张量矩阵乘法返回向量 einsum

Question

I am confused about the following example of a matrix tensor multiplication that returns a vector. At first glance I thought that it would mean multiplying the first dimension of the tensor dydx by the matrix dLdy but I don't get the expected results as depicted below. So what is the meaning of this einsum?

import torch
import numpy as np

dLdy = torch.randn(2,2)
dydx = torch.randn(2,2,2)

torch.einsum('jk,jki->i', dLdy, dydx)
tensor([0.3115, 3.7255])

dLdy
tensor([[-0.4845,  0.6838],
    [-1.1723,  1.4914]])

dydx
tensor([[[ 1.5496, -1.2722],
     [ 0.1221,  1.0495]],

    [[-1.4882,  0.0307],
     [-0.5134,  1.6276]]])

(dLdy * dydx[0]).sum()
-0.1985

Answer 1

For A and B this is contraction (sum) over the first two dimensions jk, so res(i) = sum_{j,k} A(j,k)B(j,k,i)

for example:

import torch
import numpy as np


dLdy = torch.randn(2,2)
dydx = torch.randn(2,2,2)

print(torch.einsum('jk,jki->i', dLdy, dydx))
print((dLdy * dydx[:,:,0]).sum())
print((dLdy * dydx[:,:,1]).sum())

produces

tensor([4.6025, 1.8987])
tensor(4.6025)
tensor(1.8987)

ie (dLdy * dydx[:,:,0]).sum() is the first element of the resulting vector, etc