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}:
 0.815756   1.9698    0.906113  -1.03114
 1.17761   -0.339905  0.316626   0.084276
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}:
 0.815756   1.9698    0.906113  -1.03114
 1.17761   -0.339905  0.316626   0.084276
julia> einsum!(code1, (A, B), zeros(2, 4), true, false, size_dict)  # the in-place operation2×4 Matrix{Float64}:
 0.815756   1.9698    0.906113  -1.03114
 1.17761   -0.339905  0.316626   0.084276
julia> @ein C[i,k] := A[i,j] * B[j,k]  # all-in-one macro2×4 Matrix{Float64}:
 0.815756   1.9698    0.906113  -1.03114
 1.17761   -0.339905  0.316626   0.084276

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}:
   772.037     242.855    -961.968   …    987.992    431.657     105.605
 -1362.34     1418.97      668.521        178.241    813.71     -426.068
   140.406    -297.093    -689.838        228.569   -817.923     689.101
   229.008    -710.368   -1336.9          511.542    348.596    -275.557
  -923.88     -470.409    -727.274       -891.78     360.767   -1296.85
  -297.03       13.8973   -149.46    …  -1050.8       48.4599   -813.76
  -467.228     108.213   -1118.7          680.517    -87.086    -841.93
  -301.979     444.783   -1516.13         492.875    708.983     344.039
  -216.073    -729.436    -354.813        532.492   -243.59     -761.036
   942.681    -752.188     219.081       1291.61    1590.7       377.303
     ⋮                               ⋱
   109.494    -265.966    -411.084        127.081   1529.92     -687.84
   -75.8842   1511.48      521.704        348.976  -1250.93     1204.27
  -740.917     792.635    1437.87         861.705   2375.74     -614.117
 -1763.4       256.833    1391.75        1713.88    1764.3      1602.2
 -1781.88     -945.265     414.702   …    771.2     2240.79      610.02
  1856.13    -1414.74      941.018       -169.387  -1164.72     -671.997
  -569.241     443.017    1076.9        -1308.76     597.594    -492.722
    99.6067   -703.261    1125.68        1443.36     445.565     177.958
    49.7803  -2063.14       62.1675      -673.704   -525.343   -1090.68
julia> ein"(ij,jk),(kl,lm)->im"(tensors...)  # manually specified contraction100×100 Matrix{Float64}:
   772.037     242.855    -961.968   …    987.992    431.657     105.605
 -1362.34     1418.97      668.521        178.241    813.71     -426.068
   140.406    -297.093    -689.838        228.569   -817.923     689.101
   229.008    -710.368   -1336.9          511.542    348.596    -275.557
  -923.88     -470.409    -727.274       -891.78     360.767   -1296.85
  -297.03       13.8973   -149.46    …  -1050.8       48.4599   -813.76
  -467.228     108.213   -1118.7          680.517    -87.086    -841.93
  -301.979     444.783   -1516.13         492.875    708.983     344.039
  -216.073    -729.436    -354.813        532.492   -243.59     -761.036
   942.681    -752.188     219.081       1291.61    1590.7       377.303
     ⋮                               ⋱
   109.494    -265.966    -411.084        127.081   1529.92     -687.84
   -75.8842   1511.48      521.704        348.976  -1250.93     1204.27
  -740.917     792.635    1437.87         861.705   2375.74     -614.117
 -1763.4       256.833    1391.75        1713.88    1764.3      1602.2
 -1781.88     -945.265     414.702   …    771.2     2240.79      610.02
  1856.13    -1414.74      941.018       -169.387  -1164.72     -671.997
  -569.241     443.017    1076.9        -1308.76     597.594    -492.722
    99.6067   -703.261    1125.68        1443.36     445.565     177.958
    49.7803  -2063.14       62.1675      -673.704   -525.343   -1090.68

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