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N-dimensional sparse arrays

version 1.20.0 (24.9 KB) by Matt J
Creates an N-dimensional sparse array object, for arbitrary N.


Updated 16 Mar 2021

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This submission defines a class of N-dimensional sparse arrays for N possibly greater than 2. However, it should really be thought of as a way of starting with an ordinary MATLAB sparse matrix and reshaping it to have N dimensions. In other words, the sparse data must first be able to exist as an ordinary 2D MATLAB sparse matrix before being made N-dimensional. In fact, if the intended array has dimensions MxNxP...YxZ, then the class will store it internally as an ordinary 2D sparse array of dimensions (M*N*P*...*Y)xZ. This leads to certain memory strains when using large numbers of dimensions. I find the class useful mainly for moderate dimensional things like edge detection in 3D imaging, where you often want to hold a sparse 3D edge map.


S=ndSparse(X) where X is an ordinary MATLAB sparse matrix converts X into an ndSparse object. S can be reshaped into an N-dimensional sparse array using its RESHAPE method, for arbitrary N.

S=ndSparse(X,[M,N,P,...]) is equivalent to reshape(ndSparse(X),[M,N,P,...]).

The class also has a variety of static methods that can be used to construct instances of the class. For example,,Values,[m,n,p,...],nzmax)

lets you generate an N-dimensional sparse array from a table of explicit entries. This is a generalization to N dimensions of S=sparse(i,j,s,m,n,nzmax).

Other such methods include:



>> [1 1 1; 2 1 1;2 2 2] , [50,60 70]) %Builds a 2x2x2 sparse array from table

A(:,:,1) =

(1,1) 50
(2,1) 60

A(:,:,2) =

(2,2) 70

Many of the same manipulations common to ordinary multidimensional MATLAB full arrays can be performed on the sparse 3D array A generated above. It can be permuted, summed, concatentated, and so forth e.g.,

>> B=sum( permute([A,A+10], [3,2,1]) ,2)

B(:,:,1) =

(1,1) 120
(2,1) 20

B(:,:,2) =

(1,1) 140
(2,1) 160

Other overloaded methods include BSXFUN, REPMAT, CIRCSHIFT, CONVN, FLIPDIMS, SQUEEZE, SHIFTDIM and many more. Type "methods ndSparse" for a full list and use "help ndSparse.methodname" to get details of usage.

When browsing the list of methods, note that certain common operations have different implementations, optimized for different situations. Specifically, SUM, ANY,ALL, MIN, MAX... have alternative implementations SUMML, ANYML, ALLML, MINML, MAXML which are optimized for "low-dimensional" ndSparse objects OBJ. Here, low-dimensional means that a normal N-column MATLAB sparse matrix won't consume too much memory on your platform for N=MAX(NUMEL(OBJ)./SIZE(OBJ)).

Another feature of the class is that bi-operand operations are allowed between ndSparse objects and MATLAB objects of any numeric type (single, uint16, etc...). This is not true of ordinary MATLAB sparse matrices, as of R2010b.

>> C=eye(2,'single')*B(:,:,2)

C =

(1,1) 140
(2,1) 160

>> whos A B C

Name Size Bytes Class Attributes

A 2x2x2 136 ndSparse
B 2x1x2 140 ndSparse
C 2x1 104 ndSparse

To convert back to an ordinary n-D full array, use the class' overloaded FULL method. To convert to a normal 2D sparse matrix, use the methods SPARSE or SPARSE2D. For example, SPARSE2D will convert an MxNxPx...xQ ndSparse array to the two dimensional (M*N*P*...)xQ sparse matrix in native MATLAB form.

Cite As

Matt J (2021). N-dimensional sparse arrays (, MATLAB Central File Exchange. Retrieved .

Comments and Ratings (134)

Matt J

@Yinhe Zhu,
You mean you would like to create an ndSparse array of single floats? That can't happen until regular Matlab sparse matrices offer the same, I'm afraid, and since people have been asking for that for 15 years, I've grown a bit pessimistic. In any case, it may not save you as much memory as you think. You still need whatever memory is consumed storing the (i,j) coordinates of the non-zero entries.

Yinhe Zhu

Any chance to support 'single' variable assignment?

Matt J

Hi ebon T,
You would need to do bsxfun(@times,A,B)

ebon T

Actually, if you can explain how to multiply (m,n) sparse 2D matrix with (n,n) dense matrix as I asked in the link I'll be grateful.

In brief, I want to use `sum(A_sparse*B,'all ')` as an objective function.
obj = optimproblem('Objective',sum(A_sparse*B,'all '));

ebon T

Hi Matt. How can I multiply 3D (NxNxM) sparse matrix with 2D (NxN) dense matrix to create (NxN) matrix? I want to do element-wise multiplication. A.*B or A*B doesn't work.

Xinru Li

Matt J

Hi Xinru,
You can do B=sparse(A(:,:,5)). However, you really should be able to do almost any matrix operation with a 5x5 ndSparse as with an ordinary sparse matrix.

Xinru Li

Hi Matt,
Yes. Now when I use B=A(:,:,5), B is a 5*5*1 ndSparse matrix. But I want B to be a 5*5 sparse matrix so that I can do some multiplications. Is there any way to do that?

