I have a matrix stored as a SparseArray. How can I convert it to a List of its rows, where each row is stored as a SparseArray?

Note that the matrix is huge, so the conversion should use as little memory as possible (the lower bound being the memory of the final List object).

  • 1
    $\begingroup$ Like Table[SparseArray[Band[{1, 1}] -> 1, {4, 4}][[k]], {k, 4}]? $\endgroup$ Feb 23, 2016 at 13:13
  • $\begingroup$ @J.M. Yes, that is the final representation I want. How do I convert an input sparse matrix to this form? $\endgroup$
    – a06e
    Feb 23, 2016 at 13:14
  • $\begingroup$ Well, you could replace the SparseArray[] in my snippet with yours, for example. $\endgroup$ Feb 23, 2016 at 13:16
  • $\begingroup$ @J.M. This works and gives the result that I want. However, for some reason it is very slow for me. Maybe there is a faster way? $\endgroup$
    – a06e
    Feb 23, 2016 at 13:43
  • $\begingroup$ Can you give a (minimal?) example of a sparse matrix where this goes slow? $\endgroup$ Feb 23, 2016 at 13:48

4 Answers 4


List @@ s will do the trick:

s = 1000000 // RandomInteger[{1, 1000000}, {#, 2}] -> RandomReal[1, #]& // SparseArray;

s // Head
(* SparseArray *)

s // Dimensions
(* {1000000, 1000000} *)

l = List @@ s;

l // Head
(* List *)

l // Dimensions
(* {1000000, 1000000} *)

l // First // Head
(* SparseArray *)

Take[l, 4]

sparse array screenshot

Why Does This Work?

Commentators observe that replacing the head of the (apparent) FullForm of a SparseArray ought to produce a nonsensical result. Why, then, does this work?

Despite appearances, a SparseArray is an atom:

SparseArray[Range[10]] // AtomQ
(* True *)

The full-form is therefore synthetic. Any operations that operate upon the (notionally non-existent) subparts of these atoms are implemented as special definitions. The documentation strongly implies that a SparseArray is meant to act virtually interchangeably with a regular List.

The Details section gives explicit examples where the parts of a multidimensional sparse array themselves appear as sparse arrays to operations like Map, Part, Listable, etc. In light of this, it stands to reason that replacing the notional head of a multidimensional sparse array with List would result in a list of sparse arrays. It is likely that the implementation has explicit code to handle this case.

I cannot point to a definitive statement in the documentation that guarantees the behaviour of List @@ s. But I would argue that it is so strongly implied that it would be a regression if it were changed in some future release.

  • $\begingroup$ Simple and fast! $\endgroup$
    – a06e
    Feb 23, 2016 at 16:24
  • 4
    $\begingroup$ Brilliant, but I don't understand why doest it work considering what is the FullForm of a SparseArray. Can you comment on the rationale behind? $\endgroup$
    – rhermans
    Feb 23, 2016 at 16:33
  • 3
    $\begingroup$ Actually I didn't understand either, I was just happy it worked. I second @rhermans request for further explanation. $\endgroup$
    – a06e
    Feb 23, 2016 at 16:55
  • 3
    $\begingroup$ @rhermans I added the section Why Does This Work? $\endgroup$
    – WReach
    Feb 23, 2016 at 18:33

Easy to do with Map.

This generates a random sparse matrix(array):

{m, n} = {1000, 2000};
pairs = Flatten[
   Outer[List, RandomInteger[{1, m}, Floor[0.7 m]], 
    RandomInteger[{1, n}, Floor[0.7 n]]], 1];
smat = SparseArray[
  Thread[pairs -> RandomReal[{0, 1}, Length[pairs]]], {m, n}]

enter image description here

Get the rows:

vecs = Map[# &, smat];

Verify it is all sparse arrays:

Shallow[Head /@ vecs]

(* Out[170]//Shallow= {SparseArray, SparseArray, SparseArray, \
SparseArray, SparseArray, SparseArray, SparseArray, SparseArray, \
SparseArray, SparseArray, <<990>>} *)

Here is some profiling code:

Profiling code:

memOld = MemoryInUse[];
timeOld = Date[];
vecs = Map[# &, smat];
memNew = MemoryInUse[];
timeNew = Date[];

(memNew - memOld)
(memNew - memOld)/memOld // N
timeNew - timeOld  

(* 12172576


  {0, 0, 0, 0, 0, 0.021384} *)

(I got the output on a 2.3 GHz Intel Core i7 MacBook Pro laptop with Mathematica 10.3.2.)

  • $\begingroup$ Hah, I'd done it as Identity /@ smat myself, but I wasn't sure if it was faster or slower than Table[]. $\endgroup$ Feb 23, 2016 at 14:19
  • 2
    $\begingroup$ I did some profiling: Identity /@ smat does not seem faster than #& /@ smat and it seems to be 10-20% less memory consuming. Of course the comparison should be done on a collection of sparse matrices with different sizes and densities. $\endgroup$ Feb 23, 2016 at 14:32

@WReach has provided the ultimate answer. Consider this an extended comment about the performance of ArrayRules

A large SparseArray

sa = SparseArray[
    RandomInteger[{1, 99999}, 2] -> RandomReal[1]
    , 9999]

An example with Table that takes some time even for a subset of the rows (some list will be empty).

First@AbsoluteTiming[Table[sa[[k]], {k, 9999}]]
(* 2.68629 *)

But if you can get away with ArrayRules instead of SparseArray

First@AbsoluteTiming[GatherBy[Most@ArrayRules[sa], First@*First]]
(* 0.0172612 *)
  • 1
    $\begingroup$ With the Table form dont'you extract only the first 9999 rows? $\endgroup$
    – unlikely
    Feb 23, 2016 at 15:45
  • 1
    $\begingroup$ Yes, in the Table form you are missing most rows, that's why the timing is so small I think. $\endgroup$
    – a06e
    Feb 23, 2016 at 15:47
  • 1
    $\begingroup$ not a fair timing comparison unless you add the code to reconstruct the individual sparse arrays. $\endgroup$
    – george2079
    Feb 23, 2016 at 16:21
  • $\begingroup$ Yes, this of not for benchmark, times here are only to show that what is almost impossibly slow even for a subsection of a SparseArray becomes trivial with ArrayRules for the whole set. This may or may not be of use to the OP. $\endgroup$
    – rhermans
    Feb 23, 2016 at 16:25

If m contains your SparseArray

m = SparseArray[{{1, 2} -> a, {3, 2} -> b, {3, 3} -> c}];

enter image description here

then the following will return its list of sparse array rows.

m[[#]] & /@ Range[First@Dimensions@m]

enter image description here

Hope this helps.


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