Different norms in stacked form optimisation
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I'm trying to generalise the stacked form of a minimisation problem:
$$text{argmin}_x||Ax-y||_p^p+alpha||Dx||_q^q$$
where the L2 norm is often used, so $p=q=2$. This can be brought to
$$text{argmin}_xleft|left|begin{bmatrix}A\sqrt{alpha}Dend{bmatrix}x-begin{bmatrix}y\0end{bmatrix}right|right|_2^2$$
And the left side being equal to the right side would optimise the situation, so taking both the square and the norm out, we are left with a basic matrix equation.
However, if one was to use different norms, that is $pnot=q$, I imagine the solution should change accordingly.
So given the minimisation, for example
$$text{argmin}_x||Ax-y||_2^2+alpha||Dx||_1$$
how would one start deriving the stacked form?
I tried to search for "optimisation stacked form" and some similar things, but didn't find much. My terminology may be a bit off. Maybe there's a name for this sort of approach?
matrices optimization
asked Nov 22 at 19:57
Felix
1558
add a comment |
up vote
0
down vote
favorite
I'm trying to generalise the stacked form of a minimisation problem:
$$text{argmin}_x||Ax-y||_p^p+alpha||Dx||_q^q$$
where the L2 norm is often used, so $p=q=2$. This can be brought to
$$text{argmin}_xleft|left|begin{bmatrix}A\sqrt{alpha}Dend{bmatrix}x-begin{bmatrix}y\0end{bmatrix}right|right|_2^2$$
And the left side being equal to the right side would optimise the situation, so taking both the square and the norm out, we are left with a basic matrix equation.
However, if one was to use different norms, that is $pnot=q$, I imagine the solution should change accordingly.
So given the minimisation, for example
$$text{argmin}_x||Ax-y||_2^2+alpha||Dx||_1$$
how would one start deriving the stacked form?
I tried to search for "optimisation stacked form" and some similar things, but didn't find much. My terminology may be a bit off. Maybe there's a name for this sort of approach?
matrices optimization
asked Nov 22 at 19:57
Felix
1558
1
I doubt this is possible for $pne q$.
– daw
Nov 22 at 20:34
If you think the minimum will be 0 then you can always write the system of equations $Ax=y$ and $Dx=0$. But in practice one uses regression for very overdetermined systems.
– Michal Adamaszek
Nov 23 at 7:40
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I'm trying to generalise the stacked form of a minimisation problem:
$$text{argmin}_x||Ax-y||_p^p+alpha||Dx||_q^q$$
where the L2 norm is often used, so $p=q=2$. This can be brought to
$$text{argmin}_xleft|left|begin{bmatrix}A\sqrt{alpha}Dend{bmatrix}x-begin{bmatrix}y\0end{bmatrix}right|right|_2^2$$
And the left side being equal to the right side would optimise the situation, so taking both the square and the norm out, we are left with a basic matrix equation.
However, if one was to use different norms, that is $pnot=q$, I imagine the solution should change accordingly.
So given the minimisation, for example
$$text{argmin}_x||Ax-y||_2^2+alpha||Dx||_1$$
how would one start deriving the stacked form?
I tried to search for "optimisation stacked form" and some similar things, but didn't find much. My terminology may be a bit off. Maybe there's a name for this sort of approach?
matrices optimization
asked Nov 22 at 19:57
Felix
1558
I'm trying to generalise the stacked form of a minimisation problem:
$$text{argmin}_x||Ax-y||_p^p+alpha||Dx||_q^q$$
where the L2 norm is often used, so $p=q=2$. This can be brought to
$$text{argmin}_xleft|left|begin{bmatrix}A\sqrt{alpha}Dend{bmatrix}x-begin{bmatrix}y\0end{bmatrix}right|right|_2^2$$
And the left side being equal to the right side would optimise the situation, so taking both the square and the norm out, we are left with a basic matrix equation.
However, if one was to use different norms, that is $pnot=q$, I imagine the solution should change accordingly.
So given the minimisation, for example
$$text{argmin}_x||Ax-y||_2^2+alpha||Dx||_1$$
how would one start deriving the stacked form?
I tried to search for "optimisation stacked form" and some similar things, but didn't find much. My terminology may be a bit off. Maybe there's a name for this sort of approach?
matrices optimization
matrices optimization
asked Nov 22 at 19:57
Felix
1558
asked Nov 22 at 19:57
Felix
1558
edited Nov 22 at 20:04
asked Nov 22 at 19:57
Felix
1558
asked Nov 22 at 19:57
Felix
1558
asked Nov 22 at 19:57
Felix
1558
1558
1
I doubt this is possible for $pne q$.
– daw
Nov 22 at 20:34
If you think the minimum will be 0 then you can always write the system of equations $Ax=y$ and $Dx=0$. But in practice one uses regression for very overdetermined systems.
– Michal Adamaszek
Nov 23 at 7:40
add a comment |
1
I doubt this is possible for $pne q$.
– daw
Nov 22 at 20:34
If you think the minimum will be 0 then you can always write the system of equations $Ax=y$ and $Dx=0$. But in practice one uses regression for very overdetermined systems.
– Michal Adamaszek
Nov 23 at 7:40
1
1
I doubt this is possible for $pne q$.
– daw
Nov 22 at 20:34
I doubt this is possible for $pne q$.
– daw
Nov 22 at 20:34
If you think the minimum will be 0 then you can always write the system of equations $Ax=y$ and $Dx=0$. But in practice one uses regression for very overdetermined systems.
– Michal Adamaszek
Nov 23 at 7:40
If you think the minimum will be 0 then you can always write the system of equations $Ax=y$ and $Dx=0$. But in practice one uses regression for very overdetermined systems.
– Michal Adamaszek
Nov 23 at 7:40
add a comment |
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1
I doubt this is possible for $pne q$.
– daw
Nov 22 at 20:34
If you think the minimum will be 0 then you can always write the system of equations $Ax=y$ and $Dx=0$. But in practice one uses regression for very overdetermined systems.
– Michal Adamaszek
Nov 23 at 7:40