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 OMEinsumjulia> code1 = ein"ij,jk -> ik" # the string literalij, jk -> ikjulia> 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 3julia> 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}: 0.04332 0.310376 -0.892087 0.35848 -0.19543 -0.362537 0.132169 0.0506082julia> 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' => 4julia> einsum(code1, (A, B), size_dict) # lower-level function2×4 Matrix{Float64}: 0.04332 0.310376 -0.892087 0.35848 -0.19543 -0.362537 0.132169 0.0506082julia> einsum!(code1, (A, B), zeros(2, 4), true, false, size_dict) # the in-place operation2×4 Matrix{Float64}: 0.04332 0.310376 -0.892087 0.35848 -0.19543 -0.362537 0.132169 0.0506082julia> @ein C[i,k] := A[i,j] * B[j,k] # all-in-one macro2×4 Matrix{Float64}: 0.04332 0.310376 -0.892087 0.35848 -0.19543 -0.362537 0.132169 0.0506082
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}: -147.049 59.0542 -535.491 … -508.429 1198.57 250.285 454.656 -66.4566 20.4181 -291.112 1315.5 -200.207 -1255.8 102.445 -1116.93 1938.58 -1165.59 475.279 -761.01 186.494 -1230.33 1456.38 -2031.35 -919.034 1092.38 414.261 -1242.11 1444.38 -2580.49 -964.747 -987.642 -1466.98 -969.148 … 790.347 526.017 1848.82 870.915 708.365 754.068 575.921 -1392.14 1907.14 773.563 -452.498 1175.14 2586.07 -601.826 -818.937 1478.0 -1008.87 114.986 649.344 -126.143 -1191.53 41.958 -157.358 68.7165 1729.42 1772.06 301.075 ⋮ ⋱ 256.65 -667.955 1276.98 264.204 -255.161 -418.436 -1052.4 -665.098 -1544.52 -236.829 1158.61 240.563 500.267 -535.343 846.42 -420.044 -922.657 57.4077 -1136.59 -406.651 262.808 1507.06 -885.925 1146.28 -183.93 501.363 -1806.56 … -193.938 -456.192 -1290.0 -1468.34 -280.877 -278.9 -1140.08 142.659 26.8811 185.205 -1414.0 -485.614 -689.548 -1647.18 -1877.76 571.094 1438.68 -900.909 371.953 471.885 37.2684 -206.929 -120.68 -1255.53 895.81 950.013 1363.51julia> ein"(ij,jk),(kl,lm)->im"(tensors...) # manually specified contraction100×100 Matrix{Float64}: -147.049 59.0542 -535.491 … -508.429 1198.57 250.285 454.656 -66.4566 20.4181 -291.112 1315.5 -200.207 -1255.8 102.445 -1116.93 1938.58 -1165.59 475.279 -761.01 186.494 -1230.33 1456.38 -2031.35 -919.034 1092.38 414.261 -1242.11 1444.38 -2580.49 -964.747 -987.642 -1466.98 -969.148 … 790.347 526.017 1848.82 870.915 708.365 754.068 575.921 -1392.14 1907.14 773.563 -452.498 1175.14 2586.07 -601.826 -818.937 1478.0 -1008.87 114.986 649.344 -126.143 -1191.53 41.958 -157.358 68.7165 1729.42 1772.06 301.075 ⋮ ⋱ 256.65 -667.955 1276.98 264.204 -255.161 -418.436 -1052.4 -665.098 -1544.52 -236.829 1158.61 240.563 500.267 -535.343 846.42 -420.044 -922.657 57.4077 -1136.59 -406.651 262.808 1507.06 -885.925 1146.28 -183.93 501.363 -1806.56 … -193.938 -456.192 -1290.0 -1468.34 -280.877 -278.9 -1140.08 142.659 26.8811 185.205 -1414.0 -485.614 -689.548 -1647.18 -1877.76 571.094 1438.68 -900.909 371.953 471.885 37.2684 -206.929 -120.68 -1255.53 895.81 950.013 1363.51
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 OMEinsumjulia> s = fill(1) # scalar0-dimensional Array{Int64, 0}: 1julia> w, v = [1, 2], [4, 5]; # vectorsjulia> A, B = [1 2; 3 4], [5 6; 7 8]; # matricesjulia> 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}: 3julia> ein"ij->i"(A) # sum of the rows of a matrix.2-element Vector{Int64}: 3 7julia> ein"ii->"(A) # sum of the diagonal elements of a matrix, i.e., the trace.0-dimensional Array{Int64, 0}: 5julia> ein"ij->"(A) # sum of the elements of a matrix.0-dimensional Array{Int64, 0}: 10julia> ein"i->ii"(w) # create a diagonal matrix.2×2 Matrix{Int64}: 1 0 0 2julia> ein"i->ij"(w; size_info=Dict('j'=>2)) # repeat a vector to form a matrix.2×2 Matrix{Int64}: 1 1 2 2julia> 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 50julia> ein"ijb,jkb->ikb"(T1, T2) # batch matrix multiplication.2×2×2 Array{Int64, 3}: [:, :, 1] = 39 47 58 70 [:, :, 2] = 163 187 190 218julia> ein"ij,ij->ij"(A, B) # element-wise multiplication.2×2 Matrix{Int64}: 5 12 21 32julia> ein"ij,ij->"(A, B) # sum of the element-wise multiplication.0-dimensional Array{Int64, 0}: 70julia> 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 152julia> 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