Smallest $i$ for Einstein summation $ij,ik,il→jkl$ to have a (under-determined) solution given arbitrary...
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I'm wondering what is the smallest $i$ for Einstein summation $ij,ik,il→jkl$ to have a (under-determined) solution given arbitrary jkl term (and solve for $ij$, $ik$, and $jk$ three matrices).
For example,
When $j = k = l = 2$, apparently $i = 1$ won't work since outer product of three vectors do not form all possible 3-tensors
When $j = k = l = 4$, this is more or less related to "how many element-wise multiplications are at least needed when doing 2x2 matrix multiplication". The answer I know is 7 as described by Strassen's algorithm
I've been trying to find the solution myself using numerical optimization. Unfortunately I can't keep doing that since it becomes extremely slow for $j = k = l > 10$. What I got is $2,5,7,10,14,19,24,30$ separately for $i,j,k = 2,3,4,5,6,7,8,9$. These solution might not be correct since numerical optimization is subject to floating precision (and local minima).
Thanks for any help!
matrices tensors
asked Dec 3 at 3:25
ZisIsNotZis
1064
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up vote
0
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I'm wondering what is the smallest $i$ for Einstein summation $ij,ik,il→jkl$ to have a (under-determined) solution given arbitrary jkl term (and solve for $ij$, $ik$, and $jk$ three matrices).
For example,
When $j = k = l = 2$, apparently $i = 1$ won't work since outer product of three vectors do not form all possible 3-tensors
When $j = k = l = 4$, this is more or less related to "how many element-wise multiplications are at least needed when doing 2x2 matrix multiplication". The answer I know is 7 as described by Strassen's algorithm
I've been trying to find the solution myself using numerical optimization. Unfortunately I can't keep doing that since it becomes extremely slow for $j = k = l > 10$. What I got is $2,5,7,10,14,19,24,30$ separately for $i,j,k = 2,3,4,5,6,7,8,9$. These solution might not be correct since numerical optimization is subject to floating precision (and local minima).
Thanks for any help!
matrices tensors
asked Dec 3 at 3:25
ZisIsNotZis
1064
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I'm wondering what is the smallest $i$ for Einstein summation $ij,ik,il→jkl$ to have a (under-determined) solution given arbitrary jkl term (and solve for $ij$, $ik$, and $jk$ three matrices).
For example,
When $j = k = l = 2$, apparently $i = 1$ won't work since outer product of three vectors do not form all possible 3-tensors
When $j = k = l = 4$, this is more or less related to "how many element-wise multiplications are at least needed when doing 2x2 matrix multiplication". The answer I know is 7 as described by Strassen's algorithm
I've been trying to find the solution myself using numerical optimization. Unfortunately I can't keep doing that since it becomes extremely slow for $j = k = l > 10$. What I got is $2,5,7,10,14,19,24,30$ separately for $i,j,k = 2,3,4,5,6,7,8,9$. These solution might not be correct since numerical optimization is subject to floating precision (and local minima).
Thanks for any help!
matrices tensors
asked Dec 3 at 3:25
ZisIsNotZis
1064
I'm wondering what is the smallest $i$ for Einstein summation $ij,ik,il→jkl$ to have a (under-determined) solution given arbitrary jkl term (and solve for $ij$, $ik$, and $jk$ three matrices).
For example,
When $j = k = l = 2$, apparently $i = 1$ won't work since outer product of three vectors do not form all possible 3-tensors
When $j = k = l = 4$, this is more or less related to "how many element-wise multiplications are at least needed when doing 2x2 matrix multiplication". The answer I know is 7 as described by Strassen's algorithm
I've been trying to find the solution myself using numerical optimization. Unfortunately I can't keep doing that since it becomes extremely slow for $j = k = l > 10$. What I got is $2,5,7,10,14,19,24,30$ separately for $i,j,k = 2,3,4,5,6,7,8,9$. These solution might not be correct since numerical optimization is subject to floating precision (and local minima).
Thanks for any help!
matrices tensors
matrices tensors
asked Dec 3 at 3:25
ZisIsNotZis
1064
asked Dec 3 at 3:25
ZisIsNotZis
1064
asked Dec 3 at 3:25
ZisIsNotZis
1064
asked Dec 3 at 3:25
ZisIsNotZis
1064
asked Dec 3 at 3:25
ZisIsNotZis
1064
1064
add a comment |
add a comment |
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