Does bilinear models on vectors mean dot or outer product?
$begingroup$
If I have 2 vectors $x$ and $y$ where $x in mathcal{R}^{m}$ and $y in mathcal{R}^{n}$.
Does bilinear model mean?
$f(x,y) = x^TWy$ where $W in mathcal{R}^{m*n}$
which result in a scalar
or
$f(x,y) = W(x⊗y^T)$ where ⊗ is the outer
product and $W in mathcal{R}^{m*n}$.
which result in a matrix
I checked 2 papers, the first one Low-rank Bilinear Pooling
in page 2 in equation 1 their bilinear model produce a scalar
while in Compact Bilinear Pooling in section 3.1 they said "Bilinear models take the outer product of two vectors"
linear-algebra matrices
asked Dec 19 '18 at 11:53
floydfloyd
1032
$endgroup$
add a comment |
$begingroup$
If I have 2 vectors $x$ and $y$ where $x in mathcal{R}^{m}$ and $y in mathcal{R}^{n}$.
Does bilinear model mean?
$f(x,y) = x^TWy$ where $W in mathcal{R}^{m*n}$
which result in a scalar
or
$f(x,y) = W(x⊗y^T)$ where ⊗ is the outer
product and $W in mathcal{R}^{m*n}$.
which result in a matrix
I checked 2 papers, the first one Low-rank Bilinear Pooling
in page 2 in equation 1 their bilinear model produce a scalar
while in Compact Bilinear Pooling in section 3.1 they said "Bilinear models take the outer product of two vectors"
linear-algebra matrices
asked Dec 19 '18 at 11:53
floydfloyd
1032
$endgroup$
add a comment |
$begingroup$
If I have 2 vectors $x$ and $y$ where $x in mathcal{R}^{m}$ and $y in mathcal{R}^{n}$.
Does bilinear model mean?
$f(x,y) = x^TWy$ where $W in mathcal{R}^{m*n}$
which result in a scalar
or
$f(x,y) = W(x⊗y^T)$ where ⊗ is the outer
product and $W in mathcal{R}^{m*n}$.
which result in a matrix
I checked 2 papers, the first one Low-rank Bilinear Pooling
in page 2 in equation 1 their bilinear model produce a scalar
while in Compact Bilinear Pooling in section 3.1 they said "Bilinear models take the outer product of two vectors"
linear-algebra matrices
asked Dec 19 '18 at 11:53
floydfloyd
1032
$endgroup$
If I have 2 vectors $x$ and $y$ where $x in mathcal{R}^{m}$ and $y in mathcal{R}^{n}$.
Does bilinear model mean?
$f(x,y) = x^TWy$ where $W in mathcal{R}^{m*n}$
which result in a scalar
or
$f(x,y) = W(x⊗y^T)$ where ⊗ is the outer
product and $W in mathcal{R}^{m*n}$.
which result in a matrix
I checked 2 papers, the first one Low-rank Bilinear Pooling
in page 2 in equation 1 their bilinear model produce a scalar
while in Compact Bilinear Pooling in section 3.1 they said "Bilinear models take the outer product of two vectors"
linear-algebra matrices
linear-algebra matrices
asked Dec 19 '18 at 11:53
floydfloyd
1032
asked Dec 19 '18 at 11:53
floydfloyd
1032
asked Dec 19 '18 at 11:53
floydfloyd
1032
asked Dec 19 '18 at 11:53
floydfloyd
1032
asked Dec 19 '18 at 11:53
floydfloyd
1032
1032
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
"Bilinear" is simply an adjective that you can apply to any function of two vectors to indicate that it is linear in each argument. So the linear map
$(x,y) rightarrow x^TWy$
and the tensor product
$(x,y) rightarrow xy^T$
can both be described as bilinear, even though their codomains are different.
answered Dec 19 '18 at 12:16
gandalf61gandalf61
8,826725
$endgroup$
$begingroup$
What about the element-wise product between 2 vectors? The reason I mention it is because it's different than the dot and outer product because it's doesn't take into account all the interactions between the 2 vectors (like what the dot product does).
$endgroup$
– floyd
Jan 7 at 20:20
add a comment |
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%2f3046295%2fdoes-bilinear-models-on-vectors-mean-dot-or-outer-product%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
"Bilinear" is simply an adjective that you can apply to any function of two vectors to indicate that it is linear in each argument. So the linear map
$(x,y) rightarrow x^TWy$
and the tensor product
$(x,y) rightarrow xy^T$
can both be described as bilinear, even though their codomains are different.
answered Dec 19 '18 at 12:16
gandalf61gandalf61
8,826725
$endgroup$
$begingroup$
What about the element-wise product between 2 vectors? The reason I mention it is because it's different than the dot and outer product because it's doesn't take into account all the interactions between the 2 vectors (like what the dot product does).
$endgroup$
– floyd
Jan 7 at 20:20
add a comment |
$begingroup$
"Bilinear" is simply an adjective that you can apply to any function of two vectors to indicate that it is linear in each argument. So the linear map
$(x,y) rightarrow x^TWy$
and the tensor product
$(x,y) rightarrow xy^T$
can both be described as bilinear, even though their codomains are different.
answered Dec 19 '18 at 12:16
gandalf61gandalf61
8,826725
$endgroup$
$begingroup$
What about the element-wise product between 2 vectors? The reason I mention it is because it's different than the dot and outer product because it's doesn't take into account all the interactions between the 2 vectors (like what the dot product does).
$endgroup$
– floyd
Jan 7 at 20:20
add a comment |
$begingroup$
"Bilinear" is simply an adjective that you can apply to any function of two vectors to indicate that it is linear in each argument. So the linear map
$(x,y) rightarrow x^TWy$
and the tensor product
$(x,y) rightarrow xy^T$
can both be described as bilinear, even though their codomains are different.
answered Dec 19 '18 at 12:16
gandalf61gandalf61
8,826725
$endgroup$
"Bilinear" is simply an adjective that you can apply to any function of two vectors to indicate that it is linear in each argument. So the linear map
$(x,y) rightarrow x^TWy$
and the tensor product
$(x,y) rightarrow xy^T$
can both be described as bilinear, even though their codomains are different.
answered Dec 19 '18 at 12:16
gandalf61gandalf61
8,826725
answered Dec 19 '18 at 12:16
gandalf61gandalf61
8,826725
answered Dec 19 '18 at 12:16
gandalf61gandalf61
8,826725
answered Dec 19 '18 at 12:16
gandalf61gandalf61
8,826725
8,826725
$begingroup$
What about the element-wise product between 2 vectors? The reason I mention it is because it's different than the dot and outer product because it's doesn't take into account all the interactions between the 2 vectors (like what the dot product does).
$endgroup$
– floyd
Jan 7 at 20:20
add a comment |
$begingroup$
What about the element-wise product between 2 vectors? The reason I mention it is because it's different than the dot and outer product because it's doesn't take into account all the interactions between the 2 vectors (like what the dot product does).
$endgroup$
– floyd
Jan 7 at 20:20
$begingroup$
What about the element-wise product between 2 vectors? The reason I mention it is because it's different than the dot and outer product because it's doesn't take into account all the interactions between the 2 vectors (like what the dot product does).
$endgroup$
– floyd
Jan 7 at 20:20
$begingroup$
What about the element-wise product between 2 vectors? The reason I mention it is because it's different than the dot and outer product because it's doesn't take into account all the interactions between the 2 vectors (like what the dot product does).
$endgroup$
– floyd
Jan 7 at 20:20
add a comment |
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.
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%2f3046295%2fdoes-bilinear-models-on-vectors-mean-dot-or-outer-product%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
