Julia Tensors.jl

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类型 Tensor 中最后一个参数的含义

在 Tensors.Tensor 中解释了如下的参数：

Tensor{order,dim,T<:Real}

Tensor type supported for order ∈ (1,2,4) and dim ∈ (1,2,3).

上面只解释了 3 个参数，在调用中会显示 Tensor 类型其实有 4 个参数

julia> Tensor{1,3,Float64}((1.0, 2.0, 3.0))
3-element Tensor{1,3,Float64,3}:
 1.0
 2.0
 3.0

其中最后一个参数，表示了元素的个数。

---

运算符 ⊗

Julia 中 Tensors.jl 有个很好玩的运算符 ⊗(\otimes [tab])

julia> A = rand(SymmetricTensor{2, 2});

julia> B = rand(SymmetricTensor{2, 2});

julia> A ⊗ B
2×2×2×2 SymmetricTensor{4,2,Float64,9}:
[:, :, 1, 1] =
 0.271839  0.352792
 0.352792  0.260518

[:, :, 2, 1] =
 0.469146  0.608857
 0.608857  0.449607

[:, :, 1, 2] =
 0.469146  0.608857
 0.608857  0.449607

[:, :, 2, 2] =
 0.504668  0.654957
 0.654957  0.48365

---

Automatic Differentiation

目前还是不清楚怎么求一个函数的 gradient，gradient 期望的是，下面我们想求一个简单的函数 f(x) = x^2 的导数在 xx 处的值，但是如下面几种情况 f1 直接就是错的：

julia> using Tensors

julia> xx = Vec{2}([3., 4.])
2-element Tensor{1,2,Float64,2}:
 3.0
 4.0

julia> f1(x) = x^2
f1 (generic function with 1 method)

julia> f2(x) = x[1]^2
f2 (generic function with 1 method)

julia> f3(x) = Vec{2}( (x[1]^2, x[2]^2) )
f3 (generic function with 1 method)

julia> g1 = gradient(f1,xx)
ERROR: MethodError: no method matching ^(::Tensor{1,2,ForwardDiff.Dual{Nothing,Float64,2},2}, ::Int64)
Closest candidates are:
  ^(::Float16, ::Integer) at math.jl:795
  ^(::Missing, ::Integer) at missing.jl:120
  ^(::Missing, ::Number) at missing.jl:93
  ...
Stacktrace:
 [1] macro expansion at ./none:0 [inlined]
 [2] literal_pow at ./none:0 [inlined]
 [3] f1(::Tensor{1,2,ForwardDiff.Dual{Nothing,Float64,2},2}) at ./REPL[4]:1
 [4] gradient(::typeof(f1), ::Tensor{1,2,Float64,2}) at /Users/yczhang/.julia/packages/Tensors/EOKLk/src/automatic_differentiation.jl:330
 [5] top-level scope at none:0

julia> g2 = gradient(f2,xx)
2-element Tensor{1,2,Float64,2}:
 6.0
 0.0

julia> g3 = gradient(f3,xx)
2×2 Tensor{2,2,Float64,4}:
 6.0  0.0
 0.0  8.0