Sample random vector meeting constraint
I'd like to sample a vector $mathbf{x}inmathbb{R}^k$ such that $frac{mathbf{x_i}}{|mathbf{x}|}geq c$ for all $i$ where $c in [0,frac{1}{sqrt{k}}]$. Is this possible? How can this be done?
So far, what I've been doing is randomly sampling a uniform distribution with a small interval and accepting the sample when the constraint is met.
vectors sampling
|
show 4 more comments
I'd like to sample a vector $mathbf{x}inmathbb{R}^k$ such that $frac{mathbf{x_i}}{|mathbf{x}|}geq c$ for all $i$ where $c in [0,frac{1}{sqrt{k}}]$. Is this possible? How can this be done?
So far, what I've been doing is randomly sampling a uniform distribution with a small interval and accepting the sample when the constraint is met.
vectors sampling
It depends on the value of $c$. If $c > 1$ then it won't be possible since $x_{i}/|x|leq 1$
– pwerth
Jan 4 at 18:56
How is the $mathbf{x}$ in the denominator related to the $mathbf{x}_i$?
– Calvin Godfrey
Jan 4 at 18:58
@pwerth I updated the question so that $cin[0,1]$. It should still work for my purposes.
– Collin
Jan 4 at 19:01
3
@CalvinGodfrey $x_i$ is meant to be the $i$-th dimension of $x$.
– Collin
Jan 4 at 19:01
1
Yes, $c$ is bound to be at most $1/sqrt{n}$
– dafinguzman
Jan 4 at 19:05
|
show 4 more comments
I'd like to sample a vector $mathbf{x}inmathbb{R}^k$ such that $frac{mathbf{x_i}}{|mathbf{x}|}geq c$ for all $i$ where $c in [0,frac{1}{sqrt{k}}]$. Is this possible? How can this be done?
So far, what I've been doing is randomly sampling a uniform distribution with a small interval and accepting the sample when the constraint is met.
vectors sampling
I'd like to sample a vector $mathbf{x}inmathbb{R}^k$ such that $frac{mathbf{x_i}}{|mathbf{x}|}geq c$ for all $i$ where $c in [0,frac{1}{sqrt{k}}]$. Is this possible? How can this be done?
So far, what I've been doing is randomly sampling a uniform distribution with a small interval and accepting the sample when the constraint is met.
vectors sampling
vectors sampling
edited Jan 4 at 19:17
Collin
asked Jan 4 at 18:53
CollinCollin
2201312
2201312
It depends on the value of $c$. If $c > 1$ then it won't be possible since $x_{i}/|x|leq 1$
– pwerth
Jan 4 at 18:56
How is the $mathbf{x}$ in the denominator related to the $mathbf{x}_i$?
– Calvin Godfrey
Jan 4 at 18:58
@pwerth I updated the question so that $cin[0,1]$. It should still work for my purposes.
– Collin
Jan 4 at 19:01
3
@CalvinGodfrey $x_i$ is meant to be the $i$-th dimension of $x$.
– Collin
Jan 4 at 19:01
1
Yes, $c$ is bound to be at most $1/sqrt{n}$
– dafinguzman
Jan 4 at 19:05
|
show 4 more comments
It depends on the value of $c$. If $c > 1$ then it won't be possible since $x_{i}/|x|leq 1$
– pwerth
Jan 4 at 18:56
How is the $mathbf{x}$ in the denominator related to the $mathbf{x}_i$?
– Calvin Godfrey
Jan 4 at 18:58
@pwerth I updated the question so that $cin[0,1]$. It should still work for my purposes.
– Collin
Jan 4 at 19:01
3
@CalvinGodfrey $x_i$ is meant to be the $i$-th dimension of $x$.
– Collin
Jan 4 at 19:01
1
Yes, $c$ is bound to be at most $1/sqrt{n}$
– dafinguzman
Jan 4 at 19:05
It depends on the value of $c$. If $c > 1$ then it won't be possible since $x_{i}/|x|leq 1$
– pwerth
Jan 4 at 18:56
It depends on the value of $c$. If $c > 1$ then it won't be possible since $x_{i}/|x|leq 1$
– pwerth
Jan 4 at 18:56
How is the $mathbf{x}$ in the denominator related to the $mathbf{x}_i$?
