"numpy.dot 对于尺寸 > 2"

Question

I am trying to understand how dot product works for dimensions more than 2.

Answer 1

It may be easier to visualize this using the notation of np.einsum<\/code> . Start with regular 2D matrix multiplication, which is pretty unambiguous, and follows "normal math" rules:

a = np.ones((2, 3))
b = np.ones((3, 4))
np.einsum('ij,jk->ik', a, b)  # Same as a.dot(b)

Answer 2

Here's my best attempt at making sense of the rule.

Suppose that a is ap ₁ x ... xp _m-1 xp _m array and that b is aq ₁ x ... xq _n-1 xq _n array. The dot function seems to be interpreting a as an p ₁ x ... xp _m-2 array of p _m-1 xp _m matrices, and b as aq ₁ x ... xq _n-2 array of q _n-1 xq _n matrices.

With that established, the entry dot(a,b)[i_1,...,i_(n-2),k_1,j_1,...,j_(m-2),k_2] is the k_1,k_2 entry of the matrix product dot(a[i_1,...,i_(n-2),:,:],b[j_1,...,j_(m-2),:,:]) of ap _m-1 xp _m with aq _n-1 xq _n . Notably, this product is only defined if p _m =q _n-1 .

Answer 3

Standard matrix multiplication definition:

dot(a, b)[i,j] = sum(a[i,:] * b[:,j])

Answer 4

First try to understand the case where both arrays are 2d.

"the last axis of a and the second-to-last axis of b"