Einstein summation is a concise mathematical notation that implicitly sums over repeated indices of n-dimensional arrays. Many ordinary matrix operations (e.g., transpose, matrix multiplication, scalar product, ‘diag()’, trace, etc.) can be written using Einstein notation. The notation is particularly convenient for expressing operations on arrays with more than two dimensions because the respective operators (‘tensor products’) might not have a standardized name.
You can install the package from CRAN with:
install.packages("einsum")or if you want to use the development version from GitHub:
# install.packages("devtools")
devtools::install_github("const-ae/einsum")Load the package:
library(einsum)Let’s make two matrices:
mat1 <- matrix(rnorm(n = 8 * 4), nrow = 8, ncol = 4)
mat2 <- matrix(rnorm(n = 4 * 4), nrow = 4, ncol = 4)We can use einsum() to calculate the matrix product
einsum("ij, jk -> ik", mat1, mat2)
#> [,1] [,2] [,3] [,4]
#> [1,] -0.5087677 0.6680792 0.2909357 0.49456493
#> [2,] -1.1888008 0.9411126 0.6737345 0.39054429
#> [3,] -1.4715071 1.0242759 0.2400887 -0.31436543
#> [4,] 0.3899863 -1.1212621 -0.7660189 -2.28836686
#> [5,] -0.9058902 0.5529122 0.4775118 0.18014286
#> [6,] -1.4494020 1.6341965 1.4738795 2.73834635
#> [7,] -0.5380896 0.8600228 0.4430880 -0.06022769
#> [8,] 3.0707573 -2.5552313 -1.5538108 -1.28101253which produces the same as the standard matrix multiplication
mat1 %*% mat2
#> [,1] [,2] [,3] [,4]
#> [1,] -0.5087677 0.6680792 0.2909357 0.49456493
#> [2,] -1.1888008 0.9411126 0.6737345 0.39054429
#> [3,] -1.4715071 1.0242759 0.2400887 -0.31436543
#> [4,] 0.3899863 -1.1212621 -0.7660189 -2.28836686
#> [5,] -0.9058902 0.5529122 0.4775118 0.18014286
#> [6,] -1.4494020 1.6341965 1.4738795 2.73834635
#> [7,] -0.5380896 0.8600228 0.4430880 -0.06022769
#> [8,] 3.0707573 -2.5552313 -1.5538108 -1.28101253The matrix multiplication example is straightforward, and there is little benefit of using the Einstein notation over the more familiar matrix product expression. Furthermore, ‘einsum’ is a lot slower.
However, ‘einsum’ truly shines when working with more than 2-dimensional arrays, where it can be difficult to figure out the correct kind of tensor product:
# Make three n-dimensional arrays
arr1 <- array(rnorm(3 * 9 * 2), dim = c(3, 9, 2))
arr2 <- array(rnorm(2 * 5), dim = c(2, 5))
arr3 <- array(rnorm(9 * 3), dim = c(9, 3))
# Sum across axes a, b, and c
einsum("abc, cd, ba -> d", arr1, arr2, arr3)
#> [1] -0.7015596 -4.0114655 -1.6420695 -3.4131292 0.7233701The equivalent expression using tensor products (which are not intuitive) would look like this:
tensor::tensor(tensor::tensor(arr1, arr2, alongA = 3, alongB = 1), arr3, alongA = c(2,1), alongB = c(1, 2))
#> [1] -0.7015596 -4.0114655 -1.6420695 -3.4131292 0.7233701If you need to do the same computation repeatedly, you can use
einsum_generator(), which generates and compiles an
efficient C++ function for that calculation (to see the function code,
set compile_function=FALSE). It can take a few seconds to
compile the function, but the returned function can be one or two orders
of magnitude faster than einsum().
# einsum_generator returns a function
array_prod <- einsum_generator("abc, cd, ba -> d")
array_prod(arr1, arr2, arr3)
#> [1] -0.7015596 -4.0114655 -1.6420695 -3.4131292 0.7233701bench::mark(
tensor = tensor::tensor(tensor::tensor(arr1, arr2, alongA = 3, alongB = 1),
arr3, alongA = c(2,1), alongB = c(1, 2)),
einsum = einsum("abc, cd, ba -> d", arr1, arr2, arr3),
einsum_generator = array_prod(arr1, arr2, arr3)
)
#> # A tibble: 3 × 6
#> expression min median `itr/sec` mem_alloc `gc/sec`
#> <bch:expr> <bch:tm> <bch:tm> <dbl> <bch:byt> <dbl>
#> 1 tensor 48.6µs 52.02µs 18215. 2.93KB 25.4
#> 2 einsum 380.66µs 403.17µs 2373. 2.49KB 23.5
#> 3 einsum_generator 3.85µs 4.32µs 196916. 2.49KB 19.7Lastly, you can also generate C++ code if you need an efficient implementation of some function, which you could (with proper credit) for example paste into your R package:
# The C++ code underlying the tensor product
cat(einsum_generator("abc, cd, ba -> d", compile_function = FALSE))
#> NumericVector einsum_impl_func(NumericVector array1, NumericVector array2, NumericVector array3){
#> NumericVector size(4);
#> IntegerVector array1_dim = array1.hasAttribute("dim") ? array1.attr("dim") : IntegerVector::create(array1.length());
#> IntegerVector array2_dim = array2.hasAttribute("dim") ? array2.attr("dim") : IntegerVector::create(array2.length());
#> IntegerVector array3_dim = array3.hasAttribute("dim") ? array3.attr("dim") : IntegerVector::create(array3.length());
#> size[0] = array1_dim[0];
#> if(size[0] != array3_dim[1]) stop("Dimension 2 of object array3 does not match!");
#> size[1] = array1_dim[1];
#> if(size[1] != array3_dim[0]) stop("Dimension 1 of object array3 does not match!");
#> size[2] = array1_dim[2];
#> if(size[2] != array2_dim[0]) stop("Dimension 1 of object array2 does not match!");
#> size[3] = array2_dim[1];
#>
#> NumericVector result(size[3]);
#> for(int d = 0; d < size[3]; ++d){
#> double sum = 0.0;
#> for(int a = 0; a < size[0]; ++a){
#> for(int b = 0; b < size[1]; ++b){
#> for(int c = 0; c < size[2]; ++c){
#> sum += array1[1 * (a + size[0] * (b + size[1] * (c)))] * array2[1 * (c + size[2] * (d))] * array3[1 * (b + size[1] * (a))];
#> }
#> }
#> }
#> result[1 * (d)] = sum;
#> }
#> result.attr("dim") = IntegerVector::create(size[3]);
#> return result;
#>
#> }This package is inspired by the einsum function in NumPy.