Basic Usage

In the following example, we demonstrate the einsum notation for basic tensor operations.

Einsum notation

To specify the operation, the user can either use the @ein_str-string literal or the EinCode object. For example, both the following code snippets define the matrix multiplication operation:

julia> using OMEinsum
julia> code1 = ein"ij,jk -> ik"  # the string literalij, jk -> ik
julia> ixs = [[1, 2], [2, 3]]  # the input indices2-element Vector{Vector{Int64}}:
 [1, 2]
 [2, 3]
julia> iy = [1, 3]  # the output indices2-element Vector{Int64}:
 1
 3
julia> code2 = EinCode(ixs, iy)  # the EinCode object (equivalent to the string literal)1∘2, 2∘3 -> 1∘3

The @ein_str macro can be used to define the einsum notation directly in the function call.

julia> A, B = randn(2, 3), randn(3, 4);
julia> code1(A, B)  # matrix multiplication2×4 Matrix{Float64}:
 -2.61076  -1.95243  -0.656993  1.43942
 -2.17537  -2.84521  -2.38323   3.31437
julia> size_dict = OMEinsum.get_size_dict(getixsv(code1), (A, B))  # get the size of the labelsDict{Char, Int64} with 3 entries:
  'j' => 3
  'i' => 2
  'k' => 4
julia> einsum(code1, (A, B), size_dict)  # lower-level function2×4 Matrix{Float64}:
 -2.61076  -1.95243  -0.656993  1.43942
 -2.17537  -2.84521  -2.38323   3.31437
julia> einsum!(code1, (A, B), zeros(2, 4), true, false, size_dict)  # the in-place operation2×4 Matrix{Float64}:
 -2.61076  -1.95243  -0.656993  1.43942
 -2.17537  -2.84521  -2.38323   3.31437
julia> @ein C[i,k] := A[i,j] * B[j,k]  # all-in-one macro2×4 Matrix{Float64}:
 -2.61076  -1.95243  -0.656993  1.43942
 -2.17537  -2.84521  -2.38323   3.31437

Here, we show that the @ein macro combines the einsum notation defintion and the operation in a single line, which is more convenient for simple operations. Separating the einsum notation and the operation (the first approach) can be useful for reusing the einsum notation for multiple input tensors. Lower level functions, einsum and einsum!, can be used for more control over the operation.

For more than two input tensors, the @ein_str macro does not optimize the contraction order. In such cases, the user can use the @optein_str string literal to optimize the contraction order or specify the contraction order manually.

julia> tensors = [randn(100, 100) for _ in 1:4];
julia> optein"ij,jk,kl,lm->im"(tensors...)  # optimized contraction (without knowing the size)100×100 Matrix{Float64}:
 -1996.2     -688.844     52.191   …   1717.8      -759.934   -364.637
  1147.1     -270.668   -644.16         192.212    -977.341    -15.4255
 -1038.38    -991.216   -355.294        851.935    -435.09    1458.44
  -440.975  -1195.49    -342.819        -87.772     239.591    512.456
  -349.853    129.605     25.1695      -911.206    -588.266   -824.554
  1571.02     697.185   -321.533   …   1265.89      503.258    522.084
  -933.436   -282.084   -865.232       -459.03     -866.671   -292.52
   367.692    220.019  -1085.33       -1276.33     -652.851   -907.663
  -287.869   1343.33   -1000.09        -652.037    -234.802   -129.831
   332.608    771.604   -636.148        -78.0908    457.035     14.4169
     ⋮                             ⋱
  1267.3      221.675  -1137.79        -296.931   -1293.52    1609.43
  -642.229   -951.205  -1281.86         204.266   -1250.92     150.965
   657.418   -313.463   1409.44        -936.032    -133.6     -126.27
   340.744   -419.117    792.555        832.515     695.239   2136.14
  1393.96    1037.57   -1105.43    …  -2375.31     -725.771  -2029.0
   801.296   -460.749    592.687       -879.581    -116.868   -835.627
  1413.02    1404.45     527.074      -1490.64     -162.535    969.293
  -368.059   1098.93    2335.78        -326.061     355.318    346.483
   426.73    -122.229   -498.937      -1660.01    -1515.6    -1019.39
julia> ein"(ij,jk),(kl,lm)->im"(tensors...)  # manually specified contraction100×100 Matrix{Float64}:
 -1996.2     -688.844     52.191   …   1717.8      -759.934   -364.637
  1147.1     -270.668   -644.16         192.212    -977.341    -15.4255
 -1038.38    -991.216   -355.294        851.935    -435.09    1458.44
  -440.975  -1195.49    -342.819        -87.772     239.591    512.456
  -349.853    129.605     25.1695      -911.206    -588.266   -824.554
  1571.02     697.185   -321.533   …   1265.89      503.258    522.084
  -933.436   -282.084   -865.232       -459.03     -866.671   -292.52
   367.692    220.019  -1085.33       -1276.33     -652.851   -907.663
  -287.869   1343.33   -1000.09        -652.037    -234.802   -129.831
   332.608    771.604   -636.148        -78.0908    457.035     14.4169
     ⋮                             ⋱
  1267.3      221.675  -1137.79        -296.931   -1293.52    1609.43
  -642.229   -951.205  -1281.86         204.266   -1250.92     150.965
   657.418   -313.463   1409.44        -936.032    -133.6     -126.27
   340.744   -419.117    792.555        832.515     695.239   2136.14
  1393.96    1037.57   -1105.43    …  -2375.31     -725.771  -2029.0
   801.296   -460.749    592.687       -879.581    -116.868   -835.627
  1413.02    1404.45     527.074      -1490.64     -162.535    969.293
  -368.059   1098.93    2335.78        -326.061     355.318    346.483
   426.73    -122.229   -498.937      -1660.01    -1515.6    -1019.39

