Efficiently replicate matrices in R

Question

I have a matrix and look for an efficient way to replicate it n times (where n is the number of observations in the dataset). For example, if I have a matrix A

A <- matrix(1:15, nrow=3)

then I want an output of the form

rbind(A, A, A, ...) #n times.

Obviously, there are many ways to construct such a large matrix, for example using a for loop or apply or similar functions. However, the call to the "matrix-replication-function" takes place in the very core of my optimization algorithm where it is called tens of thousands of times during one run of my program. Therefore, loops, apply-type of functions and anything similar to that are not efficient enough. (Such a solution would basically mean that a loop over n is performed tens of thousands of times, which is obviously inefficient.) I already tried to use the ordinary rep function, but haven't found a way to arrange the output of rep in a matrix of the desired format.

The solution do.call("rbind", replicate(n, A, simplify=F)) is also too inefficient because rbind is used too often in this case. (Then, about 30% of the total runtime of my program are spent performing the rbinds.)

Does anyone know a better solution?

Answer 1

Two more solutions:

The first is a modification of the example in the question

do.call("rbind", rep(list(A), n))

The second involves unrolling the matrix, replicating it, and reassembling it.

matrix(rep(t(A),n), ncol=ncol(A), byrow=TRUE)

Since efficiency is what was requested, benchmarking is necessary

library("rbenchmark")
A <- matrix(1:15, nrow=3)
n <- 10

benchmark(rbind(A, A, A, A, A, A, A, A, A, A),
          do.call("rbind", replicate(n, A, simplify=FALSE)),
          do.call("rbind", rep(list(A), n)),
          apply(A, 2, rep, n),
          matrix(rep(t(A),n), ncol=ncol(A), byrow=TRUE),
          order="relative", replications=100000)

which gives:

                                                 test replications elapsed
1                 rbind(A, A, A, A, A, A, A, A, A, A)       100000    0.91
3                   do.call("rbind", rep(list(A), n))       100000    1.42
5  matrix(rep(t(A), n), ncol = ncol(A), byrow = TRUE)       100000    2.20
2 do.call("rbind", replicate(n, A, simplify = FALSE))       100000    3.03
4                                 apply(A, 2, rep, n)       100000    7.75
  relative user.self sys.self user.child sys.child
1    1.000      0.91        0         NA        NA
3    1.560      1.42        0         NA        NA
5    2.418      2.19        0         NA        NA
2    3.330      3.03        0         NA        NA
4    8.516      7.73        0         NA        NA

So the fastest is the raw rbind call, but that assumes n is fixed and known ahead of time. If n is not fixed, then the fastest is do.call("rbind", rep(list(A), n). These were for a 3x5 matrix and 10 replications. Different sized matrices might give different orderings.

EDIT:

For n=600, the results are in a different order (leaving out the explicit rbind version):

A <- matrix(1:15, nrow=3)
n <- 600

benchmark(do.call("rbind", replicate(n, A, simplify=FALSE)),
          do.call("rbind", rep(list(A), n)),
          apply(A, 2, rep, n),
          matrix(rep(t(A),n), ncol=ncol(A), byrow=TRUE),
          order="relative", replications=10000)

giving

                                                 test replications elapsed
4  matrix(rep(t(A), n), ncol = ncol(A), byrow = TRUE)        10000    1.74
3                                 apply(A, 2, rep, n)        10000    2.57
2                   do.call("rbind", rep(list(A), n))        10000    2.79
1 do.call("rbind", replicate(n, A, simplify = FALSE))        10000    6.68
  relative user.self sys.self user.child sys.child
4    1.000      1.75        0         NA        NA
3    1.477      2.54        0         NA        NA
2    1.603      2.79        0         NA        NA
1    3.839      6.65        0         NA        NA

If you include the explicit rbind version, it is slightly faster than the do.call("rbind", rep(list(A), n)) version, but not by much, and slower than either the apply or matrix versions. So a generalization to arbitrary n does not require a loss of speed in this case.

Answer 2

Probably this is more efficient:

apply(A, 2, rep, n)