Given $n$ vectors, find partitions with closest centroids
$begingroup$
Given vectors $a_1, dots, a_nin mathbb R^d$ where $n$ is even, I want to find partitions $I$ and $J$ of $[n]$ with $|I|=|J|=frac n2$ to minimize
$$left| sum_{iin I} a_i - sum_{jin J} a_j right|.$$
This problem can be written as a binary optimization problem. Given matrix $A = [a_1 dots a_n]$, I want to minimize $|Ax|$ over $xin{-1,1}^n$ and $sum_{i=1}^n x_i=0$.
Finding exact global minimum looks NP-hard (in $d$ or $n$). Is it possible to find a nice approximate solution (like $(1+epsilon)$-approximation for $K$-means)?
Convex relaxation does not seems to work because the convex hull of the feasible region contains a trivial global minimizer $x=0$.
Any help will greatly appreciated.
discrete-optimization binary-programming
asked Jan 25 at 2:56
MayuMayu
63
$endgroup$
add a comment |
$begingroup$
Given vectors $a_1, dots, a_nin mathbb R^d$ where $n$ is even, I want to find partitions $I$ and $J$ of $[n]$ with $|I|=|J|=frac n2$ to minimize
$$left| sum_{iin I} a_i - sum_{jin J} a_j right|.$$
This problem can be written as a binary optimization problem. Given matrix $A = [a_1 dots a_n]$, I want to minimize $|Ax|$ over $xin{-1,1}^n$ and $sum_{i=1}^n x_i=0$.
Finding exact global minimum looks NP-hard (in $d$ or $n$). Is it possible to find a nice approximate solution (like $(1+epsilon)$-approximation for $K$-means)?
Convex relaxation does not seems to work because the convex hull of the feasible region contains a trivial global minimizer $x=0$.
Any help will greatly appreciated.
discrete-optimization binary-programming
asked Jan 25 at 2:56
MayuMayu
63
$endgroup$
$begingroup$
Looks like a multidimensional generalization of the partition problem (unless the restriction $|I|=|J|$ makes a big difference, not sure).
$endgroup$
– Rahul
Jan 25 at 4:13
$begingroup$
@Rahul Thanks for pointing this out. It is interesting enough to consider the problem without the restriction. Can the algorithm generalize to multidimensional case?
$endgroup$
– Mayu
Jan 25 at 5:56
add a comment |
$begingroup$
Given vectors $a_1, dots, a_nin mathbb R^d$ where $n$ is even, I want to find partitions $I$ and $J$ of $[n]$ with $|I|=|J|=frac n2$ to minimize
$$left| sum_{iin I} a_i - sum_{jin J} a_j right|.$$
This problem can be written as a binary optimization problem. Given matrix $A = [a_1 dots a_n]$, I want to minimize $|Ax|$ over $xin{-1,1}^n$ and $sum_{i=1}^n x_i=0$.
Finding exact global minimum looks NP-hard (in $d$ or $n$). Is it possible to find a nice approximate solution (like $(1+epsilon)$-approximation for $K$-means)?
Convex relaxation does not seems to work because the convex hull of the feasible region contains a trivial global minimizer $x=0$.
Any help will greatly appreciated.
discrete-optimization binary-programming
asked Jan 25 at 2:56
MayuMayu
63
$endgroup$
Given vectors $a_1, dots, a_nin mathbb R^d$ where $n$ is even, I want to find partitions $I$ and $J$ of $[n]$ with $|I|=|J|=frac n2$ to minimize
$$left| sum_{iin I} a_i - sum_{jin J} a_j right|.$$
This problem can be written as a binary optimization problem. Given matrix $A = [a_1 dots a_n]$, I want to minimize $|Ax|$ over $xin{-1,1}^n$ and $sum_{i=1}^n x_i=0$.
Finding exact global minimum looks NP-hard (in $d$ or $n$). Is it possible to find a nice approximate solution (like $(1+epsilon)$-approximation for $K$-means)?
Convex relaxation does not seems to work because the convex hull of the feasible region contains a trivial global minimizer $x=0$.
Any help will greatly appreciated.
discrete-optimization binary-programming
discrete-optimization binary-programming
asked Jan 25 at 2:56
MayuMayu
63
asked Jan 25 at 2:56
MayuMayu
63
asked Jan 25 at 2:56
MayuMayu
63
asked Jan 25 at 2:56
MayuMayu
63
asked Jan 25 at 2:56
MayuMayu
63
63
$begingroup$
Looks like a multidimensional generalization of the partition problem (unless the restriction $|I|=|J|$ makes a big difference, not sure).
$endgroup$
– Rahul
Jan 25 at 4:13
$begingroup$
@Rahul Thanks for pointing this out. It is interesting enough to consider the problem without the restriction. Can the algorithm generalize to multidimensional case?
$endgroup$
– Mayu
Jan 25 at 5:56
add a comment |
$begingroup$
Looks like a multidimensional generalization of the partition problem (unless the restriction $|I|=|J|$ makes a big difference, not sure).
$endgroup$
– Rahul
Jan 25 at 4:13
$begingroup$
@Rahul Thanks for pointing this out. It is interesting enough to consider the problem without the restriction. Can the algorithm generalize to multidimensional case?
$endgroup$
– Mayu
Jan 25 at 5:56
$begingroup$
Looks like a multidimensional generalization of the partition problem (unless the restriction $|I|=|J|$ makes a big difference, not sure).
$endgroup$
– Rahul
Jan 25 at 4:13
$begingroup$
Looks like a multidimensional generalization of the partition problem (unless the restriction $|I|=|J|$ makes a big difference, not sure).
