Clean way to compute jacobian of array summation - arrays

Question

I'm doing some optimization in R and in connection with that I need to write a function that returns a jacobian. It's a very simple jacobian -- just zeros and ones -- but I'd like to populate it quickly and cleanly. My current code works but is very sloppy.
I have a four-dimensional array of probabilities. Index the dimensions by i, j, k, l. My constraint is that, for each i, j, k, the sum of probabilities over index l must equal 1.
I compute my constraint vector like this:
get_prob_array_from_vector <- function(prob_vector, array_dim) {
return(array(prob_vector, array_dim))
}
constraint_function <- function(prob_vector, array_dim) {
prob_array <- get_prob_array_from_vector(prob_vector, array_dim)
prob_array_sums <- apply(prob_array, MARGIN=c(1, 2, 3), FUN=sum)
return(as.vector(prob_array_sums) - 1) # Should equal zero
}
My question is: what is a clean, fast way of computing the jacobian of as.vector(apply(array(my_input_vector, array_dim), MARGIN=c(1, 2, 3), FUN=sum)) -- i.e., my constraint_function in the code above -- with respect to my_input_vector?
Here is my sloppy solution (which I check for correctness against the jacobian function from the numDeriv package):
library(numDeriv)
array_dim <- c(5, 4, 3, 3)
get_prob_array_from_vector <- function(prob_vector, array_dim) {
return(array(prob_vector, array_dim))
}
constraint_function <- function(prob_vector, array_dim) {
prob_array <- get_prob_array_from_vector(prob_vector, array_dim)
prob_array_sums <- apply(prob_array, MARGIN=c(1, 2, 3), FUN=sum)
return(as.vector(prob_array_sums) - 1)
}
constraint_function_jacobian <- function(prob_vector, array_dim) {
prob_array <- get_prob_array_from_vector(prob_vector, array_dim)
jacobian <- matrix(0, Reduce("*", dim(prob_array)[1:3]), length(prob_vector))
## Must be a faster, clearner way of populating jacobian
for(i in seq_along(prob_vector)) {
dummy_vector <- rep(0, length(prob_vector))
dummy_vector[i] <- 1
dummy_array <- get_prob_array_from_vector(dummy_vector, array_dim)
dummy_array_sums <- apply(dummy_array, MARGIN=c(1, 2, 3), FUN=sum)
jacobian_row_idx <- which(dummy_array_sums != 0, arr.ind=FALSE)
stopifnot(length(jacobian_row_idx) == 1)
jacobian[jacobian_row_idx, i] <- 1
} # Is there a fast, readable one-liner that does the same as this for loop?
stopifnot(sum(jacobian) == length(prob_vector))
stopifnot(all(jacobian == 0 | jacobian == 1))
return(jacobian)
}
## Example of a probability array satisfying my constraint
my_prob_array <- array(0, array_dim)
for(i in seq_len(array_dim[1])) {
for(j in seq_len(array_dim[2])) {
my_prob_array[i, j, , ] <- diag(array_dim[3])
}
}
my_prob_array[1, 1, , ] <- 1 / array_dim[3]
my_prob_array[2, 1, , ] <- 0.25 * (1 / array_dim[3]) + 0.75 * diag(array_dim[3])
my_prob_vector <- as.vector(my_prob_array) # Flattened representation of my_prob_array
should_be_zero_vector <- constraint_function(my_prob_vector, array_dim)
is.vector(should_be_zero_vector)
all(should_be_zero_vector == 0) # Constraint is satistied
## Check constraint_function_jacobian for correctness using numDeriv
jacobian_analytical <- constraint_function_jacobian(my_prob_vector, array_dim)
jacobian_numerical <- jacobian(constraint_function, my_prob_vector, array_dim=array_dim)
max(abs(jacobian_analytical - jacobian_numerical)) # Very small
My functions take prob_vector as input -- i.e., a flattened representation of my probability array -- because optimization functions require vector arguments.

Answer 1

Spend some time to understand what you were trying to do, but here is a proposition to replace your constraint_function_jacobian:
enhanced <- function(prob_vector, array_dim) {
firstdim <- Reduce("*", array_dim[1:3])
seconddim <- length(prob_vector)
jacobian <- matrix(0, firstdim, seconddim)
idxs <- split(1:seconddim, cut(1:seconddim, array_dim[4], labels=FALSE))
for (i in seq_along(idxs)) {
diag(jacobian[, idxs[[i]] ]) <- 1
}
stopifnot(sum(jacobian) == length(prob_vector))
stopifnot(all(jacobian == 0 | jacobian == 1))
jacobian
}
Unless I'm wrong, the jacobian construction is filling diagonals with 1, as it is not a square matrix we have to split it on array_dim[4] square matrix to fill up their diagonals with 1.
I did get rid of the transformation of prob_vector to an array to then get its dim as it will be the same as array_dim, skipping this step is not a huge improvement but it simplify the code IMO.
Results are ok according to test:
identical(constraint_function_jacobian(my_prob_vector, array_dim),
enhanced(my_prob_vector, array_dim))
# [1] TRUE
According to benchmark it gives a great speedup:
microbenchmark::microbenchmark(
original=constraint_function_jacobian(my_prob_vector, array_dim),
enhanced=enhanced(my_prob_vector, array_dim), times=100)
# Unit: microseconds
# expr min lq mean median uq max neval cld
# original 16946.979 18466.491 20150.304 19066.7410 19671.4100 28148.035 100 b
# enhanced 678.222 737.948 799.005 796.3905 834.5925 1141.773 100 a