From c41eaa599fb68ab069c459088ab3d6b975ad6ad1 Mon Sep 17 00:00:00 2001
From: benedict-96 <benedikt.brantner@icloud.com>
Date: Fri, 22 Mar 2024 11:22:29 +0100
Subject: [PATCH 01/33] Optimizing in the set of PSD matrices is better.

---
 src/layers/symplectic_stiefel_layer.jl | 66 --------------------------
 1 file changed, 66 deletions(-)
 delete mode 100644 src/layers/symplectic_stiefel_layer.jl

diff --git a/src/layers/symplectic_stiefel_layer.jl b/src/layers/symplectic_stiefel_layer.jl
deleted file mode 100644
index f33a6079f..000000000
--- a/src/layers/symplectic_stiefel_layer.jl
+++ /dev/null
@@ -1,66 +0,0 @@
-
-import SparseArrays
-import Manifolds
-
-"""
-n is input dimension (small);
-N is output dimension (big)
-"""
-struct SymplecticStiefelLayer{inverse, F1, MT <: Manifolds.SymplecticStiefel} <:
-       ManifoldLayer
-    dim_n::Int
-    dim_N::Int
-    manifold::MT
-    initial_weight::F1
-    sympl_in::SparseArrays.SparseMatrixCSC
-    sympl_out::SparseArrays.SparseMatrixCSC
-end
-
-#maybe implement another random number generator; TODO: maybe rename this to reflect that the dimensions are 2n & 2N
-function SymplecticStiefelLayer(dim_n::Int, dim_N::Int; inverse::Bool = false)
-    iseven(dim_n) && iseven(dim_N) || error("Dimension must be even.")
-    dim_n ≤ dim_N || error("Output dimension must be bigger than input dimension.")
-    M = Manifolds.SymplecticStiefel(dim_N, dim_n)
-    init_weight = () -> rand(M)
-    SymplecticStiefelLayer{inverse, typeof(init_weight), typeof(M)}(dim_n, dim_N, M,
-                                                                    init_weight,
-                                                                    SymplecticMatrix(dim_n ÷
-                                                                                     2),
-                                                                    SymplecticMatrix(dim_N ÷
-                                                                                     2))
-end
-
-function Lux.initialparameters(RNG::AbstractRNG, d::SymplecticStiefelLayer)
-    (weight = d.initial_weight(),)
-end
-
-function Lux.initialstates(RNG::AbstractRNG, d::SymplecticStiefelLayer)
-    (IsManifold = true, Manifold = d.manifold)
-end
-
-function Lux.parameterlength(d::SymplecticStiefelLayer)
-    (4 * d.dim_N - 2 * d.dim_n + 1) * d.dim_n
-end
-
-Lux.statelength(d::SymplecticStiefelLayer) = 2
-
-@inline function (d::SymplecticStiefelLayer{false})(x::AbstractVecOrMat, ps::NamedTuple,
-                                                    st::NamedTuple)
-    ps.weight * x, st
-end
-
-@inline function (d::SymplecticStiefelLayer{true})(x::AbstractVecOrMat, ps::NamedTuple,
-                                                   st::NamedTuple)
-    -d.sympl_in * ps.weight' * d.sympl_out * x, st
-end
-
-#TODO: implement cayley retraction your self and also the derivative (parallel transport)
-function update!(l::SymplecticStiefelLayer, x::NamedTuple, dx::NamedTuple,
-                       η::AbstractFloat)
-    Manifolds.retract_cayley!(l.manifold, x.weight, copy(x.weight), dx.weight, η)
-end
-
-#returns symplecticity violation
-function check_symplecticity(l::SymplecticStiefelLayer, x::NamedTuple)
-    norm(x.weight' * l.sympl_out * x.weight - l.sympl_in) / (2 * l.dim_n)
-end
\ No newline at end of file

From 3856e723f318e6de08b8fe40ab2c93a18835fb5b Mon Sep 17 00:00:00 2001
From: benedict-96 <benedikt.brantner@icloud.com>
Date: Fri, 22 Mar 2024 11:27:22 +0100
Subject: [PATCH 02/33] Test still uses Lux setup.

---
 .../multihead_attention_layer_parallel.jl     | 77 -------------------
 1 file changed, 77 deletions(-)
 delete mode 100644 test/layers/multihead_attention_layer_parallel.jl

diff --git a/test/layers/multihead_attention_layer_parallel.jl b/test/layers/multihead_attention_layer_parallel.jl
deleted file mode 100644
index e58d9f247..000000000
--- a/test/layers/multihead_attention_layer_parallel.jl
+++ /dev/null
@@ -1,77 +0,0 @@
-using GeometricMachineLearning, LinearAlgebra, ProgressMeter, Test
-using GeometricMachineLearning: _add
-using CUDA
-
-import Lux, Zygote, Random, MLDatasets, Flux, Lux.gpu, KernelAbstractions
-
-backend = CUDA.CUDABackend()
-
-#MNIST images are 28×28, so a sequence_length of 16 = 4² means the image patches are of size 7² = 49
-image_dim = 28
-patch_length = 7
-n_heads = 7
-n_layers = 5
-patch_number = (image_dim÷patch_length)^2
-
-train_x, train_y = MLDatasets.MNIST(split=:train)[:]
-test_x, test_y = MLDatasets.MNIST(split=:test)[:]
-
-train_x_reshaped = zeros(Float32, patch_length^2, patch_number, size(train_x, 3))
-test_x_reshaped = zeros(Float32, patch_length^2, patch_number, size(test_x, 3))
-
-#preprocessing steps 
-for i in 1:size(train_x, 3)
-    train_x_reshaped[:, :, i] = split_and_flatten(train_x[:, :, i], patch_length)
-end
-for i in 1:size(test_x, 3)
-    test_x_reshaped[:, :, i] = split_and_flatten(test_x[:, :, i], patch_length)
-end
-
-#implement this encoding yourself!
-train_y = Flux.onehotbatch(train_y, 0:9) 
-test_y = Flux.onehotbatch(test_y, 0:9)
-
-model = MultiHeadAttention(patch_length^2, n_heads)
-ps, st = Lux.setup(CUDA.device(), Random.default_rng(), model)
-
-function main(n_data)
-
-    @time output1 = Lux.apply(model, train_x_reshaped[:,:,1:n_data] |> cu, ps, st)[1]
-
-    output2 = KernelAbstractions.zeros(backend, eltype(train_x), patch_length^2, patch_number, n_data)
-
-    @time for i in 1:n_data
-        output2[:, :, i] = Matrix(Lux.apply(model, train_x_reshaped[:, :, i] |> cu, ps, st)[1])
-    end
-
-    @test isapprox(output1, output2)
-end
-
-function main₂(n_data)
-    #this first function is for testing purposes only
-    #loss(ps) = mapreduce(i -> norm(Lux.apply(model, train_x_reshaped[:,:,1:n_data] |> cu, ps, st)[1][:,:,i]), +, 1:n_data)
-    @time output1 = Zygote.gradient(ps -> norm(Lux.apply(model, train_x_reshaped[:,:,1:n_data] |> cu, ps, st)[1]), ps)[1]
-
-    output2 = Zygote.gradient(ps -> norm(Lux.apply(model, train_x_reshaped[:,:,1] |> cu, ps, st)[1]), ps)[1]
-    @time for i in 2:n_data
-        output2 = _add(output2, Zygote.gradient(ps -> norm(Lux.apply(model, train_x_reshaped[:,:,i] |> cu, ps, st)[1]), ps)[1])
-    end
-
-    #=
-    for key1 in (Symbol("PQ"), Symbol("PK"), Symbol("PV"))
-        for head in 1:n_heads
-            key2 = Symbol("head_"*string(head))
-            @test isapprox(output1[key1][key2], output2[key1][key2])
-        end
-    end
-    output1, output2
-    =#
-end
-
-
-for n_data = 100:100:1000
-    print("n_data = ", n_data, "\n")
-    main(n_data)
-    main₂(n_data)
-    print("\n")
-end
\ No newline at end of file

From ad07fbdc607b770d2d5ab51da5f082cb8a59deed Mon Sep 17 00:00:00 2001
From: benedict-96 <benedikt.brantner@icloud.com>
Date: Fri, 22 Mar 2024 11:27:50 +0100
Subject: [PATCH 03/33] test is superfluous and not in CI anyway.

---
 test/layers/psd_like_layer.jl | 13 -------------
 1 file changed, 13 deletions(-)
 delete mode 100644 test/layers/psd_like_layer.jl

diff --git a/test/layers/psd_like_layer.jl b/test/layers/psd_like_layer.jl
deleted file mode 100644
index fc3e7a515..000000000
--- a/test/layers/psd_like_layer.jl
+++ /dev/null
@@ -1,13 +0,0 @@
-using GeometricMachineLearning, Test
-import Random, Lux
-
-N = 10
-n = 5
-
-model = PSDLayer(2*N, 2*n)
-
-ps, st = Lux.setup(Random.default_rng(), model)
-@test typeof(ps.weight) <: StiefelManifold
-
-ps, st = Lux.setup(TrivialInitRNG(), model)
-@test typeof(ps.weight) <: StiefelLieAlgHorMatrix
\ No newline at end of file

From a6a4d3ce984c574bd3eafd071963c302edae2594 Mon Sep 17 00:00:00 2001
From: benedict-96 <benedikt.brantner@icloud.com>
Date: Fri, 22 Mar 2024 11:30:50 +0100
Subject: [PATCH 04/33] Only executing on Ubuntu.

---
 .github/workflows/Latex.yml | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/.github/workflows/Latex.yml b/.github/workflows/Latex.yml
index 7ad8a0072..b263efbf0 100644
--- a/.github/workflows/Latex.yml
+++ b/.github/workflows/Latex.yml
@@ -10,7 +10,7 @@ jobs:
   LatexDocs:
     strategy:
       matrix:
-        os: [ubuntu-latest, macos-latest]
+        os: [ubuntu-latest]
     runs-on: ${{ matrix.os }}
     steps:
       - uses: actions/checkout@v4

From 9d1c10767149e85385d3cda42ec81f7d8967c290 Mon Sep 17 00:00:00 2001
From: benedict-96 <benedikt.brantner@icloud.com>
Date: Fri, 22 Mar 2024 11:42:31 +0100
Subject: [PATCH 05/33] Now using new ANN interface.