Matt J

Hi Xinru,
Are you saying you tried to do B=A(:,:,5) and the result wasn't what you expected?

Xinru Li

Hi, Matt,
Is there a way to assign some part of ndSparse matrix to a sparse matrix or sparse vector?
For example, I set A =[5,5,5]), and want to assign A(:,:,5) to a 5-by-5 sparse matrix B. Is there any way to realize it?

Yinhe Zhu

@Matt J
Thank you so much!! Very useful!

Matt J

@Yinhe Zhu,

One way is by doing,


Yinhe Zhu

Yinhe Zhu

From my naive method, I can make the current multi-vector to be full, and re-assign it using ndSparse(). It can reclaim the memory, but there is a peak of memory usage of this re-assignment.
X = ndSparse(full(X));

Yinhe Zhu

Hey Matt,
Is there any way to re-collect memory after assign some 0s back to the array? My case is, initially I created a multi-dim array, then I assigned bunch of values. But when iteration goes on, I want to set back 0s to previous entries(columns) to try to re-collect memory. But it seems not working. Even the value is 0, it still not free the size. I'm wondering is there any other way?

Matt J

@John Smith,
Here is a corrected version of the code I suggested:



Y=reshape( permute(X,[dims, setdiff(1:N,dims)]) , prod(szX(dims)),[]);

szY=szX; szY(dims)=1;


Matt J

@John Smith
The most efficient short-term workaround would probably be,



Y=reshape( permute(X,[dims, setdiff(1:N,dims)]) , args{:} );

szY=szX; szY(dims)=1;


John Smith

Great submission!
Unfortunately, it does not support the Matlab 2019+ addition of operations on dimensions vector, e.g., sum(X,[1 3 4 8]).
Would appreciate a fix!

Chang hsiung

Matt J

Hi Ken,
Your Matlab version is not compatible, I'm afraid. It might work with versions as old as R2008a, which is when Matlab introduced the classdef keyword for defining objects,



I'm using matlab R2007b. I get following errors:

- ndstest
-> A function name ("ndSparse") cannot be indexed with "."

- [1 1 1; 2 1 1;2 2 2] , [50,60 70])
-> Undefined variable "ndSparse" or class "".

Am I overlooking the obvious, or is my old matlab release is not compatible?


Biafra Ahanonu


Solve my problem, Thanks!

Kosuke Hamaguchi

Matt J

No reason why what you've shown wouldn't work, as far as I can see, although I cannot comment on what startmatlabpool may be doing, since it is not a native Matlab command.

Lee C

Pretty good! But how to use parfor to assignment???
parfor i=1:4
A(:,:,i)=ww;% A is MxNxP dimensional sparse matrix

Raphael Kaubruegger

Matt J

Hi Raphael,

ndSparse pre-dates R2016b and does not support implicit expansion, I'm afraid. The workaround is to use bsxfun, as in the following example,

>> N=1000; A=ndSparse.sprand([N,N],.001); B=ndSparse.sprand([N,N],.001);
>> C=bsxfun(@times, reshape(A,N,1,N) , B);

Raphael Kaubruegger

Hey Matt,

I would like to do an operation of the following form. I have two NxN sparse matrices A and B.
My goal is to obtain a third matrix C(i, :) = times(A(i, :), B(i, :)).
For full matrices I can achieve this by using C = reshape(A, N, 1, N) .* B.
If i do this reshaping with the ndsparse arrays i get an error during the .* multiplication 'Array dimensions must match for binary array op.'

Do you know a possible work around for this problem? This would be highly appreciated.

Paolo Barbato

Matt J

Hi Zhu,

This happens because you do not have enough zeros to make sparse storage worthwhile. In an ordinary 2D Matlab sparse matrix, you are not only storing the value of each non-zero matrix element a(i,j), but also the coordinates (i,j) of its location. So, memory consumption per non-zero element is greater than for ordinary double matrices. ndSparse is storing your data internally as a native Matlab 2D sparse matrix and so you see this effect in ndSparse as well.

I also suspect that you are using an older version of Matlab. In R2018a, sparse matrix storage is organized a bit more efficiently and, as a consequence, ndSparse will do slightly (but only slightly) better than full storage:

>> a2=ndSparse.sprand([46,28,4096],2/3); a1=full(S); whos a1 a2

Name Size Bytes Class Attributes

a1 46x28x4096 42205184 double
a2 46x28x4096 41113520 ndSparse

Zhu Zhang

I have 3D matrix with 1/3 of zeros. After using ndSparse, the size actually increased. How could that be possible? How may I fix it? I appreciate that in advance.
Name Size Bytes Class Attributes

a1 46x28x4096 42205184 double
a2 46x28x4096 46825504 ndSparse

Stefano Sorti


Hi Matt,

Is it possible to make mldivide make use of parallel computing when using the following syntax:

C=pagefun(@mldivide, A,B)

where A is a ndsparse matrix (1000x1000x1000) and B is normal matrix (1000x1x1000)



Thanks Matt, I will try it.

