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I want to find the rank of a very sparse, quite rectangular matrix mat, but I'm running out of RAM (I have 16 GB) if I try to use MatrixRank or RowReduce. The particular matrix I am considering is 44216 by 5958 and its Tally is

Tally@Flatten@Normal@mat

{{1, 89348}, {0, 263290898}, {-1, 58682}}

I have also tried to produce the Gram matrix

gram = Transpose[mat].mat

since it has the same rank, but this matrix is obviously much less sparse. I also run out of RAM trying to do MatrixRank[gram].

Also, should one take different approaches if the rank can be assumed to be small, or on the other hand, large (almost full rank)?

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  • $\begingroup$ Have you tried using SingularValueList[]? $\endgroup$
    – J. M.'s missing motivation ♦
    Commented Jul 7, 2015 at 14:09
  • $\begingroup$ The notes on internal implementations states that "PseudoInverse, NullSpace, and MatrixRank are based on SingularValueDecomposition." Doesn't that mean that I effectively have? :p I'll test it shortly. $\endgroup$
    – Marius Ladegård Meyer
    Commented Jul 7, 2015 at 14:14
  • $\begingroup$ Yes, but I wasn't asking you to compute the entire decomposition; just the singular values. :P $\endgroup$
    – J. M.'s missing motivation ♦
    Commented Jul 7, 2015 at 14:22
  • $\begingroup$ True ;) MemoryConstrained[si = SingularValueList[gram];, 10000000000] is running now (using mat didn't work as expected), and it doesn't seem to eat memory as quickly as MatrixRank. I'll post again when it completes/aborts. $\endgroup$
    – Marius Ladegård Meyer
    Commented Jul 7, 2015 at 14:27
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    $\begingroup$ Yes, something like that. What gets zeroed of course depends on the Tolerance setting. $\endgroup$
    – J. M.'s missing motivation ♦
    Commented Jul 8, 2015 at 5:44

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