In const-ae/einsum: Einstein Summation

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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 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")

Example

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)

which produces the same as the standard matrix multiplication

mat1 %*% mat2

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)

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))

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)

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)
)

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))

Credit

This package is inspired by the einsum function in NumPy.

const-ae/einsum documentation built on Sept. 3, 2023, 3:50 a.m.