---
 Project.toml                                                | 2 +-
 src/layers/attention_layer.jl                               | 4 ++--
 src/layers/bias_layer.jl                                    | 2 +-
 src/layers/classification.jl                                | 2 +-
 src/layers/grassmann_layer.jl                               | 2 +-
 src/layers/multi_head_attention.jl                          | 4 ++--
 src/layers/psd_like_layer.jl                                | 2 +-
 src/layers/resnet.jl                                        | 2 +-
 src/layers/stiefel_layer.jl                                 | 2 +-
 src/layers/sympnets.jl                                      | 6 +++---
 test/attention_layer/apply_multi_head_attention.jl          | 2 +-
 test/attention_layer/attention_setup.jl                     | 6 +++---
 test/data_loader/data_loader_optimization_step.jl           | 2 +-
 test/data_loader/mnist_utils.jl                             | 2 +-
 test/data_loader/optimizer_functor_with_adam.jl             | 2 +-
 test/layers/classification.jl                               | 2 +-
 test/layers/gradient_layer_tests.jl                         | 4 ++--
 test/layers/manifold_layers.jl                              | 4 ++--
 test/layers/resnet_tests.jl                                 | 4 ++--
 test/layers/sympnet_layers_test.jl                          | 6 +++---
 test/layers/sympnet_upscaling.jl                            | 2 +-
 test/optimizers/optimizer_convergence_tests/psd_optim.jl    | 2 +-
 test/optimizers/optimizer_convergence_tests/svd_optim.jl    | 2 +-
 test/optimizers/utils/optimization_step.jl                  | 2 +-
 .../multi_head_attention_stiefel_optim_cache.jl             | 2 +-
 .../multi_head_attention_stiefel_retraction.jl              | 2 +-
 .../multi_head_attention_stiefel_setup.jl                   | 2 +-
 test/transformer_related/transformer_application.jl         | 4 ++--
 test/transformer_related/transformer_gradient.jl            | 4 ++--
 test/transformer_related/transformer_optimizer.jl           | 2 +-
 test/transformer_related/transformer_setup.jl               | 2 +-
 31 files changed, 44 insertions(+), 44 deletions(-)

diff --git a/Project.toml b/Project.toml
index 143d4cb97..735b2131c 100644
--- a/Project.toml
+++ b/Project.toml
@@ -29,7 +29,7 @@ TimerOutputs = "a759f4b9-e2f1-59dc-863e-4aeb61b1ea8f"
 Zygote = "e88e6eb3-aa80-5325-afca-941959d7151f"
 
 [compat]
-AbstractNeuralNetworks = "0.1"
+AbstractNeuralNetworks = "0.2"
 BandedMatrices = "0.17, 1"
 ChainRules = "1"
 ChainRulesCore = "1"
diff --git a/src/layers/attention_layer.jl b/src/layers/attention_layer.jl
index 8d5be53f5..6a74a69c3 100644
--- a/src/layers/attention_layer.jl
+++ b/src/layers/attention_layer.jl
@@ -43,7 +43,7 @@ function parameterlength(d::Attention{M, M, true}) where M
     M*(M-1)
 end
 
-function initialparameters(backend::KernelAbstractions.Backend, T::Type, d::Attention{M, M, false}; rng::AbstractRNG=Random.default_rng(), initializer::AbstractNeuralNetworks.AbstractInitializer=GlorotUniform()) where {M}
+function initialparameters(d::Attention{M, M, false}, backend::KernelAbstractions.Backend, T::Type; rng::AbstractRNG=Random.default_rng(), initializer::AbstractNeuralNetworks.AbstractInitializer=GlorotUniform()) where {M}
     # transformations for queries and keys.
     PQ_weight = KernelAbstractions.allocate(backend, T, M, M)
     PK_weight = KernelAbstractions.allocate(backend, T, M, M)
@@ -52,7 +52,7 @@ function initialparameters(backend::KernelAbstractions.Backend, T::Type, d::Atte
     (PQ=PQ_weight, PK=PK_weight)
 end
 
-function initialparameters(backend::KernelAbstractions.Backend, T::Type, d::Attention{M, M, true}; rng::AbstractRNG=Random.default_rng(), initializer::AbstractNeuralNetworks.AbstractInitializer=GlorotUniform()) where {M}
+function initialparameters(d::Attention{M, M, true}, backend::KernelAbstractions.Backend, T::Type; rng::AbstractRNG=Random.default_rng(), initializer::AbstractNeuralNetworks.AbstractInitializer=GlorotUniform()) where {M}
     # projections for queries, keys and vectors.
     PQ_weight = rand(backend, rng, StiefelManifold{T}, M, M)
     PK_weight = rand(backend, rng, StiefelManifold{T}, M, M)
diff --git a/src/layers/bias_layer.jl b/src/layers/bias_layer.jl
index 6f86cb987..64122c358 100644
--- a/src/layers/bias_layer.jl
+++ b/src/layers/bias_layer.jl
@@ -8,7 +8,7 @@ function BiasLayer(M::Int)
     BiasLayer{M, M}()
 end
 
-function initialparameters(backend::Backend, ::Type{T}, ::BiasLayer{M, M}; rng::AbstractRNG = Random.default_rng(), init_bias = GlorotUniform()) where {M, T}
+function initialparameters(::BiasLayer{M, M}, backend::Backend, ::Type{T}; rng::AbstractRNG = Random.default_rng(), init_bias = GlorotUniform()) where {M, T}
     q_part = KernelAbstractions.zeros(backend, T, M÷2)
     p_part = KernelAbstractions.zeros(backend, T, M÷2)
     init_bias(rng, q_part)
diff --git a/src/layers/classification.jl b/src/layers/classification.jl
index f4204556c..c19fadc5f 100644
--- a/src/layers/classification.jl
+++ b/src/layers/classification.jl
@@ -17,7 +17,7 @@ function Classification(M::Integer, N::Integer, activation; average::Bool=false)
     Classification{M, N, average, typeof(activation)}(activation)
 end
 
-function initialparameters(device::KernelAbstractions.Backend, T::Type, ::Classification{M, N}; rng::Random.AbstractRNG=Random.default_rng(), init_weight! = GlorotUniform()) where {M, N}
+function initialparameters(::Classification{M, N}, device::KernelAbstractions.Backend, T::Type; rng::Random.AbstractRNG=Random.default_rng(), init_weight! = GlorotUniform()) where {M, N}
     weight = KernelAbstractions.allocate(device, T, N, M)
     init_weight!(rng, weight)
     (weight=weight, )
diff --git a/src/layers/grassmann_layer.jl b/src/layers/grassmann_layer.jl
index a0492ce4e..efb31f897 100644
--- a/src/layers/grassmann_layer.jl
+++ b/src/layers/grassmann_layer.jl
@@ -8,7 +8,7 @@ function GrassmannLayer(n::Integer, N::Integer, Retraction::AbstractRetraction=d
     GrassmannLayer{n, N, typeof(Retraction)}()
 end
 
-function AbstractNeuralNetworks.initialparameters(backend::KernelAbstractions.Backend, ::Type{T}, d::GrassmannLayer{N,M}; rng::AbstractRNG=Random.default_rng()) where {M,N,T}
+function AbstractNeuralNetworks.initialparameters(d::GrassmannLayer{N,M}, backend::KernelAbstractions.Backend, ::Type{T}; rng::AbstractRNG=Random.default_rng()) where {M,N,T}
     (weight = N > M ? rand(backend, rng, GrassmannManifold{T}, N, M) : rand(backend, rng, GrassmannManifold{T}, M, N), )
 end
 
diff --git a/src/layers/multi_head_attention.jl b/src/layers/multi_head_attention.jl
index e07820cb7..9624d75a6 100644
--- a/src/layers/multi_head_attention.jl
+++ b/src/layers/multi_head_attention.jl
@@ -19,7 +19,7 @@ function parameterlength(d::MultiHeadAttention{M, M, true}) where M
     Int(3*M^2 - 3*M*(M + d.n_heads)/(2*d.n_heads))
 end
 
-function initialparameters(backend::KernelAbstractions.Backend, T::Type, d::MultiHeadAttention{M, M, false}; rng::AbstractRNG=Random.default_rng(), initializer::AbstractNeuralNetworks.AbstractInitializer=GlorotUniform()) where {M}
+function initialparameters(d::MultiHeadAttention{M, M, false}, backend::KernelAbstractions.Backend, T::Type; rng::AbstractRNG=Random.default_rng(), initializer::AbstractNeuralNetworks.AbstractInitializer=GlorotUniform()) where {M}
     # number of "hidden" dimension (dimension of projection) 
     Dₕ = M ÷ d.n_heads
     # projections for queries, keys and values.
@@ -51,7 +51,7 @@ function initialparameters(backend::KernelAbstractions.Backend, T::Type, d::Mult
 end
 
 
-function initialparameters(backend::KernelAbstractions.Backend, T::Type, d::MultiHeadAttention{M, M, true}; rng::AbstractRNG=Random.default_rng(), initializer::AbstractNeuralNetworks.AbstractInitializer=GlorotUniform()) where {M}
+function initialparameters(d::MultiHeadAttention{M, M, true}, backend::KernelAbstractions.Backend, T::Type; rng::AbstractRNG=Random.default_rng(), initializer::AbstractNeuralNetworks.AbstractInitializer=GlorotUniform()) where {M}
     # number of "hidden" dimension (dimension of projection) 
     Dₕ = M ÷ d.n_heads
     # projections for queries, keys and vectors.
diff --git a/src/layers/psd_like_layer.jl b/src/layers/psd_like_layer.jl
index 7553b8266..9eeb29ebe 100644
--- a/src/layers/psd_like_layer.jl
+++ b/src/layers/psd_like_layer.jl
@@ -25,7 +25,7 @@ function parameterlength(::PSDLayer{M, N}) where {M, N}
     M÷2*(N÷2 - (M÷2+1)÷2)
 end 
 
-function initialparameters(backend::KernelAbstractions.Backend, T::Type, ::PSDLayer{M, N}, rng::AbstractRNG=Random.default_rng()) where {M, N}
+function initialparameters(::PSDLayer{M, N}, backend::KernelAbstractions.Backend, T::Type; rng::AbstractRNG=Random.default_rng()) where {M, N}
     (weight =  N > M ? rand(backend, rng, StiefelManifold{T}, N÷2, M÷2) : rand(backend, rng, StiefelManifold{T}, M÷2, N÷2), )
 end
 
diff --git a/src/layers/resnet.jl b/src/layers/resnet.jl
index 78598463c..7773329b5 100644
--- a/src/layers/resnet.jl
+++ b/src/layers/resnet.jl
@@ -6,7 +6,7 @@ function ResNet(dim::IT, activation=identity; use_bias::Bool=true) where {IT<:In
     return ResNet{dim, dim, use_bias, typeof(activation)}(activation)
 end
 
-function initialparameters(backend::KernelAbstractions.Backend, T::Type, ::ResNet{M, M, use_bias}; rng::Random.AbstractRNG=Random.default_rng(), init_weight = GlorotUniform(), init_bias = ZeroInitializer()) where {M, use_bias}
+function initialparameters(::ResNet{M, M, use_bias}, backend::KernelAbstractions.Backend, T::Type; rng::Random.AbstractRNG=Random.default_rng(), init_weight = GlorotUniform(), init_bias = ZeroInitializer()) where {M, use_bias}
     if use_bias
         weight = KernelAbstractions.allocate(backend, T, M, M)
         bias = KernelAbstractions.allocate(backend, T, M)
diff --git a/src/layers/stiefel_layer.jl b/src/layers/stiefel_layer.jl
index 58c6839bd..51492410a 100644
--- a/src/layers/stiefel_layer.jl
+++ b/src/layers/stiefel_layer.jl
@@ -8,7 +8,7 @@ function StiefelLayer(n::Integer, N::Integer; retraction::AbstractRetraction=def
     StiefelLayer{n, N, typeof(retraction)}()
 end
 
