# einsum

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.

## Installation

You can install the development version from GitHub with:

``````# install.packages("devtools")
devtools::install_github("const-ae/einsum")``````

## Example

``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 -1.28101253``````

which 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 -1.28101253``````

The 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)
#>  -0.7015596 -4.0114655 -1.6420695 -3.4131292  0.7233701``````

The 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))
#>  -0.7015596 -4.0114655 -1.6420695 -3.4131292  0.7233701``````

If 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)
#>  -0.7015596 -4.0114655 -1.6420695 -3.4131292  0.7233701``````
``````bench::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 x 6
#>   expression            min   median `itr/sec` mem_alloc `gc/sec`
#>   <bch:expr>       <bch:tm> <bch:tm>     <dbl> <bch:byt>    <dbl>
#> 1 tensor            61.22µs  71.23µs    12995.    2.93KB     84.5
#> 2 einsum            248.1µs 266.25µs     3595.    2.49KB     25.2
#> 3 einsum_generator   2.94µs   3.42µs   245344.    2.49KB     24.5``````

Lastly, 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 = array1_dim;
#> if(size != array3_dim) stop("Dimension 2 of object array3 does not match!");
#> size = array1_dim;
#> if(size != array3_dim) stop("Dimension 1 of object array3 does not match!");
#> size = array1_dim;
#> if(size != array2_dim) stop("Dimension 1 of object array2 does not match!");
#> size = array2_dim;
#>
#> NumericVector result(size);
#> for(int d = 0; d < size; ++d){
#> double sum = 0.0;
#> for(int a = 0; a < size; ++a){
#> for(int b = 0; b < size; ++b){
#> for(int c = 0; c < size; ++c){
#> sum += array1[1 * (a + size * (b + size * (c)))] * array2[1 * (c + size * (d))] * array3[1 * (b + size * (a))];
#> }
#> }
#> }
#> result[1 * (d)] = sum;
#> }
#> result.attr("dim") = IntegerVector::create(size);
#> return result;
#>
#> }``````

# Credit

This package is inspired by the einsum function in NumPy.