– Calvin Godfrey
Jan 4 at 18:58
How is the $mathbf{x}$ in the denominator related to the $mathbf{x}_i$?
– Calvin Godfrey
Jan 4 at 18:58
@pwerth I updated the question so that $cin[0,1]$. It should still work for my purposes.
– Collin
Jan 4 at 19:01
@pwerth I updated the question so that $cin[0,1]$. It should still work for my purposes.
– Collin
Jan 4 at 19:01
3
3
@CalvinGodfrey $x_i$ is meant to be the $i$-th dimension of $x$.
– Collin
Jan 4 at 19:01
@CalvinGodfrey $x_i$ is meant to be the $i$-th dimension of $x$.
– Collin
Jan 4 at 19:01
1
1
Yes, $c$ is bound to be at most $1/sqrt{n}$
– dafinguzman
Jan 4 at 19:05
Yes, $c$ is bound to be at most $1/sqrt{n}$
– dafinguzman
Jan 4 at 19:05
|
show 4 more comments
1 Answer
1
active
oldest
votes
Let $z^i$ be the vector whose coordinates are all equal to $c$, except for the $i^{th}$ coordinate which is $sqrt{1-(k-1)c^2}$. Note that $|z^i|=1$, and each coordinate of $z^i$ is at least $c$. Therefore, $z^i$ is a vector which fulfills your constraints.
Furthermore, any convex combination of the vectors $z^i$ will fulfill your constraints. To see this, note that if $lambda_i$ is a list of positive numbers summing to one, and $z=sum lambda_i z^i$, then using the triangle inequality,
$$
z_j=sum_i lambda_iz^i_jge csum lambda_i |z^i|=csum |lambda_iz^i|ge cleft|sum_i lambda_i z^iright|=c|z|
$$
Therefore, one valid method is to randomly choose $k$ positive numbers $lambda_1,dots,lambda_k$ summing to $1$, and let ${bf x}=sum_i lambda_i z_i$. To do this, see simplex sampling.
To add a little more variety, you instead let ${bf x}=Rsum_i lambda_i z_i$, where $R$ is any positive random scalar. As long as the support of $R$ is $(0,infty)$, then the support of this sampling method is the set of all admissible vectors.
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%2f3061969%2fsample-random-vector-meeting-constraint%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
Let $z^i$ be the vector whose coordinates are all equal to $c$, except for the $i^{th}$ coordinate which is $sqrt{1-(k-1)c^2}$. Note that $|z^i|=1$, and each coordinate of $z^i$ is at least $c$. Therefore, $z^i$ is a vector which fulfills your constraints.
Furthermore, any convex combination of the vectors $z^i$ will fulfill your constraints. To see this, note that if $lambda_i$ is a list of positive numbers summing to one, and $z=sum lambda_i z^i$, then using the triangle inequality,
$$
z_j=sum_i lambda_iz^i_jge csum lambda_i |z^i|=csum |lambda_iz^i|ge cleft|sum_i lambda_i z^iright|=c|z|
$$
Therefore, one valid method is to randomly choose $k$ positive numbers $lambda_1,dots,lambda_k$ summing to $1$, and let ${bf x}=sum_i lambda_i z_i$. To do this, see simplex sampling.
To add a little more variety, you instead let ${bf x}=Rsum_i lambda_i z_i$, where $R$ is any positive random scalar. As long as the support of $R$ is $(0,infty)$, then the support of this sampling method is the set of all admissible vectors.
add a comment |
Let $z^i$ be the vector whose coordinates are all equal to $c$, except for the $i^{th}$ coordinate which is $sqrt{1-(k-1)c^2}$. Note that $|z^i|=1$, and each coordinate of $z^i$ is at least $c$. Therefore, $z^i$ is a vector which fulfills your constraints.