-function AbstractNeuralNetworks.initialparameters(backend::KernelAbstractions.Backend, ::Type{T}, d::StiefelLayer{M,N}; rng::AbstractRNG=Random.default_rng()) where {M,N,T}
+function AbstractNeuralNetworks.initialparameters(d::StiefelLayer{M,N}, backend::KernelAbstractions.Backend, ::Type{T}; rng::AbstractRNG=Random.default_rng()) where {M,N,T}
     (weight = N > M ? rand(backend, rng, StiefelManifold{T}, N, M) : rand(backend, rng, StiefelManifold{T}, M, N), )
 end
 
diff --git a/src/layers/sympnets.jl b/src/layers/sympnets.jl
index d749bc7e2..9f4afe7b8 100644
--- a/src/layers/sympnets.jl
+++ b/src/layers/sympnets.jl
@@ -148,7 +148,7 @@ function Gradient(dim::Int, dim2::Int=dim, activation = identity; full_grad::Boo
         end
 end
 
-function initialparameters(backend::Backend, ::Type{T}, d::GradientLayer{M, M}; rng::AbstractRNG = Random.default_rng(), init_weight = GlorotUniform(), init_bias = ZeroInitializer(), init_scale = GlorotUniform()) where {M, T}
+function initialparameters(d::GradientLayer{M, M}, backend::Backend, ::Type{T}; rng::AbstractRNG = Random.default_rng(), init_weight = GlorotUniform(), init_bias = ZeroInitializer(), init_scale = GlorotUniform()) where {M, T}
         K = KernelAbstractions.allocate(backend, T, d.second_dim÷2, M÷2)
         b = KernelAbstractions.allocate(backend, T, d.second_dim÷2)
         a = KernelAbstractions.allocate(backend, T, d.second_dim÷2)
@@ -158,13 +158,13 @@ function initialparameters(backend::Backend, ::Type{T}, d::GradientLayer{M, M};
         return (weight=K, bias=b, scale=a)
 end
 
-function initialparameters(backend::Backend, ::Type{T}, ::ActivationLayer{M, M}; rng::AbstractRNG = Random.default_rng(), init_scale = GlorotUniform()) where {M, T}
+function initialparameters(::ActivationLayer{M, M}, backend::Backend, ::Type{T}; rng::AbstractRNG = Random.default_rng(), init_scale = GlorotUniform()) where {M, T}
         a = KernelAbstractions.zeros(backend, T, M÷2)
         init_scale(rng, a)
         return (scale = a,)
 end
 
-function initialparameters(backend::Backend, ::Type{T}, ::LinearLayer{M, M}; rng::AbstractRNG = Random.default_rng(), init_weight = GlorotUniform()) where {M, T}
+function initialparameters(::LinearLayer{M, M}, backend::Backend, ::Type{T}; rng::AbstractRNG = Random.default_rng(), init_weight = GlorotUniform()) where {M, T}
         S = KernelAbstractions.allocate(backend, T, (M÷2)*(M÷2+1)÷2)
         init_weight(rng, S)
         (weight=SymmetricMatrix(S, M÷2), )
diff --git a/test/attention_layer/apply_multi_head_attention.jl b/test/attention_layer/apply_multi_head_attention.jl
index 967411b48..bb2aa001b 100644
--- a/test/attention_layer/apply_multi_head_attention.jl
+++ b/test/attention_layer/apply_multi_head_attention.jl
@@ -11,7 +11,7 @@ function compare_attention_to_mha(N, batch_size=10, T=Float32)
     model₃ = MultiHeadAttention(N, 1, add_connection=true)
     model₄ = Attention(N, softmax, add_connection=true)
 
-    ps₂ = initialparameters(CPU(), T, model₂)
+    ps₂ = initialparameters(model₂, CPU(), T)
     ps₁ = (PQ=(head_1=ps₂.PQ,), PK=(head_1=ps₂.PK,), PV=(head_1=typeof(ps₂.PK)(I(N)),))
 
     mat = rand(T, N, N)
diff --git a/test/attention_layer/attention_setup.jl b/test/attention_layer/attention_setup.jl
index 2a8d01e7d..7ea8370e1 100644
--- a/test/attention_layer/attention_setup.jl
+++ b/test/attention_layer/attention_setup.jl
@@ -13,9 +13,9 @@ function attention_tests(N, T=Float32)
     # same as model₁, but with the add connection
     model₄ = Attention(N, Stiefel=false, add_connection=true)
 
-    ps₁ = initialparameters(CPU(), T, model₁)
-    ps₂ = initialparameters(CPU(), T, model₂)
-    ps₃ = initialparameters(CPU(), T, model₃)
+    ps₁ = initialparameters(model₁, CPU(), T)
+    ps₂ = initialparameters(model₂, CPU(), T)
+    ps₃ = initialparameters(model₃, CPU(), T)
     @test typeof(ps₂.PQ) <: StiefelManifold 
     @test typeof(ps₂.PK) <: StiefelManifold 
 
diff --git a/test/data_loader/data_loader_optimization_step.jl b/test/data_loader/data_loader_optimization_step.jl
index 3ffa5888a..7a245b0e2 100644
--- a/test/data_loader/data_loader_optimization_step.jl
+++ b/test/data_loader/data_loader_optimization_step.jl
@@ -12,7 +12,7 @@ function test_data_loader(sys_dim, n_time_steps, n_params, T=Float32)
 
     # first argument is sys_dim, second is number of heads, third is number of units
     model = Transformer(dl.input_dim, 2, 1)
-    ps = initialparameters(CPU(), T, model)
+    ps = initialparameters(model, CPU(), T)
     dx = Zygote.gradient(ps -> GeometricMachineLearning.loss(model, ps, dl), ps)[1]
     ps_copy = deepcopy(ps)
     o = Optimizer(GradientOptimizer(), ps)
diff --git a/test/data_loader/mnist_utils.jl b/test/data_loader/mnist_utils.jl
index 29e3bc87f..e65fe1ddd 100644
--- a/test/data_loader/mnist_utils.jl
+++ b/test/data_loader/mnist_utils.jl
@@ -64,7 +64,7 @@ function test_optimizer_for_classification_layer(; dim₁=28, dim₂=28, number_
     activation_function(x) = tanh.(x)
     model = Classification(patch_length * patch_length, 10, activation_function)
 
-    ps = initialparameters(CPU(), T, model)   
+    ps = initialparameters(model, CPU(), T)   
     loss₁ = GeometricMachineLearning.loss(model, ps, dl)
 
     opt = Optimizer(GradientOptimizer(), ps)
diff --git a/test/data_loader/optimizer_functor_with_adam.jl b/test/data_loader/optimizer_functor_with_adam.jl
index 267d92312..55cc6e236 100644
--- a/test/data_loader/optimizer_functor_with_adam.jl
+++ b/test/data_loader/optimizer_functor_with_adam.jl
@@ -26,7 +26,7 @@ function test_optimizer_functor_with_adam(;T=Float32, dim₁=6, dim₂=6, n_imag
     # input dim is dim₁ / patch_length * dim₂ / pach_length; the transformer is called with dim₁ / patch_length and two layers
     model = Chain(Transformer(dl.input_dim, patch_length, 2; Stiefel=true), Classification(dl.input_dim, 10, σ))
 
-    ps = initialparameters(CPU(), Float32, model)
+    ps = initialparameters(model, CPU(), Float32)
 
     loss₁ = GeometricMachineLearning.loss(model, ps, dl)
 
diff --git a/test/layers/classification.jl b/test/layers/classification.jl
index 259b6e67e..001fa758d 100644
--- a/test/layers/classification.jl
+++ b/test/layers/classification.jl
@@ -5,7 +5,7 @@ Random.seed!(1234)
 
 function test_set_up_and_application(T=Float32, sys_dim=49, output_dim=10, seq_length=16, batch_size=32; average=false)
     d = Classification(sys_dim, output_dim, σ, average=average)
-    ps = initialparameters(CPU(), T, d)
+    ps = initialparameters(d, CPU(), T)
     output₁ = d(rand(T, sys_dim, seq_length), ps)
     output₂ = d(rand(T, sys_dim, seq_length, batch_size), ps)
     @test size(output₁) == (10, 1)
diff --git a/test/layers/gradient_layer_tests.jl b/test/layers/gradient_layer_tests.jl
index a3a4da2c9..6567d2e2b 100644
--- a/test/layers/gradient_layer_tests.jl
+++ b/test/layers/gradient_layer_tests.jl
@@ -7,7 +7,7 @@ Random.seed!(1234)
 
 function test_gradient_layer_application(T, M, N, batch_size=10)
     dummy_model = GradientLayerQ(M, N, tanh)
-    ps = initialparameters(CPU(), T, dummy_model)
+    ps = initialparameters(dummy_model, CPU(), T)
 
     x = rand(T, M)
     x_applied = dummy_model(x, ps)
@@ -22,7 +22,7 @@ end
 
 function test_gradient_layer_derivative_and_update(T, M, N, batch_size=10)
     dummy_model = Chain(GradientLayerP(M, N, tanh), GradientLayerQ(M, N, tanh))
-    ps = initialparameters(CPU(), T, dummy_model)
+    ps = initialparameters(dummy_model, CPU(), T)
     o = Optimizer(AdamOptimizer(T(0.1), T(.9), T(0.999), T(3e-7)), ps)
 
