The memory use of a sparse matrix depends on its history

The amount of memory used by a sparse matrix depends on its history. In a project this week, a matrix that could have consumed a few bytes was consuming gigabytes of memory.

Demonstration:

J = sparse(zeros(2,2));
J2 = [J zeros(2,2); zeros(1,4)];
J3 = sparse(zeros(3,4)); % same data as J2
isequal(J2,J3)
%ans =
% logical
% 1
whos('J2')
% Name      Size            Bytes  Class     Attributes
% J2        3x4                88  double    sparse    
whos('J3')
% Name      Size            Bytes  Class     Attributes
% J3        3x4                56  double    sparse    
%
% 88 > 56!

Answer 1

[Steve Van Hooser]

To shrink the footprint of a sparse matrix, see https://www.mathworks.com/matlabcentral/answers/736432-how-do-i-shrink-the-memory-footprint-of-a-sparse-matrix-to-its-minimum?s_tid=answers_rc1-2_p2_BOTH

Answer 2

[Steven Lord]

The amount of memory required by a sparse matrix is not just a function of the number of rows and columns but also the number of non-zero elements stored and the number of non-zero locations allocated for storage. In the case of your J2 and J3, they are the same size but have different numbers of locations allocated for storage.

J = sparse(zeros(2,2));
J2 = [J zeros(2,2); zeros(1,4)];
J3 = sparse(zeros(3,4)); % same data as J2
numberOfNonzerosInJ2 = nnz(J2)

numberOfNonzerosInJ2 = 0

numberOfNonzerosAllocatedInJ2 = nzmax(J2)

numberOfNonzerosAllocatedInJ2 = 3

numberOfNonzerosInJ3 = nnz(J3)

numberOfNonzerosInJ3 = 0

numberOfNonzerosAllocatedInJ3 = nzmax(J3)

numberOfNonzerosAllocatedInJ3 = 1

If you know how many non-zero elements you're ultimately going to want your sparse matrix to contain, use the spalloc function to preallocate it.

J4 = spalloc(height(J2), width(J2), 2); % Matrix is same size as J2, but only 2 nonzero locations
whos J*

  Name      Size            Bytes  Class     Attributes

  J         2x2                40  double    sparse    
  J2        3x4                88  double    sparse    
  J3        3x4                56  double    sparse    
  J4        3x4                72  double    sparse    

[James Tursa]

As a follow up, when encountered in operations MATLAB will shrink the resulting sparse memory allocation to the minimum size necessary. I am unaware of any published rules for what operations will trigger this, but it does seem to happen for the "usual" stuff. E.g.,

S = spalloc(1000,1000,10000);
T = S + sparse(0);
whos

  Name         Size               Bytes  Class     Attributes

  S         1000x1000            168008  double    sparse    
  T         1000x1000              8024  double    sparse    

[Steve Van Hooser]

Thanks Steven and James