Matt J

cat(3, ndSparse(A), ndSparse(B))


Hi, I have two sparse matrices A,B of dimensions 500x500 and 500x500, Can you tell me how I can get a sparse matrix with 500x500x2 with ndSparse?

thank you.

Matt J

HI Yuzixuan Zhu,
As far as Im concerned, citing the webpage is fine.


Hi Matt, I wish to use this function in my software ( I'm wondering what would be a proper way to cite it -- Is it enough to include your license together with this function in my software package, and cite this webpage in my document? Thank you!

Matt J

Hi Tom, glad it's what you need. I do not use git, I'm afraid. I would be careful modifying some of the class conversion methods like double(), single(), logical(), etc... Matlab relies on these methods to return native Matlab data types in various situations. For example,

if ndSparse(true)
disp 'Hello'

would throw an error if the result of logical(ndSparse(true)) remained ndSparse.

Tom Deneux

Great, This is EXACTLY what i was looking for, in order to analyze movies of incoming photons (i.e. most of the pixels are zero).

I am editing the code to add a few methods (e.g. diff), corrections (e.g. the idea of 'double' is to convert values to double format, but the output should remain an ndSparse array, isn't it), efficiency improvements (e.g. when the array is already 2D, several commands such as calls to sub2ind allocate unnecessarily memory for mere copies, so i bypass them). Do you want me to share these changes, do you use git?

Matt J


Suppose I have a sparse matrix F of order 4 x 4. Now I want to store this sparse matrix in 3d matrix. What is the syntax for this representation. that is I want to assign A(:, :,1) = F. Since F is sparse it is showing an error. But if I assign A(:, :,1) = full(F), then it is working. But in that case F is not sparse. F consists of many zeros. Kindly tell me how to assign a sparse matrix in a 3d matrix.

Matt J

Hi Haihua Rao,

I have just now gained access to R2018a and have run ndstest with no issue. So, as I was saying earlier, I would need more information from you to reproduce and troubleshoot the error you are seeing.

Matt J

Hi Haihua Rao,

I do not have access to R2018 right now. Regardless, though, I will need a full copy/paste of the error message as well as instructions for how to reproduce it.

Haihua Rao

Hi Matt,
I'am trying to use your ndSparse class, but there was a error. "Error using evalc:Undefined function or variable 'data'."
I think it is because of the matlab Version. The matlab i am using is 2018.
So could you update the new Version of the ndSparse class ?



Robert Alvarez

Did you derive the algorithm from this principle or is there a reference somewhere that gives a more complete explanation?

Matt J

It's a very basic approach. Essentially, the convolution routine treats the ndSparse array as a sum of impulse functions and the convolution sifting property is used.

Robert Alvarez

Could you explain or provide a reference for the algorithm used in the convolution method convn?

Matt J

Hi Mario,

Thanks for your feedback.

I'm not sure which previous comment you are referring to, but the error you're getting isn't due to memory limitations. There are limits that must be imposed on the size of a matrix A(i,j) because there are limits on the range of indices i,j that you can generate with 64-bit double floats. MATLAB may have other criteria for limiting the size as well. In any case, the error you are seeing is coming from MATLAB (not ndSparse specifically) and happens for ordinary sparse matrices as well, e.g.,

>> A=sparse(5000^4,1);
Error using sparse
Sparse matrix sizes must be non-negative integers less than MAXSIZE as defined by COMPUTER. Use HELP
COMPUTER for more details.


I just saw that another use several years ago had a similar question which you answered that it's not possible because if too small RAM. I wonder why this is because ndSparse is actually not creating those large arrays, right? At least when i do[5000, 5000, 5000, 5000], [0])

there is no excessivly large amount of memory used (even though it would have ~10^14 entries). Thank you!


This is a really wonderful tool, thank you! I have a question: When I try to create very large arrays, such as[5000, 5000, 5000, 5000, 5000], [0])

MATLAB shows me the following error:

??? Error using ==> sparse
Sparse matrix sizes must be non-negative integers less than MAXSIZE as defined by COMPUTER. Use
HELP COMPUTER for more details.

Are you aware of a simple fix for this problem?

Pavel Vostatek

Matt J

Hi Niko,

A little more on this. It looks like the problematic line is at the very end of padarray_algo

if islogical(a)
b = logical(b);

If "b" is ndSparse, then logical(b) will convert it to an ordinary logical sparse array.

Matt J

Hi Niko,

I can't be sure, but there is no particular behavior you should expect from padarray for ndSparse. padarray is not a core MATLAB function and I did not overload it for the class, as you will see by doing

>> methods ndSparse

Pretty amazing that it didn't throw and error.

Niko K

Hi. Are you still developing this code? Look at this:

a = logical(zeros(66,66,26));
b = padarray(a,[1,1,1]);
c = ndSparse(a(:),[66,66,26]);
d = padarray(c,[1,1,1]);

Name Size Bytes Class Attributes

a 66x66x26 113256 logical
b 68x68x28 129472 logical
c 66x66x26 353 ndSparse
d 4624x28 241 logical sparse

What's going on there with matrix 'd'?


