Least square problem constrained to projection matrices












4














Some times in engineering, it is important to find an optimum subspace in which projecting on it satisfies some properties.



Let known matrices $A$ and $B$ belong to $mathbb{R}^{ptimes n}$
and $|cdot|_F$ be Frobenius norm.
How can I find the best subspace in which projecting $A$ on it is as close as possible to $B$?
In other words how can I find a solution to the following constrained optimization problem?
begin{eqnarray}
&&min_P |PA-B|_F^2 \
&&mathrm{s.t. }P^T=P, , P^2=P
end{eqnarray}




Update:



I have incorporated the symmetry property ($P^T=T$) in objective function as follows.
Since $P$ is symmetric, $P$ can decompose as $P=Y+Y^T$ where $Yin mathbb{R}^{ntimes n}$. Now, the optimization problem reduces to
begin{eqnarray}
&&min_Y |C(Y+Y^T)-D|_F^2 \
&&mathrm{s.t. }, (Y+Y^T)^2=Y+Y^T,
end{eqnarray}
Where $C=A^T$ and $D=B^T$.



Further, using "vec" operator, we get
vec$(Y^T)=$$K$vec$(Y)$, where $Kin mathbb{R}^{ntimes n}$ is a unique and known matrix.
Using "Kronecker product", our optimization problem will be reduced to
begin{eqnarray}
&&min_y |(Iotimes C)(I+K)y-d|_2^2 \
&&mathrm{s.t. }, (Y+Y^T)^2=Y+Y^T,
end{eqnarray}
where $y=$ vec$(Y)$ and $d=$ vec$(D)$.






linear-algebra matrices linear-transformations numerical-linear-algebra projection







share|cite|improve this question
























  • Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc.
    – amWhy
    Jan 3 at 21:08






  • 1




    This is reminiscent of (but not equivalent to) the low-rank approximation problem
    – Omnomnomnom
    Jan 3 at 22:35








  • 1




    How do you get the equivalence? If I have $B = USigma V^*$ then $Vert PA-B Vert_F = Vert U^*PAV - Sigma Vert_F$ but I'm not sure $U^*P$ is a projection matrix.
    – tch
    Jan 3 at 22:42






  • 1




    Sorry, I meant how do you reduce it to the case where $A$ or $B$ is diagonal with non-negative entries.
    – tch
    Jan 3 at 22:46






  • 1




    @tch My mistake. We have $$ |PA - B|_F = |(U^*PU)(U^*AV) - Sigma|_F $$ and $U^*PU$ is a projection.
    – Omnomnomnom
    Jan 3 at 22:49
















4














Some times in engineering, it is important to find an optimum subspace in which projecting on it satisfies some properties.



Let known matrices $A$ and $B$ belong to $mathbb{R}^{ptimes n}$
and $|cdot|_F$ be Frobenius norm.
How can I find the best subspace in which projecting $A$ on it is as close as possible to $B$?
In other words how can I find a solution to the following constrained optimization problem?
begin{eqnarray}
&&min_P |PA-B|_F^2 \
&&mathrm{s.t. }P^T=P, , P^2=P
end{eqnarray}




Update:



I have incorporated the symmetry property ($P^T=T$) in objective function as follows.
Since $P$ is symmetric, $P$ can decompose as $P=Y+Y^T$ where $Yin mathbb{R}^{ntimes n}$. Now, the optimization problem reduces to
begin{eqnarray}
&&min_Y |C(Y+Y^T)-D|_F^2 \
&&mathrm{s.t. }, (Y+Y^T)^2=Y+Y^T,
end{eqnarray}
Where $C=A^T$ and $D=B^T$.