     # test for vector 
diff --git a/test/layers/manifold_layers.jl b/test/layers/manifold_layers.jl
index 7e2fdb93c..492a056c5 100644
--- a/test/layers/manifold_layers.jl
+++ b/test/layers/manifold_layers.jl
@@ -5,7 +5,7 @@ Random.seed!(1234)
 
 function stiefel_layer_test(T, M, N, tol=1f-1)
     model = Chain(StiefelLayer(M, N), StiefelLayer(N, N))
-    ps = initialparameters(T, model)
+    ps = initialparameters(model, T)
     o = Optimizer(AdamOptimizer(T(1f0), T(5f-1), T(5f-1), T(3f-7)),ps)
 
     dx = ((weight=rand(T,N,M),),(weight=rand(T,N,N),))
@@ -20,7 +20,7 @@ end
 
 function grassmann_layer_test(T, M, N, tol=1f-1)
     model = Chain(GrassmannLayer(M, N), StiefelLayer(N, N))
-    ps = initialparameters(T, model)
+    ps = initialparameters(model, T)
     o = Optimizer(AdamOptimizer(T(1f0), T(5f-1), T(5f-1), T(3f-7)),ps)
 
     dx = ((weight=rand(T,N,M),),(weight=rand(T,N,N),))
diff --git a/test/layers/resnet_tests.jl b/test/layers/resnet_tests.jl
index 54aaee7df..34d5ca6f9 100644
--- a/test/layers/resnet_tests.jl
+++ b/test/layers/resnet_tests.jl
@@ -6,8 +6,8 @@ Random.seed!(1234)
 function test_resnet(M, batch_size=10, T=Float32)
     model₁ = ResNet(M, tanh, use_bias=false)
     model₂ = ResNet(M, tanh, use_bias=true)
-    ps₁ = initialparameters(CPU(), T, model₁)
-    ps₂ = initialparameters(CPU(), T, model₂) 
+    ps₁ = initialparameters(model₁, CPU(), T)
+    ps₂ = initialparameters(model₂, CPU(), T) 
     @test parameterlength(model₁) == M^2
     @test parameterlength(model₂) == M*(M+1)
     A = randn(T, M, M*2, batch_size)
diff --git a/test/layers/sympnet_layers_test.jl b/test/layers/sympnet_layers_test.jl
index a8ab61978..de05ec5e8 100644
--- a/test/layers/sympnet_layers_test.jl
+++ b/test/layers/sympnet_layers_test.jl
@@ -10,9 +10,9 @@ function sympnet_tests(N, N2=2*N, second_dim=10, third_dim=10, T=Float32)
     model₁ = Chain(LinearLayerQ(N), LinearLayerP(N))
     model₂ = Chain(ActivationLayerQ(N, tanh), ActivationLayerP(N, tanh))
     model₃ = Chain(GradientLayerQ(N, N2, tanh), GradientLayerP(N, N2, tanh))
-    ps₁ = initialparameters(CPU(), T, model₁)
-    ps₂ = initialparameters(CPU(), T, model₂)
-    ps₃ = initialparameters(CPU(), T, model₃)
+    ps₁ = initialparameters(model₁, CPU(), T)
+    ps₂ = initialparameters(model₂, CPU(), T)
+    ps₃ = initialparameters(model₃, CPU(), T)
 
     # evaluate functions 
     x_vec = rand(T, N)
diff --git a/test/layers/sympnet_upscaling.jl b/test/layers/sympnet_upscaling.jl
index 546a1d80c..1cbb99d97 100644
--- a/test/layers/sympnet_upscaling.jl
+++ b/test/layers/sympnet_upscaling.jl
@@ -2,7 +2,7 @@ using GeometricMachineLearning, Test, Zygote
 
 function test_symplecticity(N=4, N2=20, T=Float32)
     model = Chain(PSDLayer(N, N2), GradientQ(N2, 2*N2, tanh), GradientP(N2, 2*N2, tanh), PSDLayer(N2, N))
-    ps = initialparameters(CPU(), T, model)
+    ps = initialparameters(model, CPU(), T)
     x = rand(T, N)
     ten = rand(T, N, N)
     # the first and last PSD layer need to have the same weight! (else they map to a different symplectic potential)
diff --git a/test/optimizers/optimizer_convergence_tests/psd_optim.jl b/test/optimizers/optimizer_convergence_tests/psd_optim.jl
index 1a7deb786..4a73b3c32 100644
--- a/test/optimizers/optimizer_convergence_tests/psd_optim.jl
+++ b/test/optimizers/optimizer_convergence_tests/psd_optim.jl
@@ -36,7 +36,7 @@ function svd_test(A, n, train_steps=1000, tol=1e-1; retraction=Cayley())
 
     err_best = norm(A - U_result*U_result'*A)
     model = Chain(PSDLayer(2*N, 2*n, retraction=retraction), PSDLayer(2*n, 2*N, retraction=retraction))
-    ps = initialparameters(CPU(), Float64, model)
+    ps = initialparameters(model, CPU(), Float64)
 
     o₁ = Optimizer(GradientOptimizer(0.01), ps)
     o₂ = Optimizer(MomentumOptimizer(0.01), ps)
diff --git a/test/optimizers/optimizer_convergence_tests/svd_optim.jl b/test/optimizers/optimizer_convergence_tests/svd_optim.jl
index 1b4fe054a..2e39b988d 100644
--- a/test/optimizers/optimizer_convergence_tests/svd_optim.jl
+++ b/test/optimizers/optimizer_convergence_tests/svd_optim.jl
@@ -32,7 +32,7 @@ function svd_test(A, n, train_steps=1000, tol=1e-1; retraction=Cayley())
 
     err_best = norm(A - U_result*U_result'*A)
     model = Chain(StiefelLayer(N, n, retraction=retraction), StiefelLayer(n, N, retraction=retraction))
-    ps = initialparameters(CPU(), Float64, model)
+    ps = initialparameters(model, CPU(), Float64)
 
     o₁ = Optimizer(GradientOptimizer(0.01), ps)
     o₂ = Optimizer(MomentumOptimizer(0.01), ps)
diff --git a/test/optimizers/utils/optimization_step.jl b/test/optimizers/utils/optimization_step.jl
index c1e9569fc..c29ea5433 100644
--- a/test/optimizers/utils/optimization_step.jl
+++ b/test/optimizers/utils/optimization_step.jl
@@ -7,7 +7,7 @@ Random.seed!(1234)
 
 function optimization_step_test(N, n, T)
     model = Chain(StiefelLayer(N, n), Dense(N, N, tanh))
-    ps = initialparameters(KernelAbstractions.CPU(), T, model)
+    ps = initialparameters(model, KernelAbstractions.CPU(), T)
     # gradient 
     dx = ((weight=rand(Float32, N, n),), (W=rand(Float32, N, N), b=rand(Float32, N)))
     m = AdamOptimizer()
diff --git a/test/transformer_related/multi_head_attention_stiefel_optim_cache.jl b/test/transformer_related/multi_head_attention_stiefel_optim_cache.jl
index 9ac4c7f40..b2bae9f38 100644
--- a/test/transformer_related/multi_head_attention_stiefel_optim_cache.jl
+++ b/test/transformer_related/multi_head_attention_stiefel_optim_cache.jl
@@ -45,7 +45,7 @@ TODO:
 function test_cache_setups_for_optimizer_for_multihead_attention_layer(T::Type, dim::Int, n_heads::Int)
     @assert dim % n_heads == 0
     model = MultiHeadAttention(dim, n_heads, Stiefel=true)
-    ps = initialparameters(CPU(), T, model)
+    ps = initialparameters(model, CPU(), T)
 
     o₁ = Optimizer(AdamOptimizer(), ps)
     o₂ = Optimizer(MomentumOptimizer(), ps)
diff --git a/test/transformer_related/multi_head_attention_stiefel_retraction.jl b/test/transformer_related/multi_head_attention_stiefel_retraction.jl
index 2b4dbd2db..1f8ef8c60 100644
--- a/test/transformer_related/multi_head_attention_stiefel_retraction.jl
+++ b/test/transformer_related/multi_head_attention_stiefel_retraction.jl
@@ -34,7 +34,7 @@ This is a test for that checks if the retractions (geodesic and Cayley for now)
 function test_multi_head_attention_retraction(T::Type, dim, n_heads, tol=eps(T), backend=KernelAbstractions.CPU())
     model = MultiHeadAttention(dim, n_heads, Stiefel=true)
 
-    ps = initialparameters(backend, T, model)
+    ps = initialparameters(model, backend, T)
     cache = init_optimizer_cache(MomentumOptimizer(), ps)
 
     check_retraction_geodesic(cache)
diff --git a/test/transformer_related/multi_head_attention_stiefel_setup.jl b/test/transformer_related/multi_head_attention_stiefel_setup.jl
index 049fbec34..cd3deaebd 100644
--- a/test/transformer_related/multi_head_attention_stiefel_setup.jl
+++ b/test/transformer_related/multi_head_attention_stiefel_setup.jl
@@ -28,7 +28,7 @@ Check if `initialparameters` and `init_optimizer_cache` do the right thing for `
 """
 function check_multi_head_attention_stiefel_setup(T::Type, N::Int, n::Int)
     model = MultiHeadAttention(N, n, Stiefel=true)
-    ps = initialparameters(KernelAbstractions.CPU(), T, model)
+    ps = initialparameters(model, KernelAbstractions.CPU(), T)
 
     check_setup(ps)
 
diff --git a/test/transformer_related/transformer_application.jl b/test/transformer_related/transformer_application.jl
index d29408c12..89145b678 100644
--- a/test/transformer_related/transformer_application.jl
+++ b/test/transformer_related/transformer_application.jl
@@ -10,8 +10,8 @@ function transformer_application_test(T, dim, n_heads, L, seq_length=8, batch_si
     model₁ = Chain(Transformer(dim, n_heads, L, Stiefel=false), ResNet(dim))
     model₂ = Chain(Transformer(dim, n_heads, L, Stiefel=true), ResNet(dim))
 
-    ps₁ = initialparameters(KernelAbstractions.CPU(), T, model₁)
-    ps₂ = initialparameters(KernelAbstractions.CPU(), T, model₂)
+    ps₁ = initialparameters(model₁, KernelAbstractions.CPU(), T)
+    ps₂ = initialparameters(model₂, KernelAbstractions.CPU(), T)
     
     input₁ = rand(T, dim, seq_length, batch_size)
     input₂ = rand(T, dim, seq_length)
diff --git a/test/transformer_related/transformer_gradient.jl b/test/transformer_related/transformer_gradient.jl
index 713b7b0a8..f0a70bd90 100644
--- a/test/transformer_related/transformer_gradient.jl
+++ b/test/transformer_related/transformer_gradient.jl
@@ -10,8 +10,8 @@ function transformer_gradient_test(T, dim, n_heads, L, seq_length=8, batch_size=
     model₁ = Chain(Transformer(dim, n_heads, L, Stiefel=false), ResNet(dim))
     model₂ = Chain(Transformer(dim, n_heads, L, Stiefel=true), ResNet(dim))
 
-    ps₁ = initialparameters(KernelAbstractions.CPU(), T, model₁)
-    ps₂ = initialparameters(KernelAbstractions.CPU(), T, model₂)
+    ps₁ = initialparameters(model₁, KernelAbstractions.CPU(), T)
+    ps₂ = initialparameters(model₂, KernelAbstractions.CPU(), T)
     
     input₁ = rand(T, dim, seq_length, batch_size)
     input₂ = rand(T, dim, seq_length, batch_size)
diff --git a/test/transformer_related/transformer_optimizer.jl b/test/transformer_related/transformer_optimizer.jl
index 0c10815be..f31850533 100644
--- a/test/transformer_related/transformer_optimizer.jl
+++ b/test/transformer_related/transformer_optimizer.jl
@@ -10,7 +10,7 @@ function transformer_gradient_test(T, dim, n_heads, L, seq_length=8, batch_size=
     model = Chain(Transformer(dim, n_heads, L, Stiefel=true), ResNet(dim))
     model = Transformer(dim, n_heads, L, Stiefel=true)
 
-    ps = initialparameters(KernelAbstractions.CPU(), T, model)
+    ps = initialparameters(model, KernelAbstractions.CPU(), T)
     
     input = rand(T, dim, seq_length, batch_size)
     
diff --git a/test/transformer_related/transformer_setup.jl b/test/transformer_related/transformer_setup.jl
index 1311ce26c..8ac553eae 100644
--- a/test/transformer_related/transformer_setup.jl
+++ b/test/transformer_related/transformer_setup.jl
@@ -5,7 +5,7 @@ This function tests the setup of the transformer with Stiefel weights.
 """
 function transformer_setup_test(dim, n_heads, L, T)
     model = Transformer(dim, n_heads, L, Stiefel=true)
-    ps = initialparameters(KernelAbstractions.CPU(), T, model)
+    ps = initialparameters(model, KernelAbstractions.CPU(), T)
     @test typeof(ps[1].PQ.head_1) <: StiefelManifold
 end
 

From 0797839d91bd194ed183ff595008894e2419721b Mon Sep 17 00:00:00 2001
From: benedict-96 <benedikt.brantner@icloud.com>
Date: Fri, 22 Mar 2024 12:13:53 +0100
Subject: [PATCH 06/33] Gave various steps in Documenter names.

