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aesara.tensor.tensordot

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aesara.tensor.tensordot#

aesara.tensor.tensordot(a, b, axes=2)[source]#

Compute a generalized dot product over provided axes.

Given two tensors a and b, tensordot computes a generalized dot product over the provided axes. Aesara’s implementation reduces all expressions to matrix or vector dot products and is based on code from Tijmen Tieleman’s gnumpy (http://www.cs.toronto.edu/~tijmen/gnumpy.html).

Parameters:

  • a (symbolic tensor) – The first tensor variable.

  • b (symbolic tensor) – The second tensor variable

  • axes (int or array-like of length 2) –

    If an integer, the number of axes to sum over. If an array, it must have two array elements containing the axes to sum over in each tensor.

    Note that the default value of 2 is not guaranteed to work for all values of a and b, and an error will be raised if that is the case. The reason for keeping the default is to maintain the same signature as numpy’s tensordot function (and np.tensordot raises analogous errors for non-compatible inputs).

    If an integer i, it is converted to an array containing the last i dimensions of the first tensor and the first i dimensions of the second tensor:

    axes = [list(range(a.ndim - i, b.ndim)), list(range(i))]

    If an array, its two elements must contain compatible axes of the two tensors. For example, [[1, 2], [2, 0]] means sum over the 2nd and 3rd axes of a and the 3rd and 1st axes of b. (Remember axes are zero-indexed!) The 2nd axis of a and the 3rd axis of b must have the same shape; the same is true for the 3rd axis of a and the 1st axis of b.

Returns:

A tensor with shape equal to the concatenation of a’s shape (less any dimensions that were summed over) and b’s shape (less any dimensions that were summed over).

Return type:

symbolic tensor

Examples

It may be helpful to consider an example to see what tensordot does. Aesara’s implementation is identical to NumPy’s. Here a has shape (2, 3, 4) and b has shape (5, 6, 4, 3). The axes to sum over are [[1, 2], [3, 2]] – note that a.shape[1] == b.shape[3] and a.shape[2] == b.shape[2]; these axes are compatible. The resulting tensor will have shape (2, 5, 6) – the dimensions that are not being summed:

>>> a = np.random.random((2,3,4))
>>> b = np.random.random((5,6,4,3))

#tensordot >>> c = np.tensordot(a, b, [[1,2],[3,2]])

#loop replicating tensordot >>> a0, a1, a2 = a.shape >>> b0, b1, _, _ = b.shape >>> cloop = np.zeros((a0,b0,b1))

#loop over non-summed indices – these exist #in the tensor product. >>> for i in range(a0): … for j in range(b0): … for k in range(b1): … #loop over summed indices – these don’t exist … #in the tensor product. … for l in range(a1): … for m in range(a2): … cloop[i,j,k] += a[i,l,m] * b[j,k,m,l]

>>> np.allclose(c, cloop)
true

This specific implementation avoids a loop by transposing a and b such that the summed axes of a are last and the summed axes of b are first. The resulting arrays are reshaped to 2 dimensions (or left as vectors, if appropriate) and a matrix or vector dot product is taken. The result is reshaped back to the required output dimensions.

In an extreme case, no axes may be specified. The resulting tensor will have shape equal to the concatenation of the shapes of a and b:

>>> c = np.tensordot(a, b, 0)
>>> print(a.shape)
(2,3,4)
>>> print(b.shape)
(5,6,4,3)
>>> print(c.shape)
(2,3,4,5,6,4,3)

See the documentation of numpy.tensordot for more examples.

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