Further, using "vec" operator, we get
vec$(Y^T)=$$K$vec$(Y)$, where $Kin mathbb{R}^{ntimes n}$ is a unique and known matrix.
Using "Kronecker product", our optimization problem will be reduced to
begin{eqnarray}
&&min_y |(Iotimes C)(I+K)y-d|_2^2 \
&&mathrm{s.t. }, (Y+Y^T)^2=Y+Y^T,
end{eqnarray}
where $y=$ vec$(Y)$ and $d=$ vec$(D)$.






linear-algebra matrices linear-transformations numerical-linear-algebra projection







share|cite|improve this question
























  • Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc.
    – amWhy
    Jan 3 at 21:08






  • 1




    This is reminiscent of (but not equivalent to) the low-rank approximation problem
    – Omnomnomnom
    Jan 3 at 22:35








  • 1




    How do you get the equivalence? If I have $B = USigma V^*$ then $Vert PA-B Vert_F = Vert U^*PAV - Sigma Vert_F$ but I'm not sure $U^*P$ is a projection matrix.
    – tch
    Jan 3 at 22:42






  • 1




    Sorry, I meant how do you reduce it to the case where $A$ or $B$ is diagonal with non-negative entries.
    – tch
    Jan 3 at 22:46






  • 1




    @tch My mistake. We have $$ |PA - B|_F = |(U^*PU)(U^*AV) - Sigma|_F $$ and $U^*PU$ is a projection.
    – Omnomnomnom
    Jan 3 at 22:49














4












4








4


2





Some times in engineering, it is important to find an optimum subspace in which projecting on it satisfies some properties.



Let known matrices $A$ and $B$ belong to $mathbb{R}^{ptimes n}$
and $|cdot|_F$ be Frobenius norm.
How can I find the best subspace in which projecting $A$ on it is as close as possible to $B$?
In other words how can I find a solution to the following constrained optimization problem?
begin{eqnarray}
&&min_P |PA-B|_F^2 \
&&mathrm{s.t. }P^T=P, , P^2=P
end{eqnarray}




Update:



I have incorporated the symmetry property ($P^T=T$) in objective function as follows.
Since $P$ is symmetric, $P$ can decompose as $P=Y+Y^T$ where $Yin mathbb{R}^{ntimes n}$. Now, the optimization problem reduces to
begin{eqnarray}
&&min_Y |C(Y+Y^T)-D|_F^2 \
&&mathrm{s.t. }, (Y+Y^T)^2=Y+Y^T,
end{eqnarray}
Where $C=A^T$ and $D=B^T$.



Further, using "vec" operator, we get
vec$(Y^T)=$$K$vec$(Y)$, where $Kin mathbb{R}^{ntimes n}$ is a unique and known matrix.
Using "Kronecker product", our optimization problem will be reduced to
begin{eqnarray}
&&min_y |(Iotimes C)(I+K)y-d|_2^2 \
&&mathrm{s.t. }, (Y+Y^T)^2=Y+Y^T,
end{eqnarray}
where $y=$ vec$(Y)$ and $d=$ vec$(D)$.






linear-algebra matrices linear-transformations numerical-linear-algebra projection







share|cite|improve this question















Some times in engineering, it is important to find an optimum subspace in which projecting on it satisfies some properties.



Let known matrices $A$ and $B$ belong to $mathbb{R}^{ptimes n}$
and $|cdot|_F$ be Frobenius norm.
How can I find the best subspace in which projecting $A$ on it is as close as possible to $B$?
In other words how can I find a solution to the following constrained optimization problem?
begin{eqnarray}
&&min_P |PA-B|_F^2 \
&&mathrm{s.t. }P^T=P, , P^2=P
end{eqnarray}




Update:



I have incorporated the symmetry property ($P^T=T$) in objective function as follows.
Since $P$ is symmetric, $P$ can decompose as $P=Y+Y^T$ where $Yin mathbb{R}^{ntimes n}$. Now, the optimization problem reduces to
begin{eqnarray}
&&min_Y |C(Y+Y^T)-D|_F^2 \
&&mathrm{s.t. }, (Y+Y^T)^2=Y+Y^T,
end{eqnarray}
Where $C=A^T$ and $D=B^T$.



