matrix operations and component-wise addition using data.table

Question

What is the best way to do component-wise matrix addition if the number of matrices to be summed is not known in advance? More generally, is there a good way to perform matrix (or multi-dimensional array) operations in the context of data.table? I use data.table for its efficiency at sorting and grouping data by several fixed variables, or categories, each comprising a different number of observations.

For example:

1. Find the outer product of vector components given in each observation (row) of the data, returning a matrix for each row.
2. Sum the resulting matrices component-wise over all rows of each grouping of data categories.

Here illustrated with 2x2 matrices and only one category:

library(data.table)

# example data, number of rows differs by category t
N <- 5
dt <- data.table(t = rep(c("a", "b"), each = 3, len = N), 
                 x1 = rep(1:2, len = N), x2 = rep(3:5, len = N),
                 y1 = rep(1:3, len = N), y2 = rep(2:5, len = N))
setkey(dt, t)
> dt
   t x1 x2 y1 y2
1: a  1  3  1  2
2: a  2  4  2  3
3: a  1  5  3  4
4: b  2  3  1  5
5: b  1  4  2  2

I attempted a function to compute matrix sum on outer product, %o%

mat_sum <- function(x1, x2, y1, y2){
  x <- c(x1, x2) # x vector
  y <- c(y1, y2) # y vector
  xy <- x %o% y # outer product (i.e. 2x2 matrix)
  sum(xy)  # <<< THIS RETURNS A SINGLE VALUE, NOT WHAT I WANT.
  }

which, of course, does not work because sum adds up all the elements across the arrays.

I saw this answer using Reduce('+', .list) but that seems to require already having a list of all the matrices to be added. I haven't figured out how to do that within data.table, so instead I've got a cumbersome work-around:

# extract each outer product component first...
mat_comps <- function(x1, x2, y1, y2){
  x <- c(x1, x2) # x vector
  y <- c(y1, y2) # y vector
  xy <- x %o% y # outer product (i.e. 2x2 matrix)
  xy11 <- xy[1,1]
  xy21 <- xy[2,1]
  xy12 <- xy[1,2]
  xy22 <- xy[2,2]
  return(c(xy11, xy21, xy12, xy22))
}

# ...then running this function on dt, 
# taking extra step (making column 'n') to apply it row-by-row...
dt[, n := 1:nrow(dt)]
dt[, c("xy11", "xy21", "xy12", "xy22") := as.list(mat_comps(x1, x2, y1, y2)), 
   by = n]

# ...then sum them individually, now grouping by t
s <- dt[, list(s11 = sum(xy11),
               s21 = sum(xy21),
               s12 = sum(xy12),
               s22 = sum(xy22)),
        by = key(dt)]
> s
   t s11 s21 s12 s22
1: a   8  26  12  38
2: b   4  11  12  23

and that gives the summed components, which can finally be converted back to matrices.

Answer 1

In general, data.table is designed to work with columns. The more you transform your problem to col-wise operations, the more you can get out of data.table.

Here's an attempt at accomplishing this operation col-wise. Probably there are better ways. This is intended more as a template, to provide an idea on approaching the problem (even though I understand it may not be possible in all cases).

xcols <- grep("^x", names(dt))
ycols <- grep("^y", names(dt))
combs <- CJ(ycols, xcols)
len <- seq_len(nrow(combs))
cols = paste("V", len, sep="")
for (i in len) {
    c1 = combs$V2[i]
    c2 = combs$V1[i]
    set(dt, i=NULL, j=cols[i], value = dt[[c1]] * dt[[c2]])
}

#    t x1 x2 y1 y2 V1 V2 V3 V4
# 1: a  1  3  1  2  1  3  2  6
# 2: a  2  4  2  3  4  8  6 12
# 3: a  1  5  3  4  3 15  4 20
# 4: b  2  3  1  5  2  3 10 15
# 5: b  1  4  2  2  2  8  2  8

This basically applies the outer product col-wise. Now it's just a matter of aggregating it.

dt[, lapply(.SD, sum), by=t, .SDcols=cols]

#    t V1 V2 V3 V4
# 1: a  8 26 12 38
# 2: b  4 11 12 23

HTH

---

Edit: Modified cols, c1, c2 a bit to get the output with the correct order for V2 and V3.

Answer 2

EDIT: For not only 2 elements in "x"s and "y"s, a modified function could be:

ff2 = function(x_ls, y_ls)
{
   combs_ls = lapply(seq_along(x_ls[[1]]), 
                     function(i) list(sapply(x_ls, "[[", i), 
                                      sapply(y_ls, "[[", i)))
   rowSums(sapply(combs_ls, function(x) as.vector(do.call(outer, x))))
}

where, "x_ls" and "y_ls" are lists of the respective vectors.

Using it:

dt[, as.list(ff2(list(x1, x2), list(y1, y2))), by = t]
#   t V1 V2 V3 V4
#1: a  8 26 12 38
#2: b  4 11 12 23

And on other "data.frames/tables":

set.seed(101)
DF = data.frame(group = rep(letters[1:3], c(4, 2, 3)), 
                x1 = sample(1:20, 9, T), x2 = sample(1:20, 9, T), 
                x3 = sample(1:20, 9, T), x4 = sample(1:20, 9, T),
                y1 = sample(1:20, 9, T), y2 = sample(1:20, 9, T), 
                y3 = sample(1:20, 9, T), y4 = sample(1:20, 9, T))               
DT = as.data.table(DF)

DT[, as.list(ff2(list(x1, x2, x3, x4), 
                 list(y1, y2, y3, y4))), by = group]
#   group  V1  V2  V3  V4  V5  V6  V7  V8  V9 V10 V11 V12 V13 V14 V15 V16
#1:     a 338 661 457 378 551 616 652 468 460 773 536 519 416 766 442 532
#2:     b 108 261 171  99  29  77  43  29 154 386 238 146 161 313 287 121
#3:     c 345 351 432 293 401 421 425 475 492 558 621 502 510 408 479 492

I don't know, though, how would one in "data.table" not state explicitly which columns to use inside the function; i.e. how you could do the equivalent of:

do.call(rbind, lapply(split(DF[-1], DF$group), 
                      function(x) 
                          do.call(ff2, c(list(x[grep("^x", names(x))]), 
                                         list(x[grep("^y", names(x))])))))
#  [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12] [,13] [,14] [,15] [,16]
#a  338  661  457  378  551  616  652  468  460   773   536   519   416   766   442   532
#b  108  261  171   99   29   77   43   29  154   386   238   146   161   313   287   121
#c  345  351  432  293  401  421  425  475  492   558   621   502   510   408   479   492

OLD ANSWER:

Perhaps you could define your function like:

ff1 = function(x1, x2, y1, y2)
     rowSums(sapply(seq_along(x1), 
                    function(i) as.vector(c(x1[i], x2[i]) %o% c(y1[i], y2[i]))))

dt[, as.list(ff1(x1, x2, y1, y2)), by = list(t)]
#   t V1 V2 V3 V4
#1: a  8 26 12 38
#2: b  4 11 12 23