Caglayan Dicle

Matt J

Do you still have get that error? What version of MATLAB are you using? And what version of ndSparse?

You seem to be the only one with this problem, I'm afraid. It doesn't readily reproduce on my machine and I haven't received similar complaints.


Hi Matt-
Not sure why my previous comment no longer appears, but I had mentioned that I get an error message when attempting to run ndstest -- the error message being something about the inputs to kron being of the wrong type, coming from line 882 of ndtest.


Matt J

Lauren: not clear what you meant by your "comment below". As far as I can tell, you posted no prior comments.


I realized that (for my comment below) the problem is that one of the inputs being sent to kron is ndSparse, which appears to not be ok. (The error comes up when checking convn in line 882 of ndstest).


OK. I will try to find other ways to solve this issue. Thanks Matt.

Matt J

Hi Wei. At minimum, you would probably need to upgrade to a 64-bit OS with more RAM. Your array is pretty big, though. It might be beyond the scope of what ndSparse can handle :(


Hi Matt, when I ran command of[1e6,1e6,1e6]), I got message as following:
??? Error using ==> sparse
Sparse matrix sizes must be non-negative integers less than MAXSIZE as defined
by COMPUTER. Use HELP COMPUTER for more details.

Error in ==> ndSparse> at 2282

My computer is 32 bit windowsXP OS. Matlab version is 2010b. How do I solve this issue? Thanks.

Matt J

@Racquel, I've fixed the indexing bug you reported (and one or two others). Thanks for letting me know.

Raquel A

Thanks for the quick response, yes, I am aware I should be getting that, I just posted it as an example that somehow, the combination (58,58,...) gave an error but other ones that were close to 58 didn't.

Thank you for looking into it.

Matt J

Hi Raquel,

When you do

X(57,57,1) = 1

you should be getting

X(:,:,1) =

(57,57) 1

This is what I get, so I hope what you posted was some sort of typo. As for the case of X(58,58,1) = 1, I will have to investigate that. Thanks for reporting it!

Raquel A

Hi Matt, I have an issue when I run your code (thanks for submitting it though it's great)
Ok so for example I create a sparse matrix:[59,59,10])

somehow, there is an issue if I try to input in X(58,58,1). Every time I do this or any other number in the third dimension, it just fills up the full two first dimensions.

for example:

X(57,57,1) = 1

just gives me
X(:,:,1) =

(58,57) 1

BUT, if I try to do the same in

X(58,58,1) =1

I get

val(:,:,1) =

(1,1) 1
(2,1) 1
(3,1) 1
(4,1) 1
(5,1) 1
(6,1) 1
(7,1) 1
(8,1) 1
(9,1) 1
(10,1) 1
(11,1) 1
(12,1) 1
(13,1) 1
(14,1) 1
(15,1) 1
(16,1) 1
(17,1) 1
(18,1) 1
(19,1) 1
(20,1) 1
(21,1) 1
(22,1) 1


How can i fix this?

Matt J

Hi Josh,

Please give me a code sample (a short one, ideally) that let's me reproduce this.


Bug report: I have all zero ndSparse matrices that return a column vector for their size(). This causes all sorts of problems (cant' cat() them with otherwise same-dimensioned matrices, can't use full() on them...).

Matt J

Hi Yoni,

Make sure that the 1st column of your PCoords data must range between 1 and 5

Similarly, the 2nd and 3rd columns of your PCoords data must range between 1 and 50.

The 4th and 5th column of your PCoords data must range between 1 and 100.

Yoni Browning

Thanks so much for writing this!

I do just have one (hopefully) quick question: I would like to use your class to process a 5-dimensional dataset, lets call it 'P', such that size(P) = [5,50,50,100,100]. As per the recommendations for other commenters, I have been storing the coordinates of each non-zero point in P as the rows of one matrix (PCoords) and the values that should appear at each point in a sperate vector (PVals). Only at the end of my computation do I attempt to use these in a ndSparse object, which I have tried building both of these ways way:
P =,Pvals).
P =,Pvals,size(P),length(Pvals)).
Both of these causes the error:
Error using sub2ind (line 56)
Out of range subscript.

Error in ndSparse>ndCoords2IJ (line 3070)

Error in (line 2337)

Yes, I have checked, and PCoords and Pvals are the same length, dimensions, etc.

I am really sorry if I am missing something obvious, but I am having a hard time figuring out where I am going wrong from looking at your code, and I would really appreciate any direction you could give.

Thanks so much!

Matt J

Hi Mehdi,

The problem is that your posts here are being received not just by me, but also by anyone who might be watchlisting this FEX file. This is SPAM to them. They expect to be receiving news from here only about ndSparse. I've moved your question to this Newsgroup thread:


Hi Matt,

Thanks a lot for your quick reply and sorry about my lack of knowledge in this area. Since your area of expertise is image processing I thought you are the right person to ask. I need to create a 3D image dataset in which certain voxels (whose x,y,z values are my indices) will have a non-zero value.
I don't even mind if they are being assigned any value from 1 to 656535.
Because I can then segment them to two phases to get a 2-phase system.
Could you advise on what I can do to achieve this and get a 3D binary dataset made this way.

