Vectorization of this Numpy double loop

Question

How can I vectorize the following double-loop?

I have one N by A matrix and one N by B matrix, where A and B may differ and N is much smaller than A and B. I want to produce an A by B matrix as follows, but ideally without the loops:

import numpy as np

def foo(arr):
    # can be anything - just an example so that the code runs
    return np.sum(arr)

num_a = 12
num_b = 8
num_dimensions = 3

a = np.random.rand(num_dimensions, num_a)
b = np.random.rand(num_dimensions, num_b)

# this is the loop I want to eliminate:
output = np.zeros( (num_a, num_b) )
for i in xrange(num_a):
    for j in xrange(num_b):
       output[i,j] = foo(a[:,i] - b[:,j])

Any ideas?

Answer 1

First vectorise foo(), i.e. modify foo() in a way that it can correctly operate on an array of shape (N, A, B), returning an array of shape (A, B). This step is usually the difficult one. How this is done entirely depends on what foo() does. For the given example, it's very easy to do:

def foo(arr):
    return np.sum(arr, axis=0)

Now, use broadcasting rules to create a (N, A, B) array containing all the vector differences, and pass it to foo():

foo(a[:, :, np.newaxis] - b[:, np.newaxis])