Incorrect Wirtinger for prod operation

[ghost]

The wirtinger function below is taken from the Zygote/complex.

using Zygote

function jacobi(f, x)
    y, back = Zygote.pullback(f, x)
    back(1)[1], back(im)[1]
end

function wirtinger(f, x)
    du, dv = jacobi(f, x)
    (du' + im*dv')/2, (du + im*dv)/2
end

println(wirtinger(x -> prod(x), [1+2im, 3+4im]))
# (Complex{Float64}[3.0 - 4.0im 1.0 - 2.0im], Complex{Float64}[0.0 + 0.0im, 0.0 + 0.0im])

println(wirtinger(x -> x[1] * x[2], [1+2im, 3+4im]))
# (Complex{Float64}[3.0 + 4.0im 1.0 + 2.0im], Complex{Float64}[0.0 + 0.0im, 0.0 + 0.0im])


function myprod(x)
   res = one(eltype(x))
   for v in x
       res *= v
   end
   return res
end

println(wirtinger(x -> myprod(x), [1+2im, 3+4im]))
# (Complex{Float64}[3.0 + 4.0im 1.0 + 2.0im], Complex{Float64}[0.0 + 0.0im, 0.0 + 0.0im])

[mcabbott]

It's entirely possible that the gradient for prod needs to conjugate somewhere:

https://github.com/FluxML/Zygote.jl/blob/master/src/lib/array.jl#L253

[sethaxen]

It's entirely possible that the gradient for prod needs to conjugate somewhere:

https://github.com/FluxML/Zygote.jl/blob/master/src/lib/array.jl#L253

Yeah, that's probably it. Both p and xs need to be conjugated. i.e., you need

@adjoint function prod(xs::AbstractArray; dims = :)
  p = prod(xs; dims = dims)
  p, Δ -> (conj.(p ./ xs) .* Δ,)
end

But you probably also want to restrict the type of xs to xs::AbstractArray{<:Union{Real,Complex}}, because that rule assumes multiplication is commutative, which will fail for xs::AbstractArray{<:AbstractMatrix} or xs::AbstractArray{<:Quaternion}.

If someone puts together a PR, I'm happy to review.