Real inner-product in python

[Robert Kern]

In article <mailman.1043229043.27058.python-list at python.org>,
	Nadav Horesh <NadavH at VisionSense.com> writes:
> Whats about:
> 
> >>> c = N.reshape(N.arange(12), (3,2,2))
> >>> b = N.arange(3)
> >>> N.dot(b,c)
> Traceback (most recent call last):
>   File "<pyshell#18>", line 1, in ?
>     N.dot(b,c)
>   File "/usr/local/lib/python2.3/site-packages/Numeric/Numeric.py", line 
> 335, in dot
>     return multiarray.matrixproduct(a, b)
> ValueError: matrices are not aligned

[snip]

> As I see inner product between two tensors --- A of rank $n$ and B of 
> rank $m$ it should be like
> (in TeX style):
> $$
>   C = A \cdot B
> $$
> requires:
> 
> 1.  The last dimension of A must be equal to the first dimension of B, 
> and ...
> 2.
> $$
>   C_{p_1, ... p_{m-1},q_2, ... q_n} = \sum_{i=1}^{q_1} A_{p_1, ... 
> p_{m-1},i} B_{i, q_2, ... q_{n}}
> $$
> 
> Thus, I don't see the *dot* function as a proper inner product.

You are correct. Here is dot's docstring:

"""dot(a,b) returns matrix-multiplication between a and b.  The product-sum
   is over the last dimension of a and the second-to-last dimension of b.
"""

Here is my implementation of a tensor-style inner product:

def _myDot(a, b):
    """Returns the inner product.

    a_i...k * b_k...m = c_i...m     in summation notation with the ...'s 
                                    representing arbitrary, omitted indices

    The sum is over the last axis of the first argument and the first axis 
    of the last axis.

    _myDot(a, b) --> NumPy array
    """

    a = Numeric.asarray(a)
    b = Numeric.asarray(b)

    tempAxes = tuple(range(1, len(b.shape)) + [0])
    newB = Numeric.transpose(b, tempAxes)

    # innerproduct sums over the *last* axes of *both* arguments
    return Numeric.innerproduct(a, newB)

>   Nadav

-- 
Robert Kern
Ruddock House President
kern at caltech.edu

"In the fields of hell where the grass grows high
 Are the graves of dreams allowed to die."
  -- Richard Harter