Matt J

Hi Mehdi,

Since this no longer concerns ndSparse, you should probably post your issue in the Newsgroup. However, matrices and arrays are not functions. They can only be indexed by integers>=1. So, it is unclear what you mean when you say you want 1s assigned to non-integer indices.


Hi Matt,

So many thanks for your constructive comments. It all worked fine and it indeed had the issues as you mentioned.

However, I am now facing a new issue. In a similar problem, I have to assign 1's to a 3D matrix with the x,y,z values of the coordinates of the points being single-precision floating numbers rather than intergers. Then span the same range of 0<=x<=1000 0<=y<=1000 0<=z<=1800 though.

When I tried the suggestions above it simply gives an error like:

??? Attempted to access b(39.372,36.681,44.517); index must be a positive
integer or logical.

Error in ==> matrix_3D at 24
b(a(i,c-2),a(i,c-1),a(i,c)) = 1;

Could you please advise any solution to this problem? I much appreciate your time.

Matt J

Hi Mehdi,

I can tell you how to do this with ndSparse, but it sounds like your problem has simpler solutions.

First, you cannot use x,y,z coordinates that have value 0 since MATLAB indexing is 1-based. Doing so should have given you an error in your code. I'll assume henceforth that your mesh_data are really 1-based coordinates.

Second, since you say you have a powerful enough machine to hold a full double array of dimensions 1000x1000x1800 in MATLAB memory, it's not clear why simply changing the values from 0 to 1 pushes your memory limits over the edge. Changing the values of a nonsparse array doesn't even consume any extra memory. What's probably happenings is that your mesh_data are not really bounded to 1000x1000x1800 as you think, in which case MATLAB will try to grow the matrix to accomodate the new data. Growing the matrix would require extra memory and is also very speed inefficient. The solution, of course, would be to remove out-of-bounds coordinates from mesh_data or else pre-allocate a larger b to hold the coordinates you have.

Third, since your data consists only of 0s and 1s, you should probably store b as uint8, which would reduce the memory your code above is consuming by a factor of 8. You might want to see what happens if you make the following simple change to your code


In any case, to create the array you are talking about using ndSparse, you could do the following


If you have out-of-bounds mesh_data however this will not work


Hi Matt, I am trying to create a 3D array of size 1000x1000x1800 with all the elements being zero and then assign 1 instead of 0 to some specific elements in the array and finally store it as a binary file.

Although MATLAB creates and saves the 3D array quickly, while trying to change zero's with one's , it runs out of memory.

((The 4,470,000x3 array named "mesh_data" is the coordinates (x,y,z) of the points that I need them to be assigned the value "1" in the 3D array.))

I should mention that all the x,y,z values of the coordinate matrix are non-negative integer numbers with: 0<=x<=1000 0<=y<=1000 0<=z<=1800

My ultimate aim is to export the 3D matrix (after all th esubstitutions), in a binary format (other than MATLAB's default binary format) so that I can use it with other programs.
Could you guide how I can use your program for this purpose?
Any help is much appreciated.

load mesh_data;
save('3d_matrix', 'b','-v7.3');
[r c]=size(a);
for i = 1:r,
b(a(i,c-2),a(i,c-1),a(i,c)) = 1;
save('mesh_matrix', 'b','-v7.3');

Matt J

Josh - you mean you were doing an operation involving an ndSparse array and a full array of uint8s? And you were hoping to get a full array of uint8s as output? If so, I see what you mean. I'm considering redesigning the class so that it stores the non-zero values in tabular form, instead of as a MATLAB 2D sparse matrix. If I do, it will remove the need for mkCompat and you would get uint8 output or anything else you wanted. Until then, though, defaulting to doubles is the only way. A warning might not be a bad idea, though.


In mkCompat, perhaps it would be worth doing some extra checks before defaulting to doubles, or at least a warning? Memory usage can go crazy at that spot. For various reasons, I had a huge array of uint8s. But it was only 1s and 0s. Converting that to a double made my memory usage spike to 8+GB in just a second or two. Let the user know that it was because they gave suboptimal inputs, not because your code is terribly inefficient.


Thanks for the quick fix. Looking forward to your new solution.

Matt J

@Wei: I've fixed the problem, but now I think I have a better solution. I'll post it over the next few days.


found bug:
cat(2, ndSparse(rand(10,1,3)), ndSparse(rand(10,2,3)));

should be sth to do with the untrail in the permute ...

Akram Bin Sediq

Thank you so much. This class should be added to MATLAB soon!


Qiao Liyong

Thank you for your program.

Matt J

Just as an FYI, the update a just submitted should make the kind of loops over slices G(:,:,i) go significantly faster. Here's a simple test I did,

>>[882,882,82]); Z=sprand(882,882,.001);

>> tic; for ii=1:82, G(:,:,ii)=Z; end; toc,
Elapsed time is 0.149033 seconds.


This works great. I love that so many functions are overloaded.

Matt J

Hi Siavash,

If I've understood you, it sounds a lot like the issue in the comment posted above by Mircea (Aug. 27, 2011). If the application seems to hang, it probably hasn't stopped or crashed, but is probably taking a long time because of a sparse indexing bug in MATLAB (see my response Aug. 28, 2011).