Sometimes, manually optimizing the contraction order can be beneficial. Please check Contraction order optimization for more details.

Einsum examples

We first define the tensors and then demonstrate the einsum notation for various tensor operations.

julia> using OMEinsum
julia> s = fill(1)  # scalar0-dimensional Array{Int64, 0}:
1
julia> w, v = [1, 2], [4, 5];  # vectors
julia> A, B = [1 2; 3 4], [5 6; 7 8]; # matrices
julia> T1, T2 = reshape(1:8, 2, 2, 2), reshape(9:16, 2, 2, 2); # 3D tensor

Unary examples

julia> ein"i->"(w)  # sum of the elements of a vector.0-dimensional Array{Int64, 0}:
3
julia> ein"ij->i"(A)  # sum of the rows of a matrix.2-element Vector{Int64}:
 3
 7
julia> ein"ii->"(A)  # sum of the diagonal elements of a matrix, i.e., the trace.0-dimensional Array{Int64, 0}:
5
julia> ein"ij->"(A)  # sum of the elements of a matrix.0-dimensional Array{Int64, 0}:
10
julia> ein"i->ii"(w)  # create a diagonal matrix.2×2 Matrix{Int64}:
 1  0
 0  2
julia> ein"i->ij"(w; size_info=Dict('j'=>2))  # repeat a vector to form a matrix.2×2 Matrix{Int64}:
 1  1
 2  2
julia> ein"ijk->ikj"(T1)  # permute the dimensions of a tensor.2×2×2 Array{Int64, 3}:
[:, :, 1] =
 1  5
 2  6

[:, :, 2] =
 3  7
 4  8

Binary examples

julia> ein"ij, jk -> ik"(A, B)  # matrix multiplication.2×2 Matrix{Int64}:
 19  22
 43  50
julia> ein"ijb,jkb->ikb"(T1, T2)  # batch matrix multiplication.2×2×2 Array{Int64, 3}:
[:, :, 1] =
 39  47
 58  70

[:, :, 2] =
 163  187
 190  218
julia> ein"ij,ij->ij"(A, B)  # element-wise multiplication.2×2 Matrix{Int64}:
  5  12
 21  32
julia> ein"ij,ij->"(A, B)  # sum of the element-wise multiplication.0-dimensional Array{Int64, 0}:
70
julia> ein"ij,->ij"(A, s)  # element-wise multiplication by a scalar.2×2 Matrix{Int64}:
 1  2
 3  4

Nary examples

julia> optein"ai,aj,ak->ijk"(A, A, B)  # star contraction.2×2×2 Array{Int64, 3}:
[:, :, 1] =
 68   94
 94  132

[:, :, 2] =
  78  108
 108  152
julia> optein"ia,ajb,bkc,cld,dm->ijklm"(A, T1, T2, T1, A)  # tensor train contraction.2×2×2×2×2 Array{Int64, 5}:
[:, :, 1, 1, 1] =
  9500  14564
 21604  33420

[:, :, 2, 1, 1] =
 11084  17012
 25204  39036

[:, :, 1, 2, 1] =
 13644  20916
 31028  47996

[:, :, 2, 2, 1] =
 15932  24452
 36228  56108

[:, :, 1, 1, 2] =
 13214  20258
 30050  46486

[:, :, 2, 1, 2] =
 15414  23658
 35050  54286

[:, :, 1, 2, 2] =
 19430  29786
 44186  68350

[:, :, 2, 2, 2] =
 22686  34818
 51586  79894