converting einsum notation to for loops

Question

I am trying to convert the following einsum notation into for loops as I cannot fully understand what is happening to the matrices and am unable to use loops to replicate the results myself:

np.einsum('bijc,bijd->bcd', x, x)

Any help is appreciated.

Answer 1

Let's call x_1 and x_2 the two inputs:

np.einsum('bijc,bijd->bcd', x_1, x_2)

bijc,bijd->bcd boils down to ijc,ijd->cd since the first dimension is not used. Imagine you have c channels of ixj on one hand, and d channels of ixj on the other. The result we're looking for is a cxb matrix. Combining each ixj layer from x_1 (there are c in total) to each ixj layer from x_2 (there are d in total) makes a total of c*d values, this is what we're looking for. It's actually the sum of what's called the Hadamard product between the two ixj layers.

Since c is first, c will be in the first dim (the rows) while d will be the number of columns.

Here's an idea:

b_s, i_s, j_s, c_s = x_1.shape
d_s = x_2.shape[3]
y = np.zeros((b_s, c_s, d_s))
for b in range(b_s):
  for i in range(i_s):
    for j in range(j_s):
      for c in range(c_s):
        for d in range(d_s):
          y[b, c, d] += x_1[b, i, j, c]*x_2[b, i, j, d]

This post might give you a better idea

Also try the following to see what happens in a simple case with i=2, j=2, c=1 and d=1:

a = [[[1], [0]], [[0], [1]]]; b = [[[4], [1]], [[2], [2]]]
np.einsum('ijc,ijd->cd', a, b)

result is a d*c matrix of size... 1x1 (since c and d are both equal to 1). Here the result is [6]