I'm currently trying to create a tensordot using numpy for vectors. For example, let's say I have the following variables:
a = [np.array([1, 2]), np.array([3,4])]
b = [np.array([5,6]), np.array([7,8])]
and I want to compute the "tensor product of the vectors", i.e. [a[0]*b[0], a[0]*b[1], a[1]*b[0], a[1]*b[1]] which would give in our example:
a x b = [[5,12], [7,16], [15, 24], [21, 32]]
I've tried many combinations using tensordot along different axes, but it never gives me the results I want :((
For example, I tried the followings:
np.tensordot(a,b)
which gives me array(70)
or np.tensordot(a,b, axes = 0)
which gives me
array([[[[ 5, 6],
[ 7, 8]],
[[10, 12],
[14, 16]]],
[[[15, 18],
[21, 24]],
[[20, 24],
[28, 32]]]])
I also tried using different axes such as np.tensordot(a,b, axes = ([0], [1])) with no success...
Can someone please help me? :) I'm sure it's pretty trivial but I seem to miss something
Thanks.
In [663]: a = np.array([[1, 2], [3,4]]); b = np.array([[5,6], [7,8]])
A simple dot (matrix product) of these 2 arrays:
In [664]: a.dot(b)
Out[664]:
array([[19, 22],
[43, 50]])
Your desired array:
In [665]: [a[0]*b[0], a[0]*b[1], a[1]*b[0], a[1]*b[1]]
Out[665]: [array([ 5, 12]), array([ 7, 16]), array([15, 24]), array([21, 32])]
In [666]: np.array(_)
Out[666]:
array([[ 5, 12],
[ 7, 16],
[15, 24],
[21, 32]])
np.tensordot is an attempt to generalize np.dot; for 2d arrays like this it can't do anything that a few added transposes can't.
Your result isn't a tensordot in that sense. dot involves sum of products; you aren't doing any sums. Rather it looks more like an outer product, or may a variation on kron.
With a couple trials I reproduced your array with einsum:
In [673]: np.einsum('ij,kj->ikj',a,b)
Out[673]:
array([[[ 5, 12],
[ 7, 16]],
[[15, 24],
[21, 32]]])
In [674]: _.reshape(-1,2)
Out[674]:
array([[ 5, 12],
[ 7, 16],
[15, 24],
[21, 32]])
einsum like dot and tensordot is built around sums of products, but gives us a finer control over which axes are multiplied, and which are summed. Here, we don't sum any.
I can get the same 3d array with:
In [675]: a[:,None,:]*b[None,:,:]
Out[675]:
array([[[ 5, 12],
[ 7, 16]],
[[15, 24],
[21, 32]]])
According to the docs, the default value for axes is 2:
In [714]: np.tensordot(a,b)
Out[714]: array(70)
In [715]: np.tensordot(a,b,axes=2)
Out[715]: array(70)
axes = 2: (default) tensor double contraction :math:a:b
In other words, multiply the arrays, and sum over all axes. This is clearer, in my mind, with einsum notation:
In [719]: np.einsum('ij,ij',a,b)
Out[719]: 70
In [718]: np.tensordot(a,b,axes=0).shape
Out[718]: (2, 2, 2, 2)
axes = 0: tensor product :math:a\\otimes b: tensor product a\otimes b
np.einsum('ij,kl',a,b)
I can see your desired result, or at least the Out[673] version in your (2,2,2,2) array, as some sort of diagonal subset.
I don't use these scalar like axes modes of tensordot much. In a previous post or two I've puzzled over them, but I don't have a good feel. I much prefer the clarity if einsum.