---
 .github/workflows/Documenter.yml | 12 ++++++++----
 1 file changed, 8 insertions(+), 4 deletions(-)

diff --git a/.github/workflows/Documenter.yml b/.github/workflows/Documenter.yml
index 6cb97f99d..4d207605d 100644
--- a/.github/workflows/Documenter.yml
+++ b/.github/workflows/Documenter.yml
@@ -12,20 +12,24 @@ jobs:
     runs-on: ubuntu-latest
     steps:
       - uses: actions/checkout@v3
-      - run: |
+      - name: Install packages for generating tikz images
+        run: |
           sudo apt-get install imagemagick
           sudo apt-get install poppler-utils
           sudo apt-get install texlive-xetex
           sudo apt-get install texlive-science
-          make all -C docs/src/tikz
+      - name: Make tikz images
+        run: make all -C docs/src/tikz
       - uses: julia-actions/setup-julia@latest
         with:
           version: '1'
-      - run: |
+      - name: Install BrenierTwoFluid package
+        run: |
           cd docs
           make install_brenier_two_fluid test_docs
           cd .. 
-          julia --project=docs docs/make.jl html_output
+      - name: Make docs (call julia documenter)
+          run: julia --project=docs docs/make.jl html_output
       - uses: julia-actions/julia-buildpkg@v1
       - uses: julia-actions/julia-docdeploy@v1
         env:

From ec94547541690378946db5c4d0a1fce44ab0124d Mon Sep 17 00:00:00 2001
From: benedict-96 <benedikt.brantner@icloud.com>
Date: Fri, 22 Mar 2024 12:14:38 +0100
Subject: [PATCH 07/33] Fixed problem with image generation.

---
 docs/src/tikz/Makefile | 4 +++-
 1 file changed, 3 insertions(+), 1 deletion(-)

diff --git a/docs/src/tikz/Makefile b/docs/src/tikz/Makefile
index 77300f1d4..878fd12e4 100644
--- a/docs/src/tikz/Makefile
+++ b/docs/src/tikz/Makefile
@@ -1,5 +1,5 @@
 all: pdf 
-	$(MAKE) png 
+	$(MAKE) png res1=150 res2=200 res3=100 res4=180
 	$(MAKE) logo 
 	$(MAKE) clean 
 
@@ -33,6 +33,8 @@ convert_with_pdftocairo: $(MYDIR)/*.pdf
 	done
 
 png:
+	pdftocairo 	-png -r 500		-transp -singlefile	 logo_with_name.pdf								logo_with_name
+	pdftocairo 	-png -r 500		-transp -singlefile	 logo_with_name_dark.pdf						logo_with_name_dark
 	pdftocairo  -png -r $(res1) -transp -singlefile  adam_optimizer.pdf               	 			adam_optimizer  
 	pdftocairo  -png -r $(res1) -transp -singlefile  adam_optimizer_dark.pdf               	 		adam_optimizer_dark                     
 	pdftocairo  -png -r $(res1) -transp -singlefile  general_optimization.pdf		 	 			general_optimization   

From 8f4aec886b8fbfcb14ab6e49a0098cfd25ddcefa Mon Sep 17 00:00:00 2001
From: benedict-96 <benedikt.brantner@icloud.com>
Date: Fri, 22 Mar 2024 12:15:09 +0100
Subject: [PATCH 08/33] Changed tol (such that it conforms with Tobi's tests).

---
 docs/src/tutorials/grassmann_layer.md | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/docs/src/tutorials/grassmann_layer.md b/docs/src/tutorials/grassmann_layer.md
index c0f5fb5d8..8a59c0e2b 100644
--- a/docs/src/tutorials/grassmann_layer.md
+++ b/docs/src/tutorials/grassmann_layer.md
@@ -48,7 +48,7 @@ const ε = 0.1                 # entropic regularization. √ε is a length.  #
 const q = 1.0                 # annealing parameter                       # hide
 const Δ = 1.0                 # characteristic domain size                # hide
 const s = ε                   # current scale: no annealing -> equals ε   # hide
-const tol = 1e-4              # marginal condition tolerance              # hide 
+const tol = 1e-6              # marginal condition tolerance              # hide 
 const crit_it = 20            # acceleration inference                    # hide
 const p_η = 2
 

From fb8bf351e40e3bd68a4a0ec9abcef28e577bd4aa Mon Sep 17 00:00:00 2001
From: benedict-96 <benedikt.brantner@icloud.com>
Date: Fri, 22 Mar 2024 12:21:50 +0100
Subject: [PATCH 09/33] There was an indentation too many.

---
 .github/workflows/Documenter.yml | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/.github/workflows/Documenter.yml b/.github/workflows/Documenter.yml
index 4d207605d..101edc06a 100644
--- a/.github/workflows/Documenter.yml
+++ b/.github/workflows/Documenter.yml
@@ -29,7 +29,7 @@ jobs:
           make install_brenier_two_fluid test_docs
           cd .. 
       - name: Make docs (call julia documenter)
-          run: julia --project=docs docs/make.jl html_output
+        run: julia --project=docs docs/make.jl html_output
       - uses: julia-actions/julia-buildpkg@v1
       - uses: julia-actions/julia-docdeploy@v1
         env:

From 4a8ce2561585d659d2cc070fa4b81fd328392901 Mon Sep 17 00:00:00 2001
From: benedict-96 <benedikt.brantner@icloud.com>
Date: Fri, 22 Mar 2024 12:25:26 +0100
Subject: [PATCH 10/33] Got rid of tikz images that will be included with
 VolumePreservingTransformer pr.

---
 docs/src/tikz/Makefile | 4 ----
 1 file changed, 4 deletions(-)

diff --git a/docs/src/tikz/Makefile b/docs/src/tikz/Makefile
index 878fd12e4..b73b74391 100644
--- a/docs/src/tikz/Makefile
+++ b/docs/src/tikz/Makefile
@@ -57,10 +57,6 @@ png:
 	pdftocairo  -png -r $(res1) -transp -singlefile  tensor_dark.pdf 								tensor_dark
 	pdftocairo  -png -r $(res4) -transp -singlefile  tensor_sampling.pdf 							tensor_sampling
 	pdftocairo  -png -r $(res4) -transp -singlefile  tensor_sampling_dark.pdf 						tensor_sampling_dark
-	pdftocairo  -png -r $(res4) -transp -singlefile  skew_sym_visualization.pdf 					skew_sym_visualization
-	pdftocairo  -png -r $(res4) -transp -singlefile  skew_sym_visualization_dark.pdf 				skew_sym_visualization_dark
-	pdftocairo  -png -r $(res1) -transp -singlefile  vp_feedforward.pdf 							vp_feedforward
-	pdftocairo  -png -r $(res1) -transp -singlefile  vp_feedforward_dark.pdf 						vp_feedforward_dark
 
 logo:
 	cp logo_with_name.png 		../assets/logo.png

From 91a93fc7b242ad5e9c2fa955be3c8b483fe22431 Mon Sep 17 00:00:00 2001
From: benedict-96 <benedikt.brantner@icloud.com>
Date: Mon, 8 Apr 2024 12:15:41 +0200
Subject: [PATCH 11/33] Added gpu versions for parallel computation of
 inverses.

---
 src/kernels/inverses/inverse_2x2.jl | 42 ++++++++++++++++++++
 src/kernels/inverses/inverse_3x3.jl | 47 ++++++++++++++++++++++
 src/kernels/inverses/inverse_4x4.jl | 52 ++++++++++++++++++++++++
 src/kernels/inverses/inverse_5x5.jl | 61 +++++++++++++++++++++++++++++
 4 files changed, 202 insertions(+)
 create mode 100644 src/kernels/inverses/inverse_2x2.jl
 create mode 100644 src/kernels/inverses/inverse_3x3.jl
 create mode 100644 src/kernels/inverses/inverse_4x4.jl
 create mode 100644 src/kernels/inverses/inverse_5x5.jl

