Numpy einsum 沿轴计算外积

Question

I have two numpy arrays that contain compatible matrices and want to compute the element wise outer product of using numpy.einsum . The shapes of the arrays would be:

A1 = (i,j,k)
A2 = (i,k,j) 

Therefore the arrays contain i matrices of shape (k,j) and (j,k) respectively.

So given A1 would contain the matrices A,B,C and A2 would contain matrices D,E,F , the result would be:

A3 = (A(x)D,B(x)E,C(x)F)

With (x) being the outer product operator.

This would yield to my understanding based on this answer an array A3 of the following shape:

A3 = (i,j*k,j*k)

So far I have tried:

np.einsum("ijk, ilm -> ijklm", A1, A2)

But the resulting shapes do not fit correctly.

As a sanity check I am testing for this:

A = np.asarray(([1,2],[3,4]))
B = np.asarray(([5,6],[7,8]))

AB_outer = np.outer(A,B)

A_vec = np.asarray((A,A))
B_vec = np.asarray((B,B))

# this line is not correct
AB_vec = np.einsum("ijk, ilm -> ijklm", A_vec,B_vec)

np.testing.assert_array_equal(AB_outer, AB_vec[0])

This currently throws an assertion error as my einsum notation is not correct. I am also open to any suggestions that can solve this and are faster or equally fast as nymphs einsum.

Answer 1

We can extend dims and let broadcasting do the job for us -

(A1[:,:,None,:,None]*A2[:,None,:,None,:]).swapaxes(2,3)

Sample run -

In [46]: A1 = np.random.rand(3,4,4)
    ...: A2 = np.random.rand(3,4,4)

In [47]: out = (A1[:,:,None,:,None]*A2[:,None,:,None,:]).swapaxes(2,3)

In [48]: np.allclose(np.multiply.outer(A1[0],A2[0]), out[0])
Out[48]: True

In [49]: np.allclose(np.multiply.outer(A1[1],A2[1]), out[1])
Out[49]: True

In [50]: np.allclose(np.multiply.outer(A1[2],A2[2]), out[2])
Out[50]: True

The equivalent with np.einsum would be -

np.einsum('ijk,ilm->ijklm',A1,A2)

Answer 2

You can compute the result running:

result = np.einsum('ijk,ikl->ijl', A1, A2)

I checked the above code on the following test data:

A = np.arange(1, 13).reshape(3, -1)
B = np.arange(2, 14).reshape(3, -1)
C = np.arange(3, 15).reshape(3, -1)
D = np.arange(1, 13).reshape(4, -1)
E = np.arange(2, 14).reshape(4, -1)
F = np.arange(3, 15).reshape(4, -1)
A1 = np.array([A, B, C])
A2 = np.array([D, E, F])

The result is:

array([[[ 70,  80,  90],
        [158, 184, 210],
        [246, 288, 330]],

       [[106, 120, 134],
        [210, 240, 270],
        [314, 360, 406]],

       [[150, 168, 186],
        [270, 304, 338],
        [390, 440, 490]]])

Now compute 3 "partial results":

res_1 = A @ D
res_2 = B @ E
res_3 = C @ F

and check that they are just the same as consecutive sections of the result.