Column rescaling for a very large sparse matrix in R

Question

I have a very large (~500,000 x ~500,000) sparse matrix in R, and I am trying to divide each column by its sum:

sm = t(t(sm) / colSums(sm))

However, when I do this I get the following error:

# Error in evaluating the argument 'x' in selecting a method for function 't':
# Error: cannot allocate vector of size 721.1 Gb

Is there a better way to do this in R? I am able to store colSums fine, as well as compute and store the transpose of the sparse matrix, but the problem seems to arrive when trying to perform "/". Looks like sparse matrix is converted to full dense matrix here.

Any help would be greatly appreciated. Thank you!

Answer 1

This is what we can do, assuming A is a dgCMatrix:

A@x <- A@x / rep.int(colSums(A), diff(A@p))

This requires some understanding of dgCMatrix class.

1. @x stores none-zero matrix values, in a packed 1D array;
2. @p stores the cumulative number of non-zero elements by column, hence diff(A@p) gives the number of non-zero elements for each column.

We repeat each element of colSums(A) by number of none-zero elements in that column, then divide A@x by this vector. In this end, we update A@x by rescaled values. In this way, column rescaling is done in a sparse manner.

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Example:

library(Matrix)
set.seed(2); A <- Matrix(rbinom(100,10,0.05), nrow = 10)

#10 x 10 sparse Matrix of class "dgCMatrix"

# [1,] . . 1 . 2 . 1 . . 2
# [2,] 1 . . . . . 1 . 1 .
# [3,] . 1 1 1 . 1 1 . . .
# [4,] . . . 1 . 2 . . . .
# [5,] 2 . . . 2 . 1 . . .
# [6,] 2 1 . 1 1 1 . 1 1 .
# [7,] . 2 . 1 2 1 . . 2 .
# [8,] 1 . . . . 3 . 1 . .
# [9,] . . 2 1 . 1 . . 1 .
#[10,] . . . . 1 1 . . . .

diff(A@p)    ## number of non-zeros per column
# [1] 4 3 3 5 5 7 4 2 4 1

colSums(A)   ## column sums
# [1]  6  4  4  5  8 10  4  2  5  2

A@x <- A@x / rep.int(colSums(A), diff(A@p))    ## sparse column rescaling

#10 x 10 sparse Matrix of class "dgCMatrix"

# [1,] .         .    0.25 .   0.250 .   0.25 .   .   1
# [2,] 0.1666667 .    .    .   .     .   0.25 .   0.2 .
# [3,] .         0.25 0.25 0.2 .     0.1 0.25 .   .   .
# [4,] .         .    .    0.2 .     0.2 .    .   .   .
# [5,] 0.3333333 .    .    .   0.250 .   0.25 .   .   .
# [6,] 0.3333333 0.25 .    0.2 0.125 0.1 .    0.5 0.2 .
# [7,] .         0.50 .    0.2 0.250 0.1 .    .   0.4 .
# [8,] 0.1666667 .    .    .   .     0.3 .    0.5 .   .
# [9,] .         .    0.50 0.2 .     0.1 .    .   0.2 .
#[10,] .         .    .    .   0.125 0.1 .    .   .   .

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@thelatemail mentioned another method, by first converting dgCMatrix to dgTMatrix:

AA <- as(A, "dgTMatrix")
A@x <- A@x / colSumns(A)[AA@j + 1L]

For dgTMatrix class there is no @p but @j, giving the column index (0 based) for none zero matrix elements.