Further, using "vec" operator, we get
vec$(Y^T)=$$K$vec$(Y)$, where $Kin mathbb{R}^{ntimes n}$ is a unique and known matrix.
Using "Kronecker product", our optimization problem will be reduced to
begin{eqnarray}
&&min_y |(Iotimes C)(I+K)y-d|_2^2 \
&&mathrm{s.t. }, (Y+Y^T)^2=Y+Y^T,
end{eqnarray}
where $y=$ vec$(Y)$ and $d=$ vec$(D)$.






linear-algebra matrices linear-transformations numerical-linear-algebra projection




linear-algebra matrices linear-transformations numerical-linear-algebra projection






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited yesterday

























asked Jan 3 at 21:00









Bashir Sadeghi

324




324












  • Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc.
    – amWhy
    Jan 3 at 21:08






  • 1




    This is reminiscent of (but not equivalent to) the low-rank approximation problem
    – Omnomnomnom
    Jan 3 at 22:35








  • 1




    How do you get the equivalence? If I have $B = USigma V^*$ then $Vert PA-B Vert_F = Vert U^*PAV - Sigma Vert_F$ but I'm not sure $U^*P$ is a projection matrix.
    – tch
    Jan 3 at 22:42






  • 1




    Sorry, I meant how do you reduce it to the case where $A$ or $B$ is diagonal with non-negative entries.
    – tch
    Jan 3 at 22:46






  • 1




    @tch My mistake. We have $$ |PA - B|_F = |(U^*PU)(U^*AV) - Sigma|_F $$ and $U^*PU$ is a projection.
    – Omnomnomnom
    Jan 3 at 22:49


















  • Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc.
    – amWhy
    Jan 3 at 21:08






  • 1




    This is reminiscent of (but not equivalent to) the low-rank approximation problem
    – Omnomnomnom
    Jan 3 at 22:35








  • 1




    How do you get the equivalence? If I have $B = USigma V^*$ then $Vert PA-B Vert_F = Vert U^*PAV - Sigma Vert_F$ but I'm not sure $U^*P$ is a projection matrix.
    – tch
    Jan 3 at 22:42






  • 1




    Sorry, I meant how do you reduce it to the case where $A$ or $B$ is diagonal with non-negative entries.
    – tch
    Jan 3 at 22:46






  • 1




    @tch My mistake. We have $$ |PA - B|_F = |(U^*PU)(U^*AV) - Sigma|_F $$ and $U^*PU$ is a projection.
    – Omnomnomnom
    Jan 3 at 22:49
















Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc.
– amWhy
Jan 3 at 21:08




Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc.
– amWhy
Jan 3 at 21:08




1




1




This is reminiscent of (but not equivalent to) the low-rank approximation problem
– Omnomnomnom
Jan 3 at 22:35






This is reminiscent of (but not equivalent to) the low-rank approximation problem
– Omnomnomnom
Jan 3 at 22:35






1




1




How do you get the equivalence? If I have $B = USigma V^*$ then $Vert PA-B Vert_F = Vert U^*PAV - Sigma Vert_F$ but I'm not sure $U^*P$ is a projection matrix.
– tch
Jan 3 at 22:42




How do you get the equivalence? If I have $B = USigma V^*$ then $Vert PA-B Vert_F = Vert U^*PAV - Sigma Vert_F$ but I'm not sure $U^*P$ is a projection matrix.
– tch
Jan 3 at 22:42




1




1




Sorry, I meant how do you reduce it to the case where $A$ or $B$ is diagonal with non-negative entries.
– tch
Jan 3 at 22:46




Sorry, I meant how do you reduce it to the case where $A$ or $B$ is diagonal with non-negative entries.
– tch
Jan 3 at 22:46




1




1




@tch My mistake. We have $$ |PA - B|_F = |(U^*PU)(U^*AV) - Sigma|_F $$ and $U^*PU$ is a projection.
– Omnomnomnom
Jan 3 at 22:49




@tch My mistake. We have $$ |PA - B|_F = |(U^*PU)(U^*AV) - Sigma|_F $$ and $U^*PU$ is a projection.
– Omnomnomnom
Jan 3 at 22:49










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