## how to build many sparse matrices

### Yuzixuan Zhu (view profile)

on 13 Sep 2018
Latest activity Edited by Yuzixuan Zhu

on 16 Sep 2018

### Matt J (view profile)

I wish to create m = 10^5 sparse matrices of size n by n, say n = 10^4. I have been using
A = cell(m, 1);
for i = 1:m
row = ...; col = ...; val = ...; % here ... means some certain assignment in column vectors
A{i} = sparse(row, col, val, n, n);
end
But it is too slow. So I tried to use the types ndSparse (https://www.mathworks.com/matlabcentral/fileexchange/29832-n-dimensional-sparse-arrays) and sptensor (https://www.sandia.gov/~tgkolda/TensorToolbox/index-2.6.html). They do the job fast by creating m matrices all at once in 3d (n*n*m). It requires concatenating index and value vectors, where the speed is acceptable. However, I then need individual matrices for some operations that do NOT work on types ndSparse and sptensor. For example,
[R, p] = chol(A(:, :, i));
does not work. If I convert the object to Matlab sparse type as
[R, p] = chol(sparse(A(:, :, i)));
then it is even slower than creating A one by one in the for loop. Considering that Matlab does not support multidimensional sparse arrays (so I cannot reshape the abovementioned types into Matlab sparse tensor), how can I speed up creating m sparse matrices? Thank you!

Steven Lord

### Steven Lord (view profile)

on 13 Sep 2018
How are you planning to use those 1e5 sparse matrices later in your code? I want to see if it is possible to reduce the number of sparse matrices you need to create while still achieving your ultimate goal by using a different approach or algorithm.

R2015a

on 13 Sep 2018
Edited by Matt J

### Matt J (view profile)

on 13 Sep 2018

Once you have A in ndSparse form, you can then split it into the cell array form you were originally trying to get using mat2cell:
Ar=sparse( reshape(A,n,m*n) );
Acell=mat2cell(Ar,n,ones(1,m)*n);
and then
[R, p] = chol(Acell{i});

Matt J

### Matt J (view profile)

on 14 Sep 2018
Again, do you really need them all simultaneously? And do you really need them all in separate cells? It seems redundant to have the same data in two places at the same time (Ar and A2).
Yuzixuan Zhu

### Yuzixuan Zhu (view profile)

on 15 Sep 2018
I've been trying to think of a way to not use them simultaneously but haven't found a solution. However I found a way to work around building big sparse matrix (thanks for reminding me that even empty big sparse matrix can take up a lot of memory):
[C, ~, ic] = unique([row{i}; col{i}]);
len_C = length(C);
len_ic2 = length(ic)/2;
Ai = sparse(ic, [ic((len_ic2 + 1):end); ic(1:len_ic2)], val{i}, len_C, len_C);
This works for all my operations. However, the "unique" function in the first line above takes even more time than before... Is there any way to make it faster?
Yuzixuan Zhu

### Yuzixuan Zhu (view profile)

on 16 Sep 2018
I came up with a way to not keep all matrices at the same time, and codes as in the 3rd comment in this answer seem to save 50% time for the large data. Thank you!
May I know what the proper way to include "ndSparse.m" in my software is? Is it enough to include "license.txt" and cite the webpage "https://www.mathworks.com/matlabcentral/fileexchange/29832-n-dimensional-sparse-arrays" in my document?

on 13 Sep 2018
Edited by Matt J

### Matt J (view profile)

on 13 Sep 2018

It might also be a good idea, instead of constructing a 3D sparse array or a cell array of separate matrices, to instead create a big block diagonal matrix, where each n x n matrix is one of the diagonal blocks. That way you can do the entire Cholesky decomposition in a single call to CHOL.

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Matt J

### Matt J (view profile)

on 13 Sep 2018
How does it "not apply" to your application?
Making CHOL a method of ndSparse will not address the problem because the bottleneck would still be in accessing the sub-matrices A(:,:,i). Even with normal 2D sparse matrices, you are seeing that that is problem.
Matt J

### Matt J (view profile)

on 13 Sep 2018
What are the densities of these matrices nnz(A)/numel(A) ?
Yuzixuan Zhu

### Yuzixuan Zhu (view profile)

on 13 Sep 2018
Very sparse -- less then 10 nonzero elements for each matrix.

### Christine Tobler (view profile)

on 13 Sep 2018

The sparse function will often be faster if the second input, col, is sorted in ascending order. If you can cheaply construct col in a way that this is the case, that should help a bit with performance.

Yuzixuan Zhu

### Yuzixuan Zhu (view profile)

on 14 Sep 2018
Thank you for your advice! It may probably work well with denser matrices, but my matrices only have less than 10 nonzero entries each, so sorting them then creating the sparse matrix actually takes more time. :(