Promote the Orthogonality between Rows of Matrix $ S $
$begingroup$
Suppose we want to solve the following optimization problem in $S in mathbb{R}^{N times T}$, where $T gg S$,
$$min_{S} f(S) mbox{ subject to } SS^T mbox{is diagonal}$$
which means rows of the matrix $S$ are mutually orthogonal.
I am suggested to solve this alternative problem by the following method:
$$min_{S} f(S) + |mathcal{P}(SS^T)|_1$$
in which $mathcal{P}$ is a projection onto the off-diagonal indexes.
But I don't this $ell_1$ penalty will promote the orthogonality between each rows by simply promote the sparsity of the off-diagonal elements, since we are not doing any actions like block coordinate descent to promote the orthogonality.
Any suggestions?
optimization convex-optimization
edited Feb 3 at 4:07
Royi
3,52012352
asked Dec 21 '18 at 4:25
Z-HarlpetZ-Harlpet
327
$endgroup$
add a comment |
$begingroup$
Suppose we want to solve the following optimization problem in $S in mathbb{R}^{N times T}$, where $T gg S$,
$$min_{S} f(S) mbox{ subject to } SS^T mbox{is diagonal}$$
which means rows of the matrix $S$ are mutually orthogonal.
I am suggested to solve this alternative problem by the following method:
$$min_{S} f(S) + |mathcal{P}(SS^T)|_1$$
in which $mathcal{P}$ is a projection onto the off-diagonal indexes.
But I don't this $ell_1$ penalty will promote the orthogonality between each rows by simply promote the sparsity of the off-diagonal elements, since we are not doing any actions like block coordinate descent to promote the orthogonality.
Any suggestions?
optimization convex-optimization
edited Feb 3 at 4:07
Royi
3,52012352
asked Dec 21 '18 at 4:25
Z-HarlpetZ-Harlpet
327
$endgroup$
$begingroup$
What do you know about $f$? Why not introduce $n (n-1)$ quadratic equality constraints?
$endgroup$
– Rodrigo de Azevedo
Jan 19 at 16:58
$begingroup$
I think you should just rephrase the question into something like "Orthogonal Projection onto Semi Orthogonal Matrix Space". This way you'll get answers.
$endgroup$
– Royi
Feb 3 at 4:09
add a comment |
$begingroup$
Suppose we want to solve the following optimization problem in $S in mathbb{R}^{N times T}$, where $T gg S$,
$$min_{S} f(S) mbox{ subject to } SS^T mbox{is diagonal}$$
which means rows of the matrix $S$ are mutually orthogonal.
I am suggested to solve this alternative problem by the following method:
$$min_{S} f(S) + |mathcal{P}(SS^T)|_1$$
in which $mathcal{P}$ is a projection onto the off-diagonal indexes.
But I don't this $ell_1$ penalty will promote the orthogonality between each rows by simply promote the sparsity of the off-diagonal elements, since we are not doing any actions like block coordinate descent to promote the orthogonality.
Any suggestions?
optimization convex-optimization
edited Feb 3 at 4:07
Royi
3,52012352
asked Dec 21 '18 at 4:25
Z-HarlpetZ-Harlpet
327
$endgroup$
Suppose we want to solve the following optimization problem in $S in mathbb{R}^{N times T}$, where $T gg S$,
$$min_{S} f(S) mbox{ subject to } SS^T mbox{is diagonal}$$
which means rows of the matrix $S$ are mutually orthogonal.
I am suggested to solve this alternative problem by the following method:
$$min_{S} f(S) + |mathcal{P}(SS^T)|_1$$
in which $mathcal{P}$ is a projection onto the off-diagonal indexes.
But I don't this $ell_1$ penalty will promote the orthogonality between each rows by simply promote the sparsity of the off-diagonal elements, since we are not doing any actions like block coordinate descent to promote the orthogonality.
Any suggestions?
optimization convex-optimization
optimization convex-optimization
edited Feb 3 at 4:07
Royi
3,52012352
asked Dec 21 '18 at 4:25
Z-HarlpetZ-Harlpet
327
edited Feb 3 at 4:07
Royi
3,52012352
asked Dec 21 '18 at 4:25
Z-HarlpetZ-Harlpet
327
edited Feb 3 at 4:07
Royi
3,52012352
edited Feb 3 at 4:07
Royi
3,52012352
edited Feb 3 at 4:07
Royi
3,52012352
3,52012352
asked Dec 21 '18 at 4:25
Z-HarlpetZ-Harlpet
327
asked Dec 21 '18 at 4:25
Z-HarlpetZ-Harlpet
327
asked Dec 21 '18 at 4:25
Z-HarlpetZ-Harlpet
327
327
$begingroup$
What do you know about $f$? Why not introduce $n (n-1)$ quadratic equality constraints?
$endgroup$
– Rodrigo de Azevedo
Jan 19 at 16:58
$begingroup$
I think you should just rephrase the question into something like "Orthogonal Projection onto Semi Orthogonal Matrix Space". This way you'll get answers.
$endgroup$
– Royi
Feb 3 at 4:09
add a comment |
$begingroup$
What do you know about $f$? Why not introduce $n (n-1)$ quadratic equality constraints?
$endgroup$
– Rodrigo de Azevedo
Jan 19 at 16:58
$begingroup$
I think you should just rephrase the question into something like "Orthogonal Projection onto Semi Orthogonal Matrix Space". This way you'll get answers.
$endgroup$
– Royi
Feb 3 at 4:09
$begingroup$
What do you know about $f$? Why not introduce $n (n-1)$ quadratic equality constraints?
$endgroup$
– Rodrigo de Azevedo
Jan 19 at 16:58
$begingroup$
What do you know about $f$? Why not introduce $n (n-1)$ quadratic equality constraints?
$endgroup$
– Rodrigo de Azevedo
Jan 19 at 16:58
$begingroup$
I think you should just rephrase the question into something like "Orthogonal Projection onto Semi Orthogonal Matrix Space". This way you'll get answers.
$endgroup$
– Royi
Feb 3 at 4:09
$begingroup$
I think you should just rephrase the question into something like "Orthogonal Projection onto Semi Orthogonal Matrix Space". This way you'll get answers.
$endgroup$
– Royi
Feb 3 at 4:09
add a comment |
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$begingroup$
What do you know about $f$? Why not introduce $n (n-1)$ quadratic equality constraints?
$endgroup$
– Rodrigo de Azevedo
Jan 19 at 16:58
$begingroup$
I think you should just rephrase the question into something like "Orthogonal Projection onto Semi Orthogonal Matrix Space". This way you'll get answers.
$endgroup$
– Royi
Feb 3 at 4:09