Least square problem constrained to projection matrices
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)=$$K$vec$(Y)$, where $Kin mathbb{R}^{ntimes n}$ is a unique and known matrix.
Using "Kronecker product", our optimization problem will be reduced to
begin{eqnarray}
&&min_y |(Iotimes C)(I+K)y-d|_2^2 \
&&mathrm{s.t. }, (Y+Y^T)^2=Y+Y^T,
end{eqnarray}
where $y=$ vec$(Y)$ and $d=$ vec$(D)$.
linear-algebra matrices linear-transformations numerical-linear-algebra projection
asked Jan 3 at 21:00
Bashir Sadeghi
324
|
show 4 more comments
Some times in engineering, it is important to find an optimum subspace in which projecting on it satisfies some properties.
Let known matrices $A$ and $B$ belong to $mathbb{R}^{ptimes n}$
and $|cdot|_F$ be Frobenius norm.
How can I find the best subspace in which projecting $A$ on it is as close as possible to $B$?
In other words how can I find a solution to the following constrained optimization problem?
begin{eqnarray}
&&min_P |PA-B|_F^2 \
&&mathrm{s.t. }P^T=P, , P^2=P
end{eqnarray}
Update:
I have incorporated the symmetry property ($P^T=T$) in objective function as follows.
Since $P$ is symmetric, $P$ can decompose as $P=Y+Y^T$ where $Yin mathbb{R}^{ntimes n}$. Now, the optimization problem reduces to
begin{eqnarray}
&&min_Y |C(Y+Y^T)-D|_F^2 \
&&mathrm{s.t. }, (Y+Y^T)^2=Y+Y^T,
end{eqnarray}
Where $C=A^T$ and $D=B^T$.
Further, using "vec" operator, we get
vec$(Y^T)=$$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
asked Jan 3 at 21:00
Bashir Sadeghi
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
|
show 4 more comments
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)=$$K$vec$(Y)$, where $Kin mathbb{R}^{ntimes n}$ is a unique and known matrix.
Using "Kronecker product", our optimization problem will be reduced to
begin{eqnarray}
&&min_y |(Iotimes C)(I+K)y-d|_2^2 \
&&mathrm{s.t. }, (Y+Y^T)^2=Y+Y^T,
end{eqnarray}
where $y=$ vec$(Y)$ and $d=$ vec$(D)$.
linear-algebra matrices linear-transformations numerical-linear-algebra projection
asked Jan 3 at 21:00
Bashir Sadeghi
324
Some times in engineering, it is important to find an optimum subspace in which projecting on it satisfies some properties.
Let known matrices $A$ and $B$ belong to $mathbb{R}^{ptimes n}$
and $|cdot|_F$ be Frobenius norm.
How can I find the best subspace in which projecting $A$ on it is as close as possible to $B$?
In other words how can I find a solution to the following constrained optimization problem?
begin{eqnarray}
&&min_P |PA-B|_F^2 \
&&mathrm{s.t. }P^T=P, , P^2=P
end{eqnarray}
Update:
I have incorporated the symmetry property ($P^T=T$) in objective function as follows.
Since $P$ is symmetric, $P$ can decompose as $P=Y+Y^T$ where $Yin mathbb{R}^{ntimes n}$. Now, the optimization problem reduces to
begin{eqnarray}
&&min_Y |C(Y+Y^T)-D|_F^2 \
&&mathrm{s.t. }, (Y+Y^T)^2=Y+Y^T,
end{eqnarray}
Where $C=A^T$ and $D=B^T$.
Further, using "vec" operator, we get
vec$(Y^T)=$$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
asked Jan 3 at 21:00
Bashir Sadeghi
324
asked Jan 3 at 21:00
Bashir Sadeghi
324
edited yesterday
asked Jan 3 at 21:00
Bashir Sadeghi
324
asked Jan 3 at 21:00
Bashir Sadeghi
324
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
|
show 4 more comments
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
|
show 4 more comments
0
active
oldest
votes
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3060997%2fleast-square-problem-constrained-to-projection-matrices%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3060997%2fleast-square-problem-constrained-to-projection-matrices%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
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