The one thing that sounds strange in what you describe is that you say your loop runs fine up until the final iteration, kk_y. Is that correct? And if so, how do you know the earlier iterations run fine? I'll probably need to see your code to have the best picture of what's going on. You can send it through my author email link, if you like.

Beyond that, though, I want to repeat to you, what I told Mircea. Assigning into a sparse matrix (whether it be a normal MATLAB sparse matrix or an ndSparse object) is a bad practice and best kept to a minimum. Instead of loading data into the array slice-by-slice, you should generate a complete table of values in advance and then use it to build the whole 882x882x82 array in one single call to This is the approach I showed you in my comment posted yesterday.


Thanks for your prompt reply.
Actually i have a loop and in that loop I construct a 882by882 sparse matrix kk_y(82) times. and I want to store them in a 882*882*82 sparse matrix. but when it reaches to G(:,:,kk_y) = J_g_hat_kk;
it crashes! (no error, nothing, but only stop everything!).

Matt J

@Siavash - your comment is a bit hard to interpret. You haven't said what kk_y is. Also, you say you're trying to make a "882 x 882 three dimensional sparse matrix", which seems self-contradictory since 882x882 would suggest 2D, not 3D.

However, if I had to guess, I would say that you're looking for this:


G =[ii,jj,kk],ss,[n_st n_st,kk_y]);


I tried to use your code to make a 882 by 882 three dimensional sparse matrix but it did not work and the computer was crashed. Simply this is what I do:
G =[n_st n_st]);
G(:,:,kk_y) = J_g_hat_kk;
J_g_hat_kk is a 882by882 sparse matrix. and n_st is 882. Could you please help me in this respect,
Thanks a lot



I tried to use your could to make a 882 by 882 three dimensional sparse matrix but it did not work and the computer was crashed. Simply this is what I do:
G =[n_st n_st]);
G(:,:,kk_y) = J_g_hat_kk;

J_g_hat_kk is a 882by882 sparse matrix. and n_st is 882. Could you please help me in this respect,

Thanks a lot

Matt J

@Niklaus - sorry I didn't notice until now that this is happening for you in ndstest. Please post the entire error message so that I can know where in ndstest it occurs.

Matt J

Hi Niklaus - the error you describe is unrelated to how new your MATLAB version is. The code is complaining that the reshape operation you're attempting is invalid because it doesn't preserve the number of elements in the array. Like trying to reshape a 2x5 array into a 3x3 array, for example.


Hey Matt!
Sounds like a really neat code! Unfortunately I have an old version of MATLAB (R2009a). When I run ndstest, i get the following error message:

??? Error using ==> ndSparse>ndSparse.ndSparse at 126
The number of DATA elements is not consistent with the given ndShape.

Do you know why this happens? Is there any way for me to fix this? Thank you very much!


Matt J

Incidentally, if you are now happy with the reimplementation I've made of the DISPLAY method, I would appreciate it if you would change your rating. One star out of five is, in any case, a harsh rating just because you don't like 1 out of the 92 methods offered by the class.

Matt J

You talked me into it. I've now updated the distribution so that DISPLAY will print only those A(:,:,i,j,k,...) containing non-zero data.

For displaying purposes, at least, this should solve your problem. Mimicing the data you described, I'm now able to generate the following 37x37x37x37x37x37x37 data set on a 64-bit platform and get a reasonably instantaneous display:


However, I am vaguely concerned that you are pushing the limits of the class by having such a large array volume, even on a 64-bit platform. The class wasn't designed to handle arbitrarily large numel(A), just as ordinary MATLAB 2D sparse matrices were not.

I now clarify this a bit more in the description section of the FEX page.

Matt J

@Jens: The reason I don't store my own tables is because then I couldn't piggy back on the fast sparse arithmetic routines of MATLAB's underlying 2D sparse class. I would have to implement my own.

In any case, the problem with displaying that you mention is probably not strictly related to data storage. It is related to the fact that the ndSparse DISPLAY method mimics the display method for full arrays and tries to print all "sheets" of the form A(:,:,i,j,k,...) to the screen. Since you have 37^5 sheets, naturally this is going to take some time to disaply to the screen.

Upon reflection, I admit that a sparser display would be more appropriate for an ndSparse class. For example, I could modify the DISPLAY method so that it only displays non-zero sheets.

On the other hand, as a designer, I can't really anticipate how every end-user would like their data displayed. Would a display of 2D sheets
A(:,:,i,j,k,...) offer meaningful information to someone working in 7-dimensional space, like yourself, even if I trimmed it down to the non-zero ones? And would every end-user prefer this?

Also, if you only want to display the non-zero data, why not use FIND? Or create your own wrapper for FIND which displays the data the way you want?

Jens Munk Hansen

Hey Matt

Nice idea, but bad implementation. I am working with a parameter study and data is extremely sparse, dimensions are 37x37x37x37x37x37x37, but I have less than 400 non-zero elements. The display function takes more than 10 hours to execute.

Why not simple store tubles and use an intelligent search?

