Sample random vector meeting constraint












1














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







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  • 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
















1














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







share|cite|improve this question
























  • 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














1












1








1


1





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







share|cite|improve this question















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






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








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


















  • 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










1 Answer
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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.






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    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.






    share|cite|improve this answer




























      2














      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.






      share|cite|improve this answer


























        2












        2








        2






        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.






        share|cite|improve this answer














        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.







        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        edited Jan 4 at 20:50

























        answered Jan 4 at 19:55









        Mike EarnestMike Earnest

        20.7k11950




        20.7k11950






























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