NumPy Tensordot axes=2

Question

I know there are many questions about tensordot, and I've skimmed some of the 15 page mini-book answers that people I'm sure spent hours making, but I haven't found an explanation of what axes=2 does.

This made me think that np.tensordot(b,c,axes=2) == np.sum(b * c), but as an array:

b = np.array([[1,10],[100,1000]])
c = np.array([[2,3],[5,7]])
np.tensordot(b,c,axes=2)
Out: array(7532)

But then this failed:

a = np.arange(30).reshape((2,3,5))
np.tensordot(a,a,axes=2)

If anyone can provide a short, concise explanation of np.tensordot(x,y,axes=2), and only axes=2, then I would gladly accept it.

Please link to the mini books you've read. I've read some of them too, and we either didn't read the same one or you have some specific question that you're failing to articulate. — Mad Physicist, Feb 15 '21 at 20:06
I explored those single digit axis actions in a SO awhile back. They are translated into a tuple form, and on to a `dot` call. I don't think understanding this is important. `np.einsum` and `matmul` are more useful extension to `dot`. — hpaulj, Feb 15 '21 at 20:22
@hpaulj,`np.tensordot(a,a, axes=2)` gives `raise ValueError("shape-mismatch for sum")` — Joe, Feb 15 '21 at 20:30
possible duplicate https://stackoverflow.com/questions/51989572/how-does-numpy-tensordot-function-works-step-by-step — hpaulj, Feb 15 '21 at 20:34
@hpaulj, it seems like your answer to that question says `np.tensordot(a,a,axes=2)` should give me the same thing as `np.dot(a.ravel(),a.ravel())`, but `np.tensordot(a,a,axes=2)` gives me an error, while the latter gives me `np.sum(a*a)` — Joe, Feb 15 '21 at 20:40
I did not say that it was actually doing the ravel. The ravel equivalence only applies to 2d arrays, not your 3d example. — hpaulj, Feb 15 '21 at 21:25

score 0 · Accepted Answer · answered Feb 15 '21 at 21:20

In [70]: a = np.arange(24).reshape(2,3,4)
In [71]: np.tensordot(a,a,axes=2)
Traceback (most recent call last):
  File "<ipython-input-71-dbe04e46db70>", line 1, in <module>
    np.tensordot(a,a,axes=2)
  File "<__array_function__ internals>", line 5, in tensordot
  File "/usr/local/lib/python3.8/dist-packages/numpy/core/numeric.py", line 1116, in tensordot
    raise ValueError("shape-mismatch for sum")
ValueError: shape-mismatch for sum

In my previous post I deduced that axis=2 translates to axes=([-2,-1],[0,1])

How does numpy.tensordot function works step-by-step?

In [72]: np.tensordot(a,a,axes=([-2,-1],[0,1]))
Traceback (most recent call last):
  File "<ipython-input-72-efdbfe6ff0d3>", line 1, in <module>
    np.tensordot(a,a,axes=([-2,-1],[0,1]))
  File "<__array_function__ internals>", line 5, in tensordot
  File "/usr/local/lib/python3.8/dist-packages/numpy/core/numeric.py", line 1116, in tensordot
    raise ValueError("shape-mismatch for sum")
ValueError: shape-mismatch for sum

So that's trying to do a double axis reduction on the last 2 dimensions of the first a, and the first 2 dimensions of the second a. With this a that's a dimensions mismatch. Evidently this axes was intended for 2d arrays, without much thought given to 3d ones. It is not a 3 axis reduction.

These single digit axes values are something that some developer thought would be convenient, but that does not mean they were rigorously thought out or tested.

The tuple axes gives you more control:

In [74]: np.tensordot(a,a,axes=[(0,1,2),(0,1,2)])
Out[74]: array(4324)
In [75]: np.tensordot(a,a,axes=[(0,1),(0,1)])
Out[75]: 
array([[ 880,  940, 1000, 1060],
       [ 940, 1006, 1072, 1138],
       [1000, 1072, 1144, 1216],
       [1060, 1138, 1216, 1294]])

Ok, thanks. I get it now. I wouldn't say that it's not intended for higher-d arrays though. I see now that `axes=N` requires the last N dimensions of the first array to be the same as the first N dimensions of the second array, in the same order. so with `a = np.arange(720).reshape(2,3,4,5,6)` , `b = np.arange(720).reshape(5,6,8,3)` , and `c = np.arange(720).reshape(4,5,6,1,6)`, you can do `np.tensordot(a,b,axes=2)` or `np.tensordot(a,c,axes=3)`. I originally thought of `axes=2` as an option, instead of `axes=N` being the number of dimensions that must match. — Joe, Feb 15 '21 at 21:36

NumPy Tensordot axes=2

1 Answers1