Alexander Stepanov

Illustration of my issue with sensitivity:

profile on
N=1e5; % data points
data=rand(N,1); % random data
ind=NaN(N,4); % random indices

% (100x100x100) x 200000 ind , data, dims);
% (100x200000x100) x 100 circshift(ind,[0 2]) , data, circshift(dims,[0 2]));

ans1=A1(1,:,:,:); % takes about 0.09 secs
ans2=A2(:,:,1,:); % takes about 115 secs !!!!

% results are equal w.r.t. dimension shift

profile viewer
% 99% of time is spent on a line data=builtin('subsref',,E);

Matt J

This sounds different than Mircea's issue. If you can post a short example illustrating the problem, then I can possibly analyze it further

Alexander Stepanov

Ok, hope they'll fix this issue. For now I'll find a workaround.

Btw, for me the problem was not in assigning. I only create this matrix once (to store a very huge dataset), and then slice different data from it when I need. Sensitivity issue cames up on a line 478:


Anyway, thanks again, one of the best submissions I've seen on the exchange)

Matt J

Hi Alexander, thanks for your feedback. I admit the shape sensitivity thing is a drag. However, the feature you describe would be very difficult to implement and require very pervasive changes throughout the code. I would hope that TMW will fix the shape sensitivity soon (it's already been reported) so that the issue will become moot.

I want to reiterate to you, though, what I said to Mircea. Assigning into a sparse matrix (or ndSparse object) is always best kept to a minimum, even without the shape sensitivity problem.

Alexander Stepanov

Hi, this class is just wonderful. However, if you consider making another release, there's one more feature it lacks.

I think you should give user ability to decide how to group matrices to store as 2D sparse. E.g. I need a 4-D matrix MxNxPxQ, but it turns out that due to shape sensitivity discussed above, the best best way for me to store it is (MxN)x(PxQ), otherwise performance decrease dramatically. Unfortunately, this currently can't be done in your class.
Thanks for such a great submission and looking forward to your reply


Thank you again, Matt J. It is clear I was going the wrong way working with sparse matrices (this is the first time I need it to use them). Thank you for your help.

Matt J

@Mircea, the problem you're seeing is rooted in normal 2D MATLAB sparse matrices. It is related to the shape sensitivity issue being discussed in this thread:

There's not much I can do about it for now, but regardless, building sparse arrays the way you are doing is a bad idea, even for normal MATLAB sparse matrices. The efficient way is to create a complete table of non-zero entries first, and then construct the matrix using SPARSE or using

It is also a bad idea to use matrix assignment to alter the pattern of non-zeros in a sparse matrix, as illustrated by Tim Davis here:


Thank you, Matt J, this class is very usefull.

I have the following question: why does the following code

n = 500;
A =[2*n 2*n]);
A(1:2:2*n,1:2:2*n) = rand(n);

run in 0.04 seconds, while the following similar code

n = 500;
A =[2*n 2*n 2]);
A(1:2:2*n,1:2:2*n,1) = rand(n);

runs in 43 seconds (thats 1000 times slower!).

Is there a way to speed up the process of assigning values to truely N-dimensional sparse arrays (N>2)? Thank you.

Matt J

@Ohad, I think that to fix it, you just need need to change the following line




I'll do another release once I've had time to examine this a little further.


I do not seem to be able to assign matrices into arrays. e.g.:

A(:,:,1) = sparse(ones(3))

any ideas?


how do I enter "ndSparse" into the program matlab
I use matlab2010a

Matt J


My latest release includes the change of representation that we were talking about on Feb 10, 2011. That is, the data is now stored internally as a sparse matrix of dimensions (L_1*L_2*L_3...)xL_N. The memory savings should be quite noticeable.

Since the recoding was substantial, please backup your old version and give me feedback on any bugs that I may have introduced. Hopefully, ndstest was a pretty good filter, but you never know...


Thanks Matt :)

Matt J

Maigo, thanks for your feedback. Your problem is indeed version-related. See my similar comment above to Farhad on 31 Jan 2011.

However, in the .zip file you should have found a version compatible with R2009b (or later). It contains isrow and iscolumn implementations and should have been fine for you. See also my comment on 04 Feb 2011.


Thank you very much. This class helped me a lot!

But I've found a bug with the "subsref" method, which calls two functions "isrow" and "iscolumn" that are not defined.

I am using Matlab 7.10.0 (R2010a), so I don't know if this is a version-specific problem. Maybe these two function are defined in 7.11.

Anyway, I think they're pretty easy to implement, and here's the code. Just add them to end of the class file:

function bool = isrow(A)
% isrow - returns true if A is a row vector
dims = size(A);
bool = (length(dims) == 2) && (dims(1) == 1);

function bool = iscolumn(A)
%iscolumn - returns true if A is a column vector
dims = size(A);
bool = (length(dims) == 2) && (dims(2) == 1);

Matt J

Mike, the explanation that I gave you was imprecise. See Tim Davis' explanation here,

He also gives links to expanded discussions on the subject.