diff --git a/src/kernels/inverses/inverse_2x2.jl b/src/kernels/inverses/inverse_2x2.jl
new file mode 100644
index 000000000..f5933863d
--- /dev/null
+++ b/src/kernels/inverses/inverse_2x2.jl
@@ -0,0 +1,42 @@
+@kernel function inv22_kernel!(ˍ₋out::AT, A::AT) where {T, AT<:AbstractArray{T, 3}}
+    k = @index(Global)
+    begin
+         @inbounds begin
+                ˍ₋out[1, 1, k] = (/)((getindex)(A, 2, 2, k), (+)((*)((getindex)(A, 1, 1, k), (getindex)(A, 2, 2, k)), (*)((*)(-1, (getindex)(A, 1, 2, k)), (getindex)(A, 2, 1, k))))
+                ˍ₋out[2, 1, k] = (/)((*)(-1, (getindex)(A, 2, 1, k)), (+)((*)((getindex)(A, 1, 1, k), (getindex)(A, 2, 2, k)), (*)((*)(-1, (getindex)(A, 1, 2, k)), (getindex)(A, 2, 1, k))))
+                ˍ₋out[1, 2, k] = (/)((*)(-1, (getindex)(A, 1, 2, k)), (+)((*)((getindex)(A, 1, 1, k), (getindex)(A, 2, 2, k)), (*)((*)(-1, (getindex)(A, 1, 2, k)), (getindex)(A, 2, 1, k))))
+                ˍ₋out[2, 2, k] = (/)((getindex)(A, 1, 1, k), (+)((*)((getindex)(A, 1, 1, k), (getindex)(A, 2, 2, k)), (*)((*)(-1, (getindex)(A, 1, 2, k)), (getindex)(A, 2, 1, k))))
+                nothing
+            end
+    end
+end
+
+function tensor_inverse2(A::AbstractArray{T, 3}) where T 
+    out = similar(A)
+
+    tensor_inverse2!(out, A)
+    
+    out 
+end
+
+function tensor_inverse2!(out::AbstractArray{T, 3}, A::AbstractArray{T, 3}) where T 
+    @assert size(A, 1) == size(A, 2) == 2
+    @assert size(A) == size(out)
+
+    backend = get_backend(out)
+    inv22! = inv22_kernel!(backend)
+
+    inv22!(out, A, ndrange = size(A, 3))
+
+    nothing
+end
+
+function ChainRulesCore.rrule(::typeof(tensor_inverse2), A::AT) where {T, AT<:AbstractArray{T, 3}}
+    out = tensor_inverse2(A)
+    
+    function tensor_inverse_pullback(out_diff::AT)
+
+        NoTangent(), - tensor_transpose_tensor_mul(out, tensor_tensor_mul(out_diff, tensor_transpose(out)))
+    end 
+    out, tensor_inverse_pullback 
+end 
\ No newline at end of file
diff --git a/src/kernels/inverses/inverse_3x3.jl b/src/kernels/inverses/inverse_3x3.jl
new file mode 100644
index 000000000..eed5c7a65
--- /dev/null
+++ b/src/kernels/inverses/inverse_3x3.jl
@@ -0,0 +1,47 @@
+@kernel function inv33_kernel!(ˍ₋out::AT, A::AT) where {T, AT<:AbstractArray{T, 3}}
+    k = @index(Global)
+    begin
+         @inbounds begin
+                ˍ₋out[1, 1, k] = (/)((+)((*)((getindex)(A, 2, 2, k), (getindex)(A, 3, 3, k)), (*)((*)(-1, (getindex)(A, 2, 3, k)), (getindex)(A, 3, 2, k))), (+)((+)((+)((+)((+)((*)((*)((getindex)(A, 1, 1, k), (getindex)(A, 2, 2, k)), (getindex)(A, 3, 3, k)), (*)((*)((*)(-1, (getindex)(A, 1, 1, k)), (getindex)(A, 2, 3, k)), (getindex)(A, 3, 2, k))), (*)((*)((*)(-1, (getindex)(A, 1, 2, k)), (getindex)(A, 2, 1, k)), (getindex)(A, 3, 3, k))), (*)((*)((getindex)(A, 1, 2, k), (getindex)(A, 2, 3, k)), (getindex)(A, 3, 1, k))), (*)((*)((getindex)(A, 1, 3, k), (getindex)(A, 2, 1, k)), (getindex)(A, 3, 2, k))), (*)((*)((*)(-1, (getindex)(A, 1, 3, k)), (getindex)(A, 2, 2, k)), (getindex)(A, 3, 1, k))))
+                ˍ₋out[2, 1, k] = (/)((+)((*)((*)(-1, (getindex)(A, 2, 1, k)), (getindex)(A, 3, 3, k)), (*)((getindex)(A, 2, 3, k), (getindex)(A, 3, 1, k))), (+)((+)((+)((+)((+)((*)((*)((getindex)(A, 1, 1, k), (getindex)(A, 2, 2, k)), (getindex)(A, 3, 3, k)), (*)((*)((*)(-1, (getindex)(A, 1, 1, k)), (getindex)(A, 2, 3, k)), (getindex)(A, 3, 2, k))), (*)((*)((*)(-1, (getindex)(A, 1, 2, k)), (getindex)(A, 2, 1, k)), (getindex)(A, 3, 3, k))), (*)((*)((getindex)(A, 1, 2, k), (getindex)(A, 2, 3, k)), (getindex)(A, 3, 1, k))), (*)((*)((getindex)(A, 1, 3, k), (getindex)(A, 2, 1, k)), (getindex)(A, 3, 2, k))), (*)((*)((*)(-1, (getindex)(A, 1, 3, k)), (getindex)(A, 2, 2, k)), (getindex)(A, 3, 1, k))))
+                ˍ₋out[3, 1, k] = (/)((+)((*)((getindex)(A, 2, 1, k), (getindex)(A, 3, 2, k)), (*)((*)(-1, (getindex)(A, 2, 2, k)), (getindex)(A, 3, 1, k))), (+)((+)((+)((+)((+)((*)((*)((getindex)(A, 1, 1, k), (getindex)(A, 2, 2, k)), (getindex)(A, 3, 3, k)), (*)((*)((*)(-1, (getindex)(A, 1, 1, k)), (getindex)(A, 2, 3, k)), (getindex)(A, 3, 2, k))), (*)((*)((*)(-1, (getindex)(A, 1, 2, k)), (getindex)(A, 2, 1, k)), (getindex)(A, 3, 3, k))), (*)((*)((getindex)(A, 1, 2, k), (getindex)(A, 2, 3, k)), (getindex)(A, 3, 1, k))), (*)((*)((getindex)(A, 1, 3, k), (getindex)(A, 2, 1, k)), (getindex)(A, 3, 2, k))), (*)((*)((*)(-1, (getindex)(A, 1, 3, k)), (getindex)(A, 2, 2, k)), (getindex)(A, 3, 1, k))))
+                ˍ₋out[1, 2, k] = (/)((+)((*)((*)(-1, (getindex)(A, 1, 2, k)), (getindex)(A, 3, 3, k)), (*)((getindex)(A, 1, 3, k), (getindex)(A, 3, 2, k))), (+)((+)((+)((+)((+)((*)((*)((getindex)(A, 1, 1, k), (getindex)(A, 2, 2, k)), (getindex)(A, 3, 3, k)), (*)((*)((*)(-1, (getindex)(A, 1, 1, k)), (getindex)(A, 2, 3, k)), (getindex)(A, 3, 2, k))), (*)((*)((*)(-1, (getindex)(A, 1, 2, k)), (getindex)(A, 2, 1, k)), (getindex)(A, 3, 3, k))), (*)((*)((getindex)(A, 1, 2, k), (getindex)(A, 2, 3, k)), (getindex)(A, 3, 1, k))), (*)((*)((getindex)(A, 1, 3, k), (getindex)(A, 2, 1, k)), (getindex)(A, 3, 2, k))), (*)((*)((*)(-1, (getindex)(A, 1, 3, k)), (getindex)(A, 2, 2, k)), (getindex)(A, 3, 1, k))))
+                ˍ₋out[2, 2, k] = (/)((+)((*)((getindex)(A, 1, 1, k), (getindex)(A, 3, 3, k)), (*)((*)(-1, (getindex)(A, 1, 3, k)), (getindex)(A, 3, 1, k))), (+)((+)((+)((+)((+)((*)((*)((getindex)(A, 1, 1, k), (getindex)(A, 2, 2, k)), (getindex)(A, 3, 3, k)), (*)((*)((*)(-1, (getindex)(A, 1, 1, k)), (getindex)(A, 2, 3, k)), (getindex)(A, 3, 2, k))), (*)((*)((*)(-1, (getindex)(A, 1, 2, k)), (getindex)(A, 2, 1, k)), (getindex)(A, 3, 3, k))), (*)((*)((getindex)(A, 1, 2, k), (getindex)(A, 2, 3, k)), (getindex)(A, 3, 1, k))), (*)((*)((getindex)(A, 1, 3, k), (getindex)(A, 2, 1, k)), (getindex)(A, 3, 2, k))), (*)((*)((*)(-1, (getindex)(A, 1, 3, k)), (getindex)(A, 2, 2, k)), (getindex)(A, 3, 1, k))))
+                ˍ₋out[3, 2, k] = (/)((+)((*)((*)(-1, (getindex)(A, 1, 1, k)), (getindex)(A, 3, 2, k)), (*)((getindex)(A, 1, 2, k), (getindex)(A, 3, 1, k))), (+)((+)((+)((+)((+)((*)((*)((getindex)(A, 1, 1, k), (getindex)(A, 2, 2, k)), (getindex)(A, 3, 3, k)), (*)((*)((*)(-1, (getindex)(A, 1, 1, k)), (getindex)(A, 2, 3, k)), (getindex)(A, 3, 2, k))), (*)((*)((*)(-1, (getindex)(A, 1, 2, k)), (getindex)(A, 2, 1, k)), (getindex)(A, 3, 3, k))), (*)((*)((getindex)(A, 1, 2, k), (getindex)(A, 2, 3, k)), (getindex)(A, 3, 1, k))), (*)((*)((getindex)(A, 1, 3, k), (getindex)(A, 2, 1, k)), (getindex)(A, 3, 2, k))), (*)((*)((*)(-1, (getindex)(A, 1, 3, k)), (getindex)(A, 2, 2, k)), (getindex)(A, 3, 1, k))))
+                ˍ₋out[1, 3, k] = (/)((+)((*)((getindex)(A, 1, 2, k), (getindex)(A, 2, 3, k)), (*)((*)(-1, (getindex)(A, 1, 3, k)), (getindex)(A, 2, 2, k))), (+)((+)((+)((+)((+)((*)((*)((getindex)(A, 1, 1, k), (getindex)(A, 2, 2, k)), (getindex)(A, 3, 3, k)), (*)((*)((*)(-1, (getindex)(A, 1, 1, k)), (getindex)(A, 2, 3, k)), (getindex)(A, 3, 2, k))), (*)((*)((*)(-1, (getindex)(A, 1, 2, k)), (getindex)(A, 2, 1, k)), (getindex)(A, 3, 3, k))), (*)((*)((getindex)(A, 1, 2, k), (getindex)(A, 2, 3, k)), (getindex)(A, 3, 1, k))), (*)((*)((getindex)(A, 1, 3, k), (getindex)(A, 2, 1, k)), (getindex)(A, 3, 2, k))), (*)((*)((*)(-1, (getindex)(A, 1, 3, k)), (getindex)(A, 2, 2, k)), (getindex)(A, 3, 1, k))))