$endgroup$
– Rahul
Jan 25 at 4:13
$begingroup$
@Rahul Thanks for pointing this out. It is interesting enough to consider the problem without the restriction. Can the algorithm generalize to multidimensional case?
$endgroup$
– Mayu
Jan 25 at 5:56
$begingroup$
@Rahul Thanks for pointing this out. It is interesting enough to consider the problem without the restriction. Can the algorithm generalize to multidimensional case?
$endgroup$
– Mayu
Jan 25 at 5:56
add a comment |
1 Answer
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$begingroup$
What if you relax the constraint that $x in{-1, 1}^n$ into $|x|=1$? In other words, $$argmin |Ax| text{ subject to } |x|=1 text { and } langle x, underline{1}rangle = 0$$
Once you've found a solution $x^star$ to this problem, you just take the signum of its components.
I don't know how good an approximation this would provide
answered Jan 25 at 3:59
Stefan LafonStefan Lafon
2,84519
$endgroup$
$begingroup$
Thanks. I figured out this is binary quadratic programing problem. The approach you proposed is a spectral relaxation. luthuli.cs.uiuc.edu/~daf/courses/Opt-2017/Papers/0234.pdf
$endgroup$
– Mayu
Jan 25 at 8:00
add a comment |
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1 Answer
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active
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1 Answer
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active
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$begingroup$
What if you relax the constraint that $x in{-1, 1}^n$ into $|x|=1$? In other words, $$argmin |Ax| text{ subject to } |x|=1 text { and } langle x, underline{1}rangle = 0$$
Once you've found a solution $x^star$ to this problem, you just take the signum of its components.
I don't know how good an approximation this would provide
answered Jan 25 at 3:59
Stefan LafonStefan Lafon
2,84519
$endgroup$
$begingroup$
Thanks. I figured out this is binary quadratic programing problem. The approach you proposed is a spectral relaxation. luthuli.cs.uiuc.edu/~daf/courses/Opt-2017/Papers/0234.pdf
$endgroup$
– Mayu
Jan 25 at 8:00
add a comment |
$begingroup$
What if you relax the constraint that $x in{-1, 1}^n$ into $|x|=1$? In other words, $$argmin |Ax| text{ subject to } |x|=1 text { and } langle x, underline{1}rangle = 0$$
Once you've found a solution $x^star$ to this problem, you just take the signum of its components.
I don't know how good an approximation this would provide
answered Jan 25 at 3:59
Stefan LafonStefan Lafon
2,84519
$endgroup$
$begingroup$
Thanks. I figured out this is binary quadratic programing problem. The approach you proposed is a spectral relaxation. luthuli.cs.uiuc.edu/~daf/courses/Opt-2017/Papers/0234.pdf
$endgroup$
– Mayu
Jan 25 at 8:00
add a comment |
$begingroup$
What if you relax the constraint that $x in{-1, 1}^n$ into $|x|=1$? In other words, $$argmin |Ax| text{ subject to } |x|=1 text { and } langle x, underline{1}rangle = 0$$
Once you've found a solution $x^star$ to this problem, you just take the signum of its components.
I don't know how good an approximation this would provide
answered Jan 25 at 3:59
Stefan LafonStefan Lafon
2,84519
$endgroup$
What if you relax the constraint that $x in{-1, 1}^n$ into $|x|=1$? In other words, $$argmin |Ax| text{ subject to } |x|=1 text { and } langle x, underline{1}rangle = 0$$
Once you've found a solution $x^star$ to this problem, you just take the signum of its components.
I don't know how good an approximation this would provide
answered Jan 25 at 3:59
Stefan LafonStefan Lafon
2,84519
answered Jan 25 at 3:59
Stefan LafonStefan Lafon
2,84519
answered Jan 25 at 3:59
Stefan LafonStefan Lafon
2,84519
answered Jan 25 at 3:59
Stefan LafonStefan Lafon
2,84519
2,84519
$begingroup$
Thanks. I figured out this is binary quadratic programing problem. The approach you proposed is a spectral relaxation. luthuli.cs.uiuc.edu/~daf/courses/Opt-2017/Papers/0234.pdf
$endgroup$
– Mayu
Jan 25 at 8:00
add a comment |
$begingroup$
Thanks. I figured out this is binary quadratic programing problem. The approach you proposed is a spectral relaxation. luthuli.cs.uiuc.edu/~daf/courses/Opt-2017/Papers/0234.pdf
$endgroup$
– Mayu
Jan 25 at 8:00
$begingroup$
Thanks. I figured out this is binary quadratic programing problem. The approach you proposed is a spectral relaxation. luthuli.cs.uiuc.edu/~daf/courses/Opt-2017/Papers/0234.pdf
$endgroup$
– Mayu
Jan 25 at 8:00
$begingroup$
Thanks. I figured out this is binary quadratic programing problem. The approach you proposed is a spectral relaxation. luthuli.cs.uiuc.edu/~daf/courses/Opt-2017/Papers/0234.pdf
$endgroup$
– Mayu
Jan 25 at 8:00
add a comment |
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$begingroup$
Looks like a multidimensional generalization of the partition problem (unless the restriction $|I|=|J|$ makes a big difference, not sure).
$endgroup$
– Rahul
Jan 25 at 4:13
$begingroup$
@Rahul Thanks for pointing this out. It is interesting enough to consider the problem without the restriction. Can the algorithm generalize to multidimensional case?
$endgroup$
– Mayu
Jan 25 at 5:56