Matt: no problem. out of curiosity, how were you able to find out that matlab stores the # of elmnts for each column? on wiki it claims that matlab sparse uses COO format, which includes only the coordinates of each non-zero entry (so this is clearly a lie!). spinoffs of the Yale format seem very popular and do depend on the length of one of the dimensions and so seem a likely candidate, but they store something different from number of entries/column or row. when i go to sparse.mat i see only the help file, and the help file says nothing about what's going on behind the scenes, so just curious how you got to the bottom of it.

Matt J

Mike: "But given that the size of matlab sparse matrices is proportional to the length of the 2nd dimension, I would vote strongly in favor of going to the (L_1*L_2*L_3...)xL_N sparse matrix representation..."

Yes, I've been thinking in that direction myself for a few weeks now. Unfortunately, it will require pervasive changes to the code, and will probably take me a few weeks at least :(


Matt, thanks, that clarifies things. i still have a hard time thinking through how number of elements in each column helps on things like matrix multiplication, but perhaps i'm not being imaginative enough.

But given that the size of matlab sparse matrices is proportional to the length of the 2nd dimension, I would vote strongly in favor of going to the (L_1*L_2*L_3...)xL_N sparse matrix representation you mention unless there's a substantial slowdown in basic linear algebra. In the case of very large, very sparse, and very high dimensional matrices (of interest to me), the difference in storage space will be enormous.

Matt J

"but you're saying the amount of memory of sparse matrices in both cases in linearly proportional to the total size of the matrix being represented??"

The main thing to understand is that every normal MxN sparse matrix internally stores a table of the number of non-zero elements in each column. In other words, it will always contain at least the same amount of data as in rand(1,N).

That's why the following two sparse vectors, even though they both contain all zeros, have very different memory consumptions.

>> A=sparse(zeros(1,1e6)); At=A.';
>> whos A At
Name Size Bytes Class Attributes

A 1x1000000 4000016 double sparse
At 1000000x1 20 double sparse

Pathological? Well, keep in mind that MATLAB sparse matrices were intended mainly for doing fast linear algebra. Apparently, these operations are faster if you know a priori the number of nonzeros in each column, instead of having to recompute them all the time on-the-fly.


I see.. but then I'm confused why the memory consumed by normal sparse matrices doesn't depend entirely on the number of non-zero elements, as a sparse matrix is completely characterized by the coordinates and values of each non-zero element (plus,e.g., a pair of integers specifying the overall size in the 2D case). but you're saying the amount of memory of sparse matrices in both cases in linearly proportional to the total size of the matrix being represented?? if so, i think this is the far more pathological problem with matlab sparse matrices than them being limited to 2 dimensions!

Matt J

Mike, just as a follow-up point, one thing that would reduce memory consumption is if I store the data internally as an (M*N)xP sparse matrix as opposed to a Mx(N*P) matrix.

Since MATLAB sparse matrices with fewer columns consume less memory, this would indeed help with memory consumption. I've been considering this for future releases, but I need to see how pervasive the effect would be.

Matt J

Mike, the amount of memory consumed by an MxNxP ndSparse array A, will essenitally be the same as the ordinary 2D MATLAB sparse matrix of dimensions
Mx(N*P) generated by doing

You can see this better in a larger version of your example,


>>whos A As
Name Size Bytes Class Attributes

A 20x40x60 9696 ndSparse
As 20x2400 9616 double sparse

In fact, the way it works is that the object A is storing its data internally as the 2D matrix As. There is a difference of about 80 bytes due to other minor ndSparse class data.

So the answer to your question is, there is no memory inefficiency in ndSparse that isn't in normal MATLAB matrices. Since mine is built on top of theirs, there's not much I can do about it.

One thing you should be aware of, though, is that the memory consumption of sparse matrices, both mine and MATLAB's, is shape-sensitive. Permuting the dimensions can significantly reduce memory consumption, e.g.,

>> Ap=permute(A,[3 2 1]); whos Ap

Name Size Bytes Class Attributes

Ap 60x40x20 3296 ndSparse


if i do "a = ndSparse([])" and then "a(1,2,3)=4", i get back:
"a(:,:,1) =
All zero sparse: 1-by-2
a(:,:,2) =
All zero sparse: 1-by-2
a(:,:,3) =
(1,2) 5"
each of these "all zero sparse" matrices take up memory and don't seem worth keeping track of. is it possible to eliminate this behavior?

Matt J

@Farhad, I've now included a version of ndSparse that is compatible with R2009b. See the UpdateNotes.txt file for details.

Matt J

@Farhad - You don't have a recent enough version of MATLAB, and so are missing some of its newer functions. You should upgrade to R2010b. Alternatively, you could write your own iscolumn.m

function bool=iscolumn(v)

bool=isequal(size(v), size(v:));

I can't be sure that that will be the only missing function, but I'm pretty optimistic...

Farhad M

Hi, thanks! it must be very much useful for my purpose. However, I couldn't run ndstest since it gives the following error:

??? Undefined function or method 'iscolumn' for input arguments of type 'ndSparse'.

Could you help me with that?

Matt J

@vikranth, I'm not sure I understand your question. What details are you looking for beyond the description above, the GettingStarted file in the .zip, or in the help doc?

vikranth akkidas

can u give some details about the topic

Michael Völker

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