Interpreting triple product summation as matrix multiplication
$begingroup$
I've been wondering about this for a little while and I can't seem to come up with a definite answer. I'm leaning towards "this is not possible", but I figured I'd ask around to see if anybody else has an answer.
Problem statement
Given the expression (where $a_{ij}, b_{ij}$ and $c_{ij}$ are all real numbers)
$$sum_{i} a_{ij}b_{ik}c_{il}
$$can this be expressed in terms of matrix/tensor multiplication? If there were only two terms, such as in
$$sum_i a_{ij} b_{ik}= (A^TB)_{jk}$$
then this would be a straightforward problem.
If there isn't a way to map this to a matrix multiplication problem, is there some other some other computationally efficient procedure for calculating such a summation with triple/quadruple terms? Or has anybody encountered some literature/terms for further research?
Literature
Equations of this nature come up all the time in the context of Corner Transfer Matrix Renormalisation Group (CTMRG). For example, equations (1) and (2) in https://arxiv.org/pdf/1107.1677.pdf
linear-algebra matrices
asked Dec 5 '18 at 5:10
ZxvZxv
265
$endgroup$
|
show 3 more comments
$begingroup$
I've been wondering about this for a little while and I can't seem to come up with a definite answer. I'm leaning towards "this is not possible", but I figured I'd ask around to see if anybody else has an answer.
Problem statement
Given the expression (where $a_{ij}, b_{ij}$ and $c_{ij}$ are all real numbers)
$$sum_{i} a_{ij}b_{ik}c_{il}
$$can this be expressed in terms of matrix/tensor multiplication? If there were only two terms, such as in
$$sum_i a_{ij} b_{ik}= (A^TB)_{jk}$$
then this would be a straightforward problem.
If there isn't a way to map this to a matrix multiplication problem, is there some other some other computationally efficient procedure for calculating such a summation with triple/quadruple terms? Or has anybody encountered some literature/terms for further research?
Literature
Equations of this nature come up all the time in the context of Corner Transfer Matrix Renormalisation Group (CTMRG). For example, equations (1) and (2) in https://arxiv.org/pdf/1107.1677.pdf
linear-algebra matrices
asked Dec 5 '18 at 5:10
ZxvZxv
265
$endgroup$
1
$begingroup$
I think a solution, if there exist one, is by using Kronecker product operations.
$endgroup$
– Jean Marie
Dec 5 '18 at 7:50
2
$begingroup$
Interesting. I can form a matrix $Aotimes B$ using the Kronecker product. A subset of this matrix will be elements of the form $a_{ij}b_{ik}$. If I pick out those elements and form a matrix $widetilde{{AB}}$ with that, the remaining product will correspond to $sum_i widetilde{AB}_{ijk}c_{il}$, which is a standard matrix product. However, forming the full Kronecker product is a bit overkill if I just need a small subset. I wonder if there's a way to setup or trim the matrices $A$ and $B$ so that the Kronecker operation only computes what I need.
$endgroup$
– Zxv
Dec 12 '18 at 3:10
1
$begingroup$
I would also suggest trying to use partial trace operator. It sounds like it could be useful here.
$endgroup$
– Alex Smart
Dec 12 '18 at 7:52
$begingroup$
I thought I had a solution but I have erased it. A remark : let $M$ having all its entries $0$ except $M_{jk}=1$. $A^TMB$ is a matricial expression for your $(A^TB)_{jk}$. If $N$ is the same kind of matrix as $M$ but for indices $l$ and $m$, we could consider $(A^TMB)^TNC$...
$endgroup$
– Jean Marie
Dec 12 '18 at 8:27
$begingroup$
If I understand correctly your input are three matrices, but your output is a three-dimensional cube of numbers, where the number in position (j, k, l) of the cube is the outcome of the sum from your post for fixed j, k and l?
$endgroup$
– Vincent
Dec 12 '18 at 12:50
|
show 3 more comments
$begingroup$
I've been wondering about this for a little while and I can't seem to come up with a definite answer. I'm leaning towards "this is not possible", but I figured I'd ask around to see if anybody else has an answer.
Problem statement
Given the expression (where $a_{ij}, b_{ij}$ and $c_{ij}$ are all real numbers)
$$sum_{i} a_{ij}b_{ik}c_{il}
$$can this be expressed in terms of matrix/tensor multiplication? If there were only two terms, such as in
$$sum_i a_{ij} b_{ik}= (A^TB)_{jk}$$
then this would be a straightforward problem.
If there isn't a way to map this to a matrix multiplication problem, is there some other some other computationally efficient procedure for calculating such a summation with triple/quadruple terms? Or has anybody encountered some literature/terms for further research?