+                ˍ₋out[2, 3, k] = (/)((+)((*)((*)(-1, (getindex)(A, 1, 1, k)), (getindex)(A, 2, 3, k)), (*)((getindex)(A, 1, 3, k), (getindex)(A, 2, 1, k))), (+)((+)((+)((+)((+)((*)((*)((getindex)(A, 1, 1, k), (getindex)(A, 2, 2, k)), (getindex)(A, 3, 3, k)), (*)((*)((*)(-1, (getindex)(A, 1, 1, k)), (getindex)(A, 2, 3, k)), (getindex)(A, 3, 2, k))), (*)((*)((*)(-1, (getindex)(A, 1, 2, k)), (getindex)(A, 2, 1, k)), (getindex)(A, 3, 3, k))), (*)((*)((getindex)(A, 1, 2, k), (getindex)(A, 2, 3, k)), (getindex)(A, 3, 1, k))), (*)((*)((getindex)(A, 1, 3, k), (getindex)(A, 2, 1, k)), (getindex)(A, 3, 2, k))), (*)((*)((*)(-1, (getindex)(A, 1, 3, k)), (getindex)(A, 2, 2, k)), (getindex)(A, 3, 1, k))))
+                ˍ₋out[3, 3, k] = (/)((+)((*)((getindex)(A, 1, 1, k), (getindex)(A, 2, 2, k)), (*)((*)(-1, (getindex)(A, 1, 2, k)), (getindex)(A, 2, 1, k))), (+)((+)((+)((+)((+)((*)((*)((getindex)(A, 1, 1, k), (getindex)(A, 2, 2, k)), (getindex)(A, 3, 3, k)), (*)((*)((*)(-1, (getindex)(A, 1, 1, k)), (getindex)(A, 2, 3, k)), (getindex)(A, 3, 2, k))), (*)((*)((*)(-1, (getindex)(A, 1, 2, k)), (getindex)(A, 2, 1, k)), (getindex)(A, 3, 3, k))), (*)((*)((getindex)(A, 1, 2, k), (getindex)(A, 2, 3, k)), (getindex)(A, 3, 1, k))), (*)((*)((getindex)(A, 1, 3, k), (getindex)(A, 2, 1, k)), (getindex)(A, 3, 2, k))), (*)((*)((*)(-1, (getindex)(A, 1, 3, k)), (getindex)(A, 2, 2, k)), (getindex)(A, 3, 1, k))))
+                nothing
+            end
+    end
+end
+
+function tensor_inverse3(A::AbstractArray{T, 3}) where T 
+    out = similar(A)
+
+    tensor_inverse3!(out, A)
+    
+    out 
+end
+
+function tensor_inverse3!(out::AbstractArray{T, 3}, A::AbstractArray{T, 3}) where T 
+    @assert size(A, 1) == size(A, 2) == 3
+    @assert size(A) == size(out)
+
+    backend = get_backend(out)
+    inv33! = inv33_kernel!(backend)
+
+    inv33!(out, A, ndrange = size(A, 3))
+
+    nothing
+end
+
+function ChainRulesCore.rrule(::typeof(tensor_inverse3), A::AT) where {T, AT<:AbstractArray{T, 3}}
+    out = tensor_inverse3(A)
+    
+    function tensor_inverse_pullback(out_diff::AT)
+
+        NoTangent(), - tensor_transpose_tensor_mul(out, tensor_tensor_mul(out_diff, tensor_transpose(out)))
+    end 
+    out, tensor_inverse_pullback 
+end 
\ No newline at end of file
diff --git a/src/kernels/inverses/inverse_4x4.jl b/src/kernels/inverses/inverse_4x4.jl
new file mode 100644
index 000000000..fe3f4156b
--- /dev/null
+++ b/src/kernels/inverses/inverse_4x4.jl
@@ -0,0 +1,52 @@
+@kernel function inv44_kernel!(ˍ₋out::AT, A::AT) where {T, AT<:AbstractArray{T, 3}}
+    k = @index(Global)
+    @inbounds begin
+        ˍ₋out[1, 1, k] = (/)((+)((+)((*)((getindex)(A, 2, 2, k), (+)((*)((getindex)(A, 3, 3, k), (getindex)(A, 4, 4, k)), (*)((*)(-1, (getindex)(A, 3, 4, k)), (getindex)(A, 4, 3, k)))), (*)((*)(-1, (getindex)(A, 2, 3, k)), (+)((*)((getindex)(A, 3, 2, k), (getindex)(A, 4, 4, k)), (*)((*)(-1, (getindex)(A, 3, 4, k)), (getindex)(A, 4, 2, k))))), (*)((getindex)(A, 2, 4, k), (+)((*)((getindex)(A, 3, 2, k), (getindex)(A, 4, 3, k)), (*)((*)(-1, (getindex)(A, 3, 3, k)), (getindex)(A, 4, 2, k))))), (+)((+)((+)((*)((getindex)(A, 1, 1, k), (+)((+)((*)((getindex)(A, 2, 2, k), (+)((*)((getindex)(A, 3, 3, k), (getindex)(A, 4, 4, k)), (*)((*)(-1, (getindex)(A, 3, 4, k)), (getindex)(A, 4, 3, k)))), (*)((*)(-1, (getindex)(A, 2, 3, k)), (+)((*)((getindex)(A, 3, 2, k), (getindex)(A, 4, 4, k)), (*)((*)(-1, (getindex)(A, 3, 4, k)), (getindex)(A, 4, 2, k))))), (*)((getindex)(A, 2, 4, k), (+)((*)((getindex)(A, 3,2, k), (getindex)(A, 4, 3, k)), (*)((*)(-1, (getindex)(A, 3, 3, k)), (getindex)(A, 4,2, k)))))), (*)((*)(-1, (getindex)(A, 1,2, k)), (+)((+)((*)((getindex)(A, 2, 1, k), (+)((*)((getindex)(A, 3, 3, k), (getindex)(A, 4, 4, k)), (*)((*)(-1, (getindex)(A, 3, 4, k)), (getindex)(A, 4, 3, k)))), (*)((*)(-1, (getindex)(A, 2, 3, k)), (+)((*)((getindex)(A, 3, 1, k), (getindex)(A, 4, 4, k)), (*)((*)(-1, (getindex)(A, 3, 4, k)), (getindex)(A, 4, 1, k))))), (*)((getindex)(A, 2, 4, k), (+)((*)((getindex)(A, 3, 1, k), (getindex)(A, 4, 3, k)), (*)((*)(-1, (getindex)(A, 3, 3, k)), (getindex)(A, 4, 1, k))))))), (*)((getindex)(A, 1, 3, k), (+)((+)((*)((getindex)(A, 2, 1, k), (+)((*)((getindex)(A, 3,2, k), (getindex)(A, 4, 4, k)), (*)((*)(-1, (getindex)(A, 3, 4, k)), (getindex)(A, 4,2, k)))), (*)((*)(-1, (getindex)(A, 2,2, k)), (+)((*)((getindex)(A, 3, 1, k), (getindex)(A, 4, 4, k)), (*)((*)(-1, (getindex)(A, 3, 4, k)), (getindex)(A, 4, 1, k))))), (*)((getindex)(A, 2, 4, k), (+)((*)((getindex)(A, 3, 1, k), (getindex)(A, 4,2, k)), (*)((*)(-1, (getindex)(A, 3,2, k)), (getindex)(A, 4, 1, k))))))), (*)((*)(-1, (getindex)(A, 1, 4, k)), (+)((+)((*)((getindex)(A, 2, 1, k), (+)((*)((getindex)(A, 3,2, k), (getindex)(A, 4, 3, k)), (*)((*)(-1, (getindex)(A, 3, 3, k)), (getindex)(A, 4,2, k)))), (*)((*)(-1, (getindex)(A, 2,2, k)), (+)((*)((getindex)(A, 3, 1, k), (getindex)(A, 4, 3, k)), (*)((*)(-1, (getindex)(A, 3, 3, k)), (getindex)(A, 4, 1, k))))), (*)((getindex)(A, 2, 3, k), (+)((*)((getindex)(A, 3, 1, k), (getindex)(A, 4,2, k)), (*)((*)(-1, (getindex)(A, 3,2, k)), (getindex)(A, 4, 1, k))))))))
+        ˍ₋out[2, 1, k] = (/)((+)((+)((*)((*)(-1, (getindex)(A, 2, 1, k)), (+)((*)((getindex)(A, 3, 3, k), (getindex)(A, 4, 4, k)), (*)((*)(-1, (getindex)(A, 3, 4, k)), (getindex)(A, 4, 3, k)))), (*)((getindex)(A, 2, 3, k), (+)((*)((getindex)(A, 3, 1, k), (getindex)(A, 4, 4, k)), (*)((*)(-1, (getindex)(A, 3, 4, k)), (getindex)(A, 4, 1, k))))), (*)((*)(-1, (getindex)(A, 2, 4, k)), (+)((*)((getindex)(A, 3, 1, k), (getindex)(A, 4, 3, k)), (*)((*)(-1, (getindex)(A, 3, 3, k)), (getindex)(A, 4, 1, k))))), (+)((+)((+)((*)((getindex)(A, 1, 1, k), (+)((+)((*)((getindex)(A, 2,2, k), (+)((*)((getindex)(A, 3, 3, k), (getindex)(A, 4, 4, k)), (*)((*)(-1, (getindex)(A, 3, 4, k)), (getindex)(A, 4, 3, k)))), (*)((*)(-1, (getindex)(A, 2, 3, k)), (+)((*)((getindex)(A, 3,2, k), (getindex)(A, 4, 4, k)), (*)((*)(-1, (getindex)(A, 3, 4, k)), (getindex)(A, 4,2, k))))), (*)((getindex)(A, 2, 4, k), (+)((*)((getindex)(A, 3,2, k), (getindex)(A, 4, 3, k)), (*)((*)(-1, (getindex)(A, 3, 3, k)), (getindex)(A, 4,2, k)))))), (*)((*)(-1, (getindex)(A, 1,2, k)), (+)((+)((*)((getindex)(A, 2, 1, k), (+)((*)((getindex)(A, 3, 3, k), (getindex)(A, 4, 4, k)), (*)((*)(-1, (getindex)(A, 3, 4, k)), (getindex)(A, 4, 3, k)))), (*)((*)(-1, (getindex)(A, 2, 3, k)), (+)((*)((getindex)(A, 3, 1, k), (getindex)(A, 4, 4, k)), (*)((*)(-1, (getindex)(A, 3, 4, k)), (getindex)(A, 4, 1, k))))), (*)((getindex)(A, 2, 4, k), (+)((*)((getindex)(A, 3, 1, k), (getindex)(A, 4, 3, k)), (*)((*)(-1, (getindex)(A, 3, 3, k)), (getindex)(A, 4, 1, k))))))), (*)((getindex)(A, 1, 3, k), (+)((+)((*)((getindex)(A, 2, 1, k), (+)((*)((getindex)(A, 3,2, k), (getindex)(A, 4, 4, k)), (*)((*)(-1, (getindex)(A, 3, 4, k)), (getindex)(A, 4,2, k)))), (*)((*)(-1, (getindex)(A, 2,2, k)), (+)((*)((getindex)(A, 3, 1, k), (getindex)(A, 4, 4, k)), (*)((*)(-1, (getindex)(A, 3, 4, k)), (getindex)(A, 4, 1, k))))), (*)((getindex)(A, 2, 4, k), (+)((*)((getindex)(A, 3, 1, k), (getindex)(A, 4,2, k)), (*)((*)(-1, (getindex)(A, 3,2, k)), (getindex)(A, 4, 1, k))))))), (*)((*)(-1, (getindex)(A, 1, 4, k)), (+)((+)((*)((getindex)(A, 2, 1, k), (+)((*)((getindex)(A, 3,2, k), (getindex)(A, 4, 3, k)), (*)((*)(-1, (getindex)(A, 3, 3, k)), (getindex)(A, 4,2, k)))), (*)((*)(-1, (getindex)(A, 2,2, k)), (+)((*)((getindex)(A, 3, 1, k), (getindex)(A, 4, 3, k)), (*)((*)(-1, (getindex)(A, 3, 3, k)), (getindex)(A, 4, 1, k))))), (*)((getindex)(A, 2, 3, k), (+)((*)((getindex)(A, 3, 1, k), (getindex)(A, 4,2, k)), (*)((*)(-1, (getindex)(A, 3,2, k)), (getindex)(A, 4, 1, k))))))))