Furthermore, any convex combination of the vectors $z^i$ will fulfill your constraints. To see this, note that if $lambda_i$ is a list of positive numbers summing to one, and $z=sum lambda_i z^i$, then using the triangle inequality,
$$
z_j=sum_i lambda_iz^i_jge csum lambda_i |z^i|=csum |lambda_iz^i|ge cleft|sum_i lambda_i z^iright|=c|z|
$$
Therefore, one valid method is to randomly choose $k$ positive numbers $lambda_1,dots,lambda_k$ summing to $1$, and let ${bf x}=sum_i lambda_i z_i$. To do this, see simplex sampling.
To add a little more variety, you instead let ${bf x}=Rsum_i lambda_i z_i$, where $R$ is any positive random scalar. As long as the support of $R$ is $(0,infty)$, then the support of this sampling method is the set of all admissible vectors.
add a comment |
Let $z^i$ be the vector whose coordinates are all equal to $c$, except for the $i^{th}$ coordinate which is $sqrt{1-(k-1)c^2}$. Note that $|z^i|=1$, and each coordinate of $z^i$ is at least $c$. Therefore, $z^i$ is a vector which fulfills your constraints.
Furthermore, any convex combination of the vectors $z^i$ will fulfill your constraints. To see this, note that if $lambda_i$ is a list of positive numbers summing to one, and $z=sum lambda_i z^i$, then using the triangle inequality,
$$
z_j=sum_i lambda_iz^i_jge csum lambda_i |z^i|=csum |lambda_iz^i|ge cleft|sum_i lambda_i z^iright|=c|z|
$$
Therefore, one valid method is to randomly choose $k$ positive numbers $lambda_1,dots,lambda_k$ summing to $1$, and let ${bf x}=sum_i lambda_i z_i$. To do this, see simplex sampling.
To add a little more variety, you instead let ${bf x}=Rsum_i lambda_i z_i$, where $R$ is any positive random scalar. As long as the support of $R$ is $(0,infty)$, then the support of this sampling method is the set of all admissible vectors.
Let $z^i$ be the vector whose coordinates are all equal to $c$, except for the $i^{th}$ coordinate which is $sqrt{1-(k-1)c^2}$. Note that $|z^i|=1$, and each coordinate of $z^i$ is at least $c$. Therefore, $z^i$ is a vector which fulfills your constraints.
Furthermore, any convex combination of the vectors $z^i$ will fulfill your constraints. To see this, note that if $lambda_i$ is a list of positive numbers summing to one, and $z=sum lambda_i z^i$, then using the triangle inequality,
$$
z_j=sum_i lambda_iz^i_jge csum lambda_i |z^i|=csum |lambda_iz^i|ge cleft|sum_i lambda_i z^iright|=c|z|
$$
Therefore, one valid method is to randomly choose $k$ positive numbers $lambda_1,dots,lambda_k$ summing to $1$, and let ${bf x}=sum_i lambda_i z_i$. To do this, see simplex sampling.
To add a little more variety, you instead let ${bf x}=Rsum_i lambda_i z_i$, where $R$ is any positive random scalar. As long as the support of $R$ is $(0,infty)$, then the support of this sampling method is the set of all admissible vectors.
edited Jan 4 at 20:50
answered Jan 4 at 19:55
Mike EarnestMike Earnest
20.7k11950
20.7k11950
add a comment |
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.
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%2f3061969%2fsample-random-vector-meeting-constraint%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
It depends on the value of $c$. If $c > 1$ then it won't be possible since $x_{i}/|x|leq 1$
– pwerth
Jan 4 at 18:56
How is the $mathbf{x}$ in the denominator related to the $mathbf{x}_i$?
– Calvin Godfrey
Jan 4 at 18:58
@pwerth I updated the question so that $cin[0,1]$. It should still work for my purposes.
– Collin
Jan 4 at 19:01
3
@CalvinGodfrey $x_i$ is meant to be the $i$-th dimension of $x$.
– Collin
Jan 4 at 19:01
1
Yes, $c$ is bound to be at most $1/sqrt{n}$
– dafinguzman
Jan 4 at 19:05