Literature
Equations of this nature come up all the time in the context of Corner Transfer Matrix Renormalisation Group (CTMRG). For example, equations (1) and (2) in https://arxiv.org/pdf/1107.1677.pdf
linear-algebra matrices
asked Dec 5 '18 at 5:10
ZxvZxv
265
$endgroup$
I've been wondering about this for a little while and I can't seem to come up with a definite answer. I'm leaning towards "this is not possible", but I figured I'd ask around to see if anybody else has an answer.
Problem statement
Given the expression (where $a_{ij}, b_{ij}$ and $c_{ij}$ are all real numbers)
$$sum_{i} a_{ij}b_{ik}c_{il}
$$can this be expressed in terms of matrix/tensor multiplication? If there were only two terms, such as in
$$sum_i a_{ij} b_{ik}= (A^TB)_{jk}$$
then this would be a straightforward problem.
If there isn't a way to map this to a matrix multiplication problem, is there some other some other computationally efficient procedure for calculating such a summation with triple/quadruple terms? Or has anybody encountered some literature/terms for further research?
Literature
Equations of this nature come up all the time in the context of Corner Transfer Matrix Renormalisation Group (CTMRG). For example, equations (1) and (2) in https://arxiv.org/pdf/1107.1677.pdf
linear-algebra matrices
linear-algebra matrices
asked Dec 5 '18 at 5:10
ZxvZxv
265
asked Dec 5 '18 at 5:10
ZxvZxv
265
asked Dec 5 '18 at 5:10
ZxvZxv
265
asked Dec 5 '18 at 5:10
ZxvZxv
265
asked Dec 5 '18 at 5:10
ZxvZxv
265
265
1
$begingroup$
I think a solution, if there exist one, is by using Kronecker product operations.
$endgroup$
– Jean Marie
Dec 5 '18 at 7:50
2
$begingroup$
Interesting. I can form a matrix $Aotimes B$ using the Kronecker product. A subset of this matrix will be elements of the form $a_{ij}b_{ik}$. If I pick out those elements and form a matrix $widetilde{{AB}}$ with that, the remaining product will correspond to $sum_i widetilde{AB}_{ijk}c_{il}$, which is a standard matrix product. However, forming the full Kronecker product is a bit overkill if I just need a small subset. I wonder if there's a way to setup or trim the matrices $A$ and $B$ so that the Kronecker operation only computes what I need.
$endgroup$
– Zxv
Dec 12 '18 at 3:10
1
$begingroup$
I would also suggest trying to use partial trace operator. It sounds like it could be useful here.
$endgroup$
– Alex Smart
Dec 12 '18 at 7:52
$begingroup$
I thought I had a solution but I have erased it. A remark : let $M$ having all its entries $0$ except $M_{jk}=1$. $A^TMB$ is a matricial expression for your $(A^TB)_{jk}$. If $N$ is the same kind of matrix as $M$ but for indices $l$ and $m$, we could consider $(A^TMB)^TNC$...
$endgroup$
– Jean Marie
Dec 12 '18 at 8:27
$begingroup$
If I understand correctly your input are three matrices, but your output is a three-dimensional cube of numbers, where the number in position (j, k, l) of the cube is the outcome of the sum from your post for fixed j, k and l?
$endgroup$
– Vincent
Dec 12 '18 at 12:50
|
show 3 more comments
1
$begingroup$
I think a solution, if there exist one, is by using Kronecker product operations.
$endgroup$
– Jean Marie
Dec 5 '18 at 7:50
2
$begingroup$
Interesting. I can form a matrix $Aotimes B$ using the Kronecker product. A subset of this matrix will be elements of the form $a_{ij}b_{ik}$. If I pick out those elements and form a matrix $widetilde{{AB}}$ with that, the remaining product will correspond to $sum_i widetilde{AB}_{ijk}c_{il}$, which is a standard matrix product. However, forming the full Kronecker product is a bit overkill if I just need a small subset. I wonder if there's a way to setup or trim the matrices $A$ and $B$ so that the Kronecker operation only computes what I need.
$endgroup$
– Zxv
Dec 12 '18 at 3:10
1
$begingroup$
I would also suggest trying to use partial trace operator. It sounds like it could be useful here.
$endgroup$
– Alex Smart
Dec 12 '18 at 7:52
$begingroup$
I thought I had a solution but I have erased it. A remark : let $M$ having all its entries $0$ except $M_{jk}=1$. $A^TMB$ is a matricial expression for your $(A^TB)_{jk}$. If $N$ is the same kind of matrix as $M$ but for indices $l$ and $m$, we could consider $(A^TMB)^TNC$...
$endgroup$
– Jean Marie
Dec 12 '18 at 8:27
$begingroup$
If I understand correctly your input are three matrices, but your output is a three-dimensional cube of numbers, where the number in position (j, k, l) of the cube is the outcome of the sum from your post for fixed j, k and l?
$endgroup$
– Vincent
Dec 12 '18 at 12:50
1
1
$begingroup$
I think a solution, if there exist one, is by using Kronecker product operations.
