Sum of sparse vectors: bug or feature?

Question

I recently stumbled upon the following behavior in MATLAB R2022a:

>> a = sparse(1,2,1)
a =
   (1,2)        1

>> b = sparse(2,1,18)
b =
   (2,1)       18

>> a+b
ans =
   (2,1)       18
   (1,2)        1
   (2,2)       19

The presence of the (2,2) element with value 19 is quite puzzling. Intuitively, I would have expected to get either a zero (no element) or an error indicating that the vectors' sizes are not compatible. I couldn't find an explanation for this behavior in the documentation.

So, is this a bug or a feature?

Answer 1

This is due to implicit broadcasting and expected behaviour, also for full() arrays. Compare:

bsxfun(@plus, [0 1], [0; 18])

ans =

     0     1
    18    19

(I'm running R2007b, so need bsxfun() instead of implicit broadcasting).

What happens with unequal-sized vectors is that they are broadcast ("extended") into the appropriate size for addition, see e.g. this blog post on an in-depth explanation.

Verbosely writing our toy example

a = [0 1]
b = [ 0
     18]

a + b = [0 1] + [ 0
                 18]

      % Is broadcast to
      = [0 1     [ 0  0
         0 1] +   18 18]

      % element wise addition
      =  [ 0  1
          18 19]