Minimizing the operator norm induced by $ell _{2}$ and $ell _{infty}$-norm for vectors to solve for diagonal...
$begingroup$
Given an operator represented in matrix form as $B$ of size $ntimes m$ such that $m < n$ and square matrix A of size $m times m$, how do we minimize the operator norms induced by $ell _{2}$ and $ell _{infty}$-norm for vectors to solve for diagonal matrix D?
$$min_{D} Vert A - B^HDB Vert_{op,2}$$
$$min_{D} Vert A - B^HDB Vert_{op, infty}$$
My approach: Does it make sense to use vec-operator identity i.e., $$vec(B^HDB) = (B^Totimes B^H)vec(D) $$ in the beginning and then proceed with the minimization as it might tranform my problem into a standard optimization problem: $$min_{x}Vert b- Hx Vert_N $$
such that,
begin{align}
H &= (B^Totimes B^H)\
x &= vec(D)\
b &= vec(A)\
N &= norm(any)
end{align}
I am not sure that this is the right approach as for any matrix $P $ and its vectorized form $vec(P)$, the matrix and vecotor 2-norms are not equal, i.e., $$ Vert P Vert_{op,2} neq Vert vec(P) Vert_{2} = Vert vec(P) Vert_{F}$$
Also,$$ Vert P Vert_{op,infty} neq Vert vec(P) Vert_{infty} $$
Note: $Vert .Vert_{F}$ implies frobenius norm, $Vert .Vert_{OP,infty}$ implies operator norm induced by $ell _{infty}$- and $Vert .Vert_{OP,2}$ implies operator norm induced by $ell _{2}$-norm for vectors.
functional-analysis operator-theory operator-algebras
asked Jan 3 at 11:43
Christina Christina
61
$endgroup$
add a comment |
$begingroup$
Given an operator represented in matrix form as $B$ of size $ntimes m$ such that $m < n$ and square matrix A of size $m times m$, how do we minimize the operator norms induced by $ell _{2}$ and $ell _{infty}$-norm for vectors to solve for diagonal matrix D?
$$min_{D} Vert A - B^HDB Vert_{op,2}$$
$$min_{D} Vert A - B^HDB Vert_{op, infty}$$
My approach: Does it make sense to use vec-operator identity i.e., $$vec(B^HDB) = (B^Totimes B^H)vec(D) $$ in the beginning and then proceed with the minimization as it might tranform my problem into a standard optimization problem: $$min_{x}Vert b- Hx Vert_N $$
such that,
begin{align}
H &= (B^Totimes B^H)\
x &= vec(D)\
b &= vec(A)\
N &= norm(any)
end{align}
I am not sure that this is the right approach as for any matrix $P $ and its vectorized form $vec(P)$, the matrix and vecotor 2-norms are not equal, i.e., $$ Vert P Vert_{op,2} neq Vert vec(P) Vert_{2} = Vert vec(P) Vert_{F}$$
Also,$$ Vert P Vert_{op,infty} neq Vert vec(P) Vert_{infty} $$
Note: $Vert .Vert_{F}$ implies frobenius norm, $Vert .Vert_{OP,infty}$ implies operator norm induced by $ell _{infty}$- and $Vert .Vert_{OP,2}$ implies operator norm induced by $ell _{2}$-norm for vectors.
functional-analysis operator-theory operator-algebras
asked Jan 3 at 11:43
Christina Christina
61
$endgroup$
add a comment |
$begingroup$
Given an operator represented in matrix form as $B$ of size $ntimes m$ such that $m < n$ and square matrix A of size $m times m$, how do we minimize the operator norms induced by $ell _{2}$ and $ell _{infty}$-norm for vectors to solve for diagonal matrix D?
$$min_{D} Vert A - B^HDB Vert_{op,2}$$
$$min_{D} Vert A - B^HDB Vert_{op, infty}$$
My approach: Does it make sense to use vec-operator identity i.e., $$vec(B^HDB) = (B^Totimes B^H)vec(D) $$ in the beginning and then proceed with the minimization as it might tranform my problem into a standard optimization problem: $$min_{x}Vert b- Hx Vert_N $$
such that,
begin{align}
H &= (B^Totimes B^H)\
x &= vec(D)\
b &= vec(A)\
N &= norm(any)
end{align}
I am not sure that this is the right approach as for any matrix $P $ and its vectorized form $vec(P)$, the matrix and vecotor 2-norms are not equal, i.e., $$ Vert P Vert_{op,2} neq Vert vec(P) Vert_{2} = Vert vec(P) Vert_{F}$$
Also,$$ Vert P Vert_{op,infty} neq Vert vec(P) Vert_{infty} $$
Note: $Vert .Vert_{F}$ implies frobenius norm, $Vert .Vert_{OP,infty}$ implies operator norm induced by $ell _{infty}$- and $Vert .Vert_{OP,2}$ implies operator norm induced by $ell _{2}$-norm for vectors.
functional-analysis operator-theory operator-algebras
asked Jan 3 at 11:43
Christina Christina
61
$endgroup$
Given an operator represented in matrix form as $B$ of size $ntimes m$ such that $m < n$ and square matrix A of size $m times m$, how do we minimize the operator norms induced by $ell _{2}$ and $ell _{infty}$-norm for vectors to solve for diagonal matrix D?
$$min_{D} Vert A - B^HDB Vert_{op,2}$$
$$min_{D} Vert A - B^HDB Vert_{op, infty}$$
My approach: Does it make sense to use vec-operator identity i.e., $$vec(B^HDB) = (B^Totimes B^H)vec(D) $$ in the beginning and then proceed with the minimization as it might tranform my problem into a standard optimization problem: $$min_{x}Vert b- Hx Vert_N $$
such that,
begin{align}
H &= (B^Totimes B^H)\
x &= vec(D)\
b &= vec(A)\
N &= norm(any)
end{align}
I am not sure that this is the right approach as for any matrix $P $ and its vectorized form $vec(P)$, the matrix and vecotor 2-norms are not equal, i.e., $$ Vert P Vert_{op,2} neq Vert vec(P) Vert_{2} = Vert vec(P) Vert_{F}$$
Also,$$ Vert P Vert_{op,infty} neq Vert vec(P) Vert_{infty} $$
Note: $Vert .Vert_{F}$ implies frobenius norm, $Vert .Vert_{OP,infty}$ implies operator norm induced by $ell _{infty}$- and $Vert .Vert_{OP,2}$ implies operator norm induced by $ell _{2}$-norm for vectors.
functional-analysis operator-theory operator-algebras
functional-analysis operator-theory operator-algebras
asked Jan 3 at 11:43
Christina Christina
61
asked Jan 3 at 11:43
Christina Christina
61
asked Jan 3 at 11:43
Christina Christina
61
asked Jan 3 at 11:43
Christina Christina
61
asked Jan 3 at 11:43
Christina Christina
61
61
add a comment |
add a comment |
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