$endgroup$
– Jean Marie
Dec 5 '18 at 7:50
$begingroup$
I think a solution, if there exist one, is by using Kronecker product operations.
$endgroup$
– Jean Marie
Dec 5 '18 at 7:50
2
2
$begingroup$
Interesting. I can form a matrix $Aotimes B$ using the Kronecker product. A subset of this matrix will be elements of the form $a_{ij}b_{ik}$. If I pick out those elements and form a matrix $widetilde{{AB}}$ with that, the remaining product will correspond to $sum_i widetilde{AB}_{ijk}c_{il}$, which is a standard matrix product. However, forming the full Kronecker product is a bit overkill if I just need a small subset. I wonder if there's a way to setup or trim the matrices $A$ and $B$ so that the Kronecker operation only computes what I need.
$endgroup$
– Zxv
Dec 12 '18 at 3:10
$begingroup$
Interesting. I can form a matrix $Aotimes B$ using the Kronecker product. A subset of this matrix will be elements of the form $a_{ij}b_{ik}$. If I pick out those elements and form a matrix $widetilde{{AB}}$ with that, the remaining product will correspond to $sum_i widetilde{AB}_{ijk}c_{il}$, which is a standard matrix product. However, forming the full Kronecker product is a bit overkill if I just need a small subset. I wonder if there's a way to setup or trim the matrices $A$ and $B$ so that the Kronecker operation only computes what I need.
$endgroup$
– Zxv
Dec 12 '18 at 3:10
1
1
$begingroup$
I would also suggest trying to use partial trace operator. It sounds like it could be useful here.
$endgroup$
– Alex Smart
Dec 12 '18 at 7:52
$begingroup$
I would also suggest trying to use partial trace operator. It sounds like it could be useful here.
$endgroup$
– Alex Smart
Dec 12 '18 at 7:52
$begingroup$
I thought I had a solution but I have erased it. A remark : let $M$ having all its entries $0$ except $M_{jk}=1$. $A^TMB$ is a matricial expression for your $(A^TB)_{jk}$. If $N$ is the same kind of matrix as $M$ but for indices $l$ and $m$, we could consider $(A^TMB)^TNC$...
$endgroup$
– Jean Marie
Dec 12 '18 at 8:27
$begingroup$
I thought I had a solution but I have erased it. A remark : let $M$ having all its entries $0$ except $M_{jk}=1$. $A^TMB$ is a matricial expression for your $(A^TB)_{jk}$. If $N$ is the same kind of matrix as $M$ but for indices $l$ and $m$, we could consider $(A^TMB)^TNC$...
$endgroup$
– Jean Marie
Dec 12 '18 at 8:27
$begingroup$
If I understand correctly your input are three matrices, but your output is a three-dimensional cube of numbers, where the number in position (j, k, l) of the cube is the outcome of the sum from your post for fixed j, k and l?
$endgroup$
– Vincent
Dec 12 '18 at 12:50
$begingroup$
If I understand correctly your input are three matrices, but your output is a three-dimensional cube of numbers, where the number in position (j, k, l) of the cube is the outcome of the sum from your post for fixed j, k and l?
$endgroup$
– Vincent
Dec 12 '18 at 12:50
|
show 3 more comments
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1
$begingroup$
I think a solution, if there exist one, is by using Kronecker product operations.
$endgroup$
– Jean Marie
Dec 5 '18 at 7:50
2
$begingroup$
Interesting. I can form a matrix $Aotimes B$ using the Kronecker product. A subset of this matrix will be elements of the form $a_{ij}b_{ik}$. If I pick out those elements and form a matrix $widetilde{{AB}}$ with that, the remaining product will correspond to $sum_i widetilde{AB}_{ijk}c_{il}$, which is a standard matrix product. However, forming the full Kronecker product is a bit overkill if I just need a small subset. I wonder if there's a way to setup or trim the matrices $A$ and $B$ so that the Kronecker operation only computes what I need.
$endgroup$
– Zxv
Dec 12 '18 at 3:10
1
$begingroup$
I would also suggest trying to use partial trace operator. It sounds like it could be useful here.
$endgroup$
– Alex Smart
Dec 12 '18 at 7:52
$begingroup$
I thought I had a solution but I have erased it. A remark : let $M$ having all its entries $0$ except $M_{jk}=1$. $A^TMB$ is a matricial expression for your $(A^TB)_{jk}$. If $N$ is the same kind of matrix as $M$ but for indices $l$ and $m$, we could consider $(A^TMB)^TNC$...
$endgroup$
– Jean Marie
Dec 12 '18 at 8:27
$begingroup$
If I understand correctly your input are three matrices, but your output is a three-dimensional cube of numbers, where the number in position (j, k, l) of the cube is the outcome of the sum from your post for fixed j, k and l?
$endgroup$
– Vincent
Dec 12 '18 at 12:50