, this doesn't give you a way to constrain the sparsity of $epsilon$ and $M_u$.
– Brian Borchers
Sep 5 at 12:08
add a comment |Â
up vote
1
down vote
up vote
1
down vote
There's a body of research on finding a sparse matrix that is an approximate inverse for a given matrix. This is used to find an approximate inverse for a sample covariance matrix which typically isn't of full rank and thus doesn't have an exact inverse.
See for example:
Sun, Tingni, and Cun-Hui Zhang. "Sparse matrix inversion with scaled lasso." The Journal of Machine Learning Research 14.1 (2013): 3385-3418.
Let
$U=left[
beginarrayc
M_k \
0
endarray
right]
$
Then find a sparse matrix $V$ such that
$UV approx I$.
Here "approximately" is measured by the spectral norm, $| UV - I |$. The $V$ matrix would then be your $M^-1$.
Note that although $UV$ is approximately $I$, this doesn't mean that $V^-1$ will be very close to $U$ or that $V^-1=M$ would necessarily be sparse.
answered Sep 4 at 16:08
Brian Borchers
5,24111119
There's a body of research on finding a sparse matrix that is an approximate inverse for a given matrix. This is used to find an approximate inverse for a sample covariance matrix which typically isn't of full rank and thus doesn't have an exact inverse.
See for example:
Sun, Tingni, and Cun-Hui Zhang. "Sparse matrix inversion with scaled lasso." The Journal of Machine Learning Research 14.1 (2013): 3385-3418.
Let
$U=left[
beginarrayc
M_k \
0
endarray
right]
$
Then find a sparse matrix $V$ such that
$UV approx I$.
Here "approximately" is measured by the spectral norm, $| UV - I |$. The $V$ matrix would then be your $M^-1$.
Note that although $UV$ is approximately $I$, this doesn't mean that $V^-1$ will be very close to $U$ or that $V^-1=M$ would necessarily be sparse.
answered Sep 4 at 16:08
Brian Borchers
5,24111119
answered Sep 4 at 16:08
Brian Borchers
5,24111119
answered Sep 4 at 16:08
Brian Borchers
5,24111119
answered Sep 4 at 16:08
Brian Borchers
5,24111119
5,24111119
Interesting. But this won't let us find $bf M_u$, will it?
– mathreadler
Sep 5 at 9:40
$epsilon$ and $M_u}$ would be the difference between $V^-1$ and $M_k$.
– Brian Borchers
Sep 5 at 11:54
But if I also want to put constraints on them? Is it handled in the paper?
– mathreadler
Sep 5 at 12:00
As I wrote in the answer, this doesn't give you a way to constrain the sparsity of $epsilon$ and $M_u$.
– Brian Borchers
Sep 5 at 12:08
add a comment |Â
Interesting. But this won't let us find $bf M_u$, will it?
– mathreadler
Sep 5 at 9:40
$epsilon$ and $M_u}$ would be the difference between $V^-1$ and $M_k$.
– Brian Borchers
Sep 5 at 11:54
But if I also want to put constraints on them? Is it handled in the paper?
– mathreadler
Sep 5 at 12:00
As I wrote in the answer, this doesn't give you a way to constrain the sparsity of $epsilon$ and $M_u$.
– Brian Borchers
Sep 5 at 12:08
Interesting. But this won't let us find $bf M_u$, will it?
– mathreadler
Sep 5 at 9:40
Interesting. But this won't let us find $bf M_u$, will it?
– mathreadler
Sep 5 at 9:40
$epsilon$ and $M_u}$ would be the difference between $V^-1$ and $M_k$.
– Brian Borchers
Sep 5 at 11:54
$epsilon$ and $M_u}$ would be the difference between $V^-1$ and $M_k$.
– Brian Borchers
Sep 5 at 11:54
But if I also want to put constraints on them? Is it handled in the paper?
– mathreadler
Sep 5 at 12:00
But if I also want to put constraints on them? Is it handled in the paper?
– mathreadler
Sep 5 at 12:00
As I wrote in the answer, this doesn't give you a way to constrain the sparsity of $epsilon$ and $M_u$.
– Brian Borchers
Sep 5 at 12:08
As I wrote in the answer, this doesn't give you a way to constrain the sparsity of $epsilon$ and $M_u$.
– Brian Borchers
Sep 5 at 12:08
add a comment |Â
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Just to be clear, $M_k$ is fixed and $epsilon$ and $M_u$ are variables to be optimized over, right?
– Brian Borchers
Sep 4 at 14:29
@BrianBorchers Yes.
– mathreadler
Sep 4 at 14:47