The convolution is an operation on two functions of a real- valued argument and is typically denoted with an asterisk:
s(t) = (x ∗ w)(t)
It is a special kind of linear operation which is also commutative, so:
s(t) = (x ∗ w)(t) is also equal to
s(t) = (w ∗ x)(t).
In convolutional neural networks we usually use convolutions over multiple axis. In a two-dimensional image we want to use 2 convolutional operations over two axis at a time. We also use a 2-D kernel as follows:
I is the input, and
K is the Kernel.
It's commutative equivalent is:
As per www.deeplearningbook.org,
Usually the latter formula is more straightforward to implement in a machine learning library, because there is less variation in the range of valid values of m and n.
I am struggling to comprehend this sentence and thus, my question is, why would there be less variation in the range of valid values of m and n when they are commutative equivalents?