CODEWITHSUNDEEP
pythonc++eigentensornumpy-einsum
lelemmen
lelemmen

Reputation: 13

What causes different results by interchanging the tensors in an Eigen::Tensor contraction?

I am working on a C++ library that relies on tensor contractions. I won't post the full application here, but I've distilled it down to the following.

We define a toy rank-4 tensor, which is nothing but (0, 1, ..., 15) reshaped: 

Eigen::Tensor<double, 4> T (2, 2, 2, 2);
for (size_t i = 0; i < 2; i++) {
    for (size_t j = 0; j < 2; j++) {
        for (size_t k = 0; k < 2; k++) {
            for (size_t l = 0; l < 2; l++) {
                T(i, j, k, l) = l + 2 * k + 4 * j + 8 * i;
            }
        }
    }
}

and a rank-2 tensor to contract with, which is nothing but (1, 2, 3, 4) reshaped:

Eigen::Tensor<double, 2> A (2, 2);
for (size_t i = 0; i < 2; i++) {
    for (size_t j = 0; j < 2; j++) {
        A(i, j) = 1 + j + 2 * i;
    }
}

To contract two tensors in Eigen, we have to specify a contraction pair. Our goal is to contract the first two indices of the tensors, as in T(ijkl)*A(ib)=M(bjkl). With my current understanding of the Tensor module in Eigen, we will write the contraction pair as

Eigen::array<Eigen::IndexPair<int>, 1> contraction_pair = {Eigen::IndexPair<int>(0, 0)};

However, I think it should be possible to use the exact same contraction pair to perform the contraction A(ib)*T(ijkl)=N(bjkl). Unfortunately, this is not the case, and the elements of M are

0 0 0 0  24
0 0 0 1  32
0 0 1 0  28
0 0 1 1  38
0 1 0 0  32
0 1 0 1  44
0 1 1 0  36
0 1 1 1  50
1 0 0 0  40
1 0 0 1  56
1 0 1 0  44
1 0 1 1  62
1 1 0 0  48
1 1 0 1  68
1 1 1 0  52
1 1 1 1  74

while these of N are

0 0 0 0  24
0 0 0 1  28
0 0 1 0  32
0 0 1 1  36
0 1 0 0  40
0 1 0 1  44
0 1 1 0  48
0 1 1 1  52
1 0 0 0  32
1 0 0 1  38
1 0 1 0  44
1 0 1 1  50
1 1 0 0  56
1 1 0 1  62
1 1 1 0  68
1 1 1 1  74

I have tested the same toy tensors in numpy, using einsum:

T = np.arange(16).reshape(2, 2, 2, 2)
A = np.arange(1, 5).reshape(2, 2)

contraction1 = np.einsum('ijkl,ia->ajkl', integrals, C)
contraction2 = np.einsum('ia,ijkl->ajkl', C, integrals)

and both contraction1 and contraction2 are

0   0   0   0   24
0   0   0   1   28
0   0   1   0   32
0   0   1   1   36
0   1   0   0   40
0   1   0   1   44
0   1   1   0   48
0   1   1   1   52
1   0   0   0   32
1   0   0   1   38
1   0   1   0   44
1   0   1   1   50
1   1   0   0   56
1   1   0   1   62
1   1   1   0   68
1   1   1   1   74

which coincide with the case A(ib)*T(ijkl)=N(bjkl) in Eigen. What causes Eigen to not give the same result in both cases?

Upvotes: 1

Views: 276

Answers (1)

Paul Panzer
Paul Panzer

Reputation: 53089

The Eigen interface appears to take only the contraction axes for specification. It therefore has to decide itself how to arrange the non-contracted axes. The obvious way would be to keep the original axes in order, first the first argument's then second's.

To confirm this we can either

  • use np.einsum specifying different output layouts and compare to the output of Eigen:

.

import numpy as np

T = np.arange(16).reshape(2, 2, 2, 2)
A = np.arange(1, 5).reshape(2, 2)

print(np.einsum('ijkl,ia->ajkl', T, A))
# [[[[24 28]
#    [32 36]]

#   [[40 44]
#    [48 52]]]


#  [[[32 38]
#    [44 50]]

#   [[56 62]
#    [68 74]]]]

print(np.einsum('ijkl,ia->jkla', T, A))
# [[[[24 32]
#    [28 38]]

#   [[32 44]
#    [36 50]]]


#  [[[40 56]
#    [44 62]]

#   [[48 68]
#    [52 74]]]]
  • or shuffle directly in Eigen (thanks @lelemmen):

.

M.shuffle(Eigen::array<int, 4> {3, 0, 1, 2})
(M_shuffled == N).all()

# 1

Upvotes: 1

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