# Apply hessian of neural network to array

**URL:** <https://discourse.julialang.org/t/apply-hessian-of-neural-network-to-array/57602>\
**Category:** Machine Learning\
**Created:** [March 20, 2021, 1:56pm UTC](https://discourse.julialang.org/t/apply-hessian-of-neural-network-to-array/57602 "2021-03-20T13:56:35Z")\
**Posts on this page:** 3\
**Page:** 1

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**Author:** ![Sunny](https://avatars.discourse-cdn.com/v4/letter/s/7c8e57/32.png) [@Sunny](https://discourse.julialang.org/u/Sunny)\
**Post date:** [March 20, 2021, 1:56pm UTC](https://discourse.julialang.org/t/apply-hessian-of-neural-network-to-array/57602/1 "2021-03-20T13:56:35Z")

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I can simply take a hessian of some function in multiple points

```julia
hess = x -> (Zygote.hessian.(x->sum(x.^3), x))
hess(rand(2,3))

2×3 Array{Float64,2}:
 2.67795 3.8318 4.9022
 0.216379 5.67439 4.96389

```

But I’m always getting in troubles when I’m trying to implement the same operation for a neural net:

```julia
g = Dense(2,1)
hess = x -> Zygote.hessian.(x->sum(g(x)), x)
hess(rand(2,3))

```

Error:

```julia
MethodError: no method matching (::Dense{typeof(softplus),Array{Float32,2},Array{Float32,1}})(::ForwardDiff.Dual{ForwardDiff.Tag{Zygote.var"#74#75"{var"#206#208"},Float64},Float64,1})
Closest candidates are:
  Any(::AbstractArray{T,N} where N) where {T<:Union{Float32, Float64}, W<:(AbstractArray{T,N} where N)} at /home/solar/.julia/packages/Flux/goUGu/src/layers/basic.jl:134
  Any(::AbstractArray{var"#s127",N} where N where var"#s127"<:AbstractFloat) where {T<:Union{Float32, Float64}, W<:(AbstractArray{T,N} where N)} at /home/solar/.julia/packages/Flux/goUGu/src/layers/basic.jl:137
  Any(::AbstractArray) at /home/solar/.julia/packages/Flux/goUGu/src/layers/basic.jl:121

```

Is this possible to get the hessian of neural network and keep its dimensions as follows (2,2,3) or  
(2,2,m) in general case?

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<div class="post-metadata">

**Author:** ![arlk](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/arlk/32/2465_2.png) [@arlk](https://discourse.julialang.org/u/arlk)\
**Post date:** [March 20, 2021, 2:28pm UTC](https://discourse.julialang.org/t/apply-hessian-of-neural-network-to-array/57602/2 "2021-03-20T14:28:13Z")

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In the first example, do you mean to take the hessian w.r.t each column of the matrix? Because the hessian of a function from \mathbb{R}^2 \to \mathbb{R} should be in \mathbb{R}^{2 \times 2}. I think you perhaps meant to do this:

```julia
julia> hess = x -> (Zygote.hessian(x->sum(x.^3), x)) # note that it's without broadcasting
julia> hess.(eachcol(rand(2, 3)))
3-element Array{Array{Float64,2},1}:
 [1.6979767651037765 0.0; 0.0 3.4428948492135]
 [4.9107531966282725 0.0; 0.0 0.1841352060564092]
 [1.7349169169298424 0.0; 0.0 2.680030582634431]

```

Otherwise in your code you were simply calculating the double derivatives of a scalar function at 6 points. Similarly

```julia
julia> g = Dense(2, 1, softplus)
julia> hess = x -> Zygote.hessian(x->sum(g(x)), x) # again without broadcasting
julia> hess.(eachcol(rand(2, 3)))
3-element Array{Array{Float64,2},1}:
 [0.16390895842649078 -0.0006955633608079606; -0.0006955633608079607 2.9516897279012466e-6]
 [0.15883581950267744 -0.0006740350099870638; -0.0006740350099870637 2.8603321096637323e-6]
 [0.14517203907327184 -0.0006160514493076655; -0.0006160514493076656 2.614273317484523e-6]

```

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<div class="post-metadata">

**Author:** ![Sunny](https://avatars.discourse-cdn.com/v4/letter/s/7c8e57/32.png) [@Sunny](https://discourse.julialang.org/u/Sunny)\
**Post date:** [March 20, 2021, 2:35pm UTC](https://discourse.julialang.org/t/apply-hessian-of-neural-network-to-array/57602/3 "2021-03-20T14:35:37Z")

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This is really what I meant and what I looked for.  
I appreciate your help, thank you