Why aren't sparse matrices padded with zeros to the highest sizes?

The following code

A = sparse(1:3, 1:3, 1:3)
B = sparse(4,4,4)
A+B

gives the error

Error using +

Matrix dimensions must agree.

because the matrices have different sizes:

size(A) % returns ans = [3 3]
size(B) % returns ans = [4 4]

why aren't sparse matrices zero padded to the highest dimension in order to naturally allow for operations such as A+B ?

In my (beginner) understanding, I don't see any difference in sparse(1:3, 1:3, 1:3), sparse(1:3, 1:3, 1:3, 4, 4) or sparse(1:3, 1:3, 1:3, 100, 100), but apparently they differ in sizes. Which details am I missing? Where do I find them in the documentation?

Answer 1

[Guillaume]

sparse is just a method for efficiently storing matrices that contain lots of zeros. It doesn't change the fact that matrices have a fixed size and it doesn't change the mathematics that adding two differently sized matrices is undefined.

apparently they differ in sizes

Well, yes, as documented:

• S = sparse(i,j,v) generates a sparse matrix S from the triplets i, j, and v [..] The max(i)-by-max(j) output matrix

This applies to your sparse(1:3, 1:3, 1:3), so the matrix is 3x3

• S = sparse(i,j,v,m,n) specifies the size of S as m-by-n.

This applies to all your other examples, where the matrix is exactly the size you specify (4x4, 100x100)

You can't add a 3x3 full matrix with a 4x4 full matrix. The mathematics don't change with sparse.

[Nicola Dalla Pozza]

However, there are exceptions where sizes are automatically extended. Following your reasoning

• S = sparse(i,j,v) generates a sparse matrix S from the triplets i, j, and v [..] The max(i)-by-max(j) output matrix

you may think that C = sparse(1:3) (or better C = sparse(1:3,[1 1 1],1:3)) is a 1x3 matrix, but it works if summed with D = sparse(1,1,1,3,3) or D = sparse(1,1,1,10,3). 1x1 matrix can be added to 4x4 matrices (note: technically 1x1 matrix and scalar are different "objects").

BTW, I just wonder if the zero padding would simplify the conversion from code with full matrices to sparse ones.

[Bruno Luong]

You can even do this:

sparse(1:3,[1 1 1],1:3) + sparse([1 1 1],1:3,1:3)

But again, this has nothing to do with SPARSE but auto-expansion I mention in my answer below. And this feature applies for sparse as well as full matrix

>> [1 1 1] + [1;1;1] % row + column

ans =

     2     2     2
     2     2     2
     2     2     2

Auto expansion (good feature) is NOT auto-padding (bad feature).

[Steven Lord]

you may think that C = sparse(1:3) (or better C = sparse(1:3,[1 1 1],1:3)) is a 1x3 matrix

The former is a 1-by-3 matrix because it uses a different syntax of the sparse function. From the Description section of that function's documentation page:

S = sparse(A) converts a full matrix into sparse form by squeezing out any zero elements. If a matrix contains many zeros, converting the matrix to sparse storage saves memory.

The latter is not a 1-by-3 matrix. It is a max(1:3)-by-max([1 1 1]) matrix which makes it a 3-by-1 matrix.

[Guillaume]

Yes, as Bruno says, your example works because for one of the matrices the mismatched dimension is 1. You get auto expension of singletong dimensions since R2016b. It breaks as soon as neither of the two matrices (sparses or full) have singleton mismatched dimension.

C = spares(1:3, [2 2 2], 1:3)

again can't be added to a 10x3.

Also note that C = sparse(1:3,[1 1 1],1:3) is a 3x1 matrix, not a 1x3 matrix.

"I just wonder if the zero padding would simplify the conversion from code with full matrices to sparse ones"

Both behave exactly the same way. There is nothin to simplify. Note that both support extending the size of the matrix if and only if you assign a value to indices outside the original dimensions:

s = sparse(3, 3);  %a 3x3 sparse matrix
s(4,5) = 1;        %becomes a 4x5 matrix

f = zeros(3, 3);   %a 3x3 full matrix
f(4, 5) = 1;       %becomes a 4x5 matrix

[Nicola Dalla Pozza]

Right, should I edit the typo... ?

[Stephen23]

'note: technically 1x1 matrix and scalar are different "objects" '

MATLAB does not have a "scalar" numeric class, all scalars are 1x1x1x1x... arrays:

https://www.mathworks.com/help/matlab/matlab_prog/fundamental-matlab-classes.html

Answer 2

[Bruno Luong]

Not specific to sparse, similar such operator on full matrices also returns error.

A sparse matrix is designed to be equivalent to a full matrix with the same size, just the internal storage is different.

You can only add/substract matrix/array with the same sizes.

Exception is auto-expansion of "singleton" dimension(s).

That the way it is, if you want to add, you have to do your own padding first. And IMO this is a good design to avoid user doing stupid things.