+        ˍ₋out[3, 1, k] = (/)((+)((+)((*)((getindex)(A, 2, 1, k), (+)((*)((getindex)(A, 3,2, k), (getindex)(A, 4, 4, k)), (*)((*)(-1, (getindex)(A, 3, 4, k)), (getindex)(A, 4,2, k)))), (*)((*)(-1, (getindex)(A, 2,2, k)), (+)((*)((getindex)(A, 3, 1, k), (getindex)(A, 4, 4, k)), (*)((*)(-1, (getindex)(A, 3, 4, k)), (getindex)(A, 4, 1, k))))), (*)((getindex)(A, 2, 4, k), (+)((*)((getindex)(A, 3, 1, k), (getindex)(A, 4,2, k)), (*)((*)(-1, (getindex)(A, 3,2, k)), (getindex)(A, 4, 1, k))))), (+)((+)((+)((*)((getindex)(A, 1, 1, k), (+)((+)((*)((getindex)(A, 2,2, k), (+)((*)((getindex)(A, 3, 3, k), (getindex)(A, 4, 4, k)), (*)((*)(-1, (getindex)(A, 3, 4, k)), (getindex)(A, 4, 3, k)))), (*)((*)(-1, (getindex)(A, 2, 3, k)), (+)((*)((getindex)(A, 3,2, k), (getindex)(A, 4, 4, k)), (*)((*)(-1, (getindex)(A, 3, 4, k)), (getindex)(A, 4,2, k))))), (*)((getindex)(A, 2, 4, k), (+)((*)((getindex)(A, 3,2, k), (getindex)(A, 4, 3, k)), (*)((*)(-1, (getindex)(A, 3, 3, k)), (getindex)(A, 4,2, k)))))), (*)((*)(-1, (getindex)(A, 1,2, k)), (+)((+)((*)((getindex)(A, 2, 1, k), (+)((*)((getindex)(A, 3, 3, k), (getindex)(A, 4, 4, k)), (*)((*)(-1, (getindex)(A, 3, 4, k)), (getindex)(A, 4, 3, k)))), (*)((*)(-1, (getindex)(A, 2, 3, k)), (+)((*)((getindex)(A, 3, 1, k), (getindex)(A, 4, 4, k)), (*)((*)(-1, (getindex)(A, 3, 4, k)), (getindex)(A, 4, 1, k))))), (*)((getindex)(A, 2, 4, k), (+)((*)((getindex)(A, 3, 1, k), (getindex)(A, 4, 3, k)), (*)((*)(-1, (getindex)(A, 3, 3, k)), (getindex)(A, 4, 1, k))))))), (*)((getindex)(A, 1, 3, k), (+)((+)((*)((getindex)(A, 2, 1, k), (+)((*)((getindex)(A, 3,2, k), (getindex)(A, 4, 4, k)), (*)((*)(-1, (getindex)(A, 3, 4, k)), (getindex)(A, 4,2, k)))), (*)((*)(-1, (getindex)(A, 2,2, k)), (+)((*)((getindex)(A, 3, 1, k), (getindex)(A, 4, 4, k)), (*)((*)(-1, (getindex)(A, 3, 4, k)), (getindex)(A, 4, 1, k))))), (*)((getindex)(A, 2, 4, k), (+)((*)((getindex)(A, 3, 1, k), (getindex)(A, 4,2, k)), (*)((*)(-1, (getindex)(A, 3,2, k)), (getindex)(A, 4, 1, k))))))), (*)((*)(-1, (getindex)(A, 1, 4, k)), (+)((+)((*)((getindex)(A, 2, 1, k), (+)((*)((getindex)(A, 3,2, k), (getindex)(A, 4, 3, k)), (*)((*)(-1, (getindex)(A, 3, 3, k)), (getindex)(A, 4,2, k)))), (*)((*)(-1, (getindex)(A, 2,2, k)), (+)((*)((getindex)(A, 3, 1, k), (getindex)(A, 4, 3, k)), (*)((*)(-1, (getindex)(A, 3, 3, k)), (getindex)(A, 4, 1, k))))), (*)((getindex)(A, 2, 3, k), (+)((*)((getindex)(A, 3, 1, k), (getindex)(A, 4,2, k)), (*)((*)(-1, (getindex)(A, 3,2, k)), (getindex)(A, 4, 1, k))))))))
+        ˍ₋out[4, 1, k] = (/)((+)((+)((*)((*)(-1, (getindex)(A, 2, 1, k)), (+)((*)((getindex)(A, 3,2, k), (getindex)(A, 4, 3, k)), (*)((*)(-1, (getindex)(A, 3, 3, k)), (getindex)(A, 4,2, k)))), (*)((getindex)(A, 2,2, k), (+)((*)((getindex)(A, 3, 1, k), (getindex)(A, 4, 3, k)), (*)((*)(-1, (getindex)(A, 3, 3, k)), (getindex)(A, 4, 1, k))))), (*)((*)(-1, (getindex)(A, 2, 3, k)), (+)((*)((getindex)(A, 3, 1, k), (getindex)(A, 4,2, k)), (*)((*)(-1, (getindex)(A, 3,2, k)), (getindex)(A, 4, 1, k))))), (+)((+)((+)((*)((getindex)(A, 1, 1, k), (+)((+)((*)((getindex)(A, 2,2, k), (+)((*)((getindex)(A, 3, 3, k), (getindex)(A, 4, 4, k)), (*)((*)(-1, (getindex)(A, 3, 4, k)), (getindex)(A, 4, 3, k)))), (*)((*)(-1, (getindex)(A, 2, 3, k)), (+)((*)((getindex)(A, 3,2, k), (getindex)(A, 4, 4, k)), (*)((*)(-1, (getindex)(A, 3, 4, k)), (getindex)(A, 4,2, k))))), (*)((getindex)(A, 2, 4, k), (+)((*)((getindex)(A, 3,2, k), (getindex)(A, 4, 3, k)), (*)((*)(-1, (getindex)(A, 3, 3, k)), (getindex)(A, 4,2, k)))))), (*)((*)(-1, (getindex)(A, 1,2, k)), (+)((+)((*)((getindex)(A, 2, 1, k), (+)((*)((getindex)(A, 3, 3, k), (getindex)(A, 4, 4, k)), (*)((*)(-1, (getindex)(A, 3, 4, k)), (getindex)(A, 4, 3, k)))), (*)((*)(-1, (getindex)(A, 2, 3, k)), (+)((*)((getindex)(A, 3, 1, k), (getindex)(A, 4, 4, k)), (*)((*)(-1, (getindex)(A, 3, 4, k)), (getindex)(A, 4, 1, k))))), (*)((getindex)(A, 2, 4, k), (+)((*)((getindex)(A, 3, 1, k), (getindex)(A, 4, 3, k)), (*)((*)(-1, (getindex)(A, 3, 3, k)), (getindex)(A, 4, 1, k))))))), (*)((getindex)(A, 1, 3, k), (+)((+)((*)((getindex)(A, 2, 1, k), (+)((*)((getindex)(A, 3,2, k), (getindex)(A, 4, 4, k)), (*)((*)(-1, (getindex)(A, 3, 4, k)), (getindex)(A, 4,2, k)))), (*)((*)(-1, (getindex)(A, 2,2, k)), (+)((*)((getindex)(A, 3, 1, k), (getindex)(A, 4, 4, k)), (*)((*)(-1, (getindex)(A, 3, 4, k)), (getindex)(A, 4, 1, k))))), (*)((getindex)(A, 2, 4, k), (+)((*)((getindex)(A, 3, 1, k), (getindex)(A, 4,2, k)), (*)((*)(-1, (getindex)(A, 3,2, k)), (getindex)(A, 4, 1, k))))))), (*)((*)(-1, (getindex)(A, 1, 4, k)), (+)((+)((*)((getindex)(A, 2, 1, k), (+)((*)((getindex)(A, 3,2, k), (getindex)(A, 4, 3, k)), (*)((*)(-1, (getindex)(A, 3, 3, k)), (getindex)(A, 4,2, k)))), (*)((*)(-1, (getindex)(A, 2,2, k)), (+)((*)((getindex)(A, 3, 1, k), (getindex)(A, 4, 3, k)), (*)((*)(-1, (getindex)(A, 3, 3, k)), (getindex)(A, 4, 1, k))))), (*)((getindex)(A, 2, 3, k), (+)((*)((getindex)(A, 3, 1, k), (getindex)(A, 4,2, k)), (*)((*)(-1, (getindex)(A, 3,2, k)), (getindex)(A, 4, 1, k))))))))
+        ˍ₋out[1, 2, k] = (/)((+)((+)((*)((*)(-1, (getindex)(A, 1,2, k)), (+)((*)((getindex)(A, 3, 3, k), (getindex)(A, 4, 4, k)), (*)((*)(-1, (getindex)(A, 3, 4, k)), (getindex)(A, 4, 3, k)))), (*)((getindex)(A, 1, 3, k), (+)((*)((getindex)(A, 3,2, k), (getindex)(A, 4, 4, k)), (*)((*)(-1, (getindex)(A, 3, 4, k)), (getindex)(A, 4,2, k))))), (*)((*)(-1, (getindex)(A, 1, 4, k)), (+)((*)((getindex)(A, 3,2, k), (getindex)(A, 4, 3, k)), (*)((*)(-1, (getindex)(A, 3, 3, k)), (getindex)(A, 4,2, k))))), (+)((+)((+)((*)((getindex)(A, 1, 1, k), (+)((+)((*)((getindex)(A, 2,2, k), (+)((*)((getindex)(A, 3, 3, k), (getindex)(A, 4, 4, k)), (*)((*)(-1, (getindex)(A, 3, 4, k)), (getindex)(A, 4, 3, k)))), (*)((*)(-1, (getindex)(A, 2, 3, k)), (+)((*)((getindex)(A, 3,2, k), (getindex)(A, 4, 4, k)), (*)((*)(-1, (getindex)(A, 3, 4, k)), (getindex)(A, 4,2, k))))), (*)((getindex)(A, 2, 4, k), (+)((*)((getindex)(A, 3,2, k), (getindex)(A, 4, 3, k)), (*)((*)(-1, (getindex)(A, 3, 3, k)), (getindex)(A, 4,2, k)))))), (*)((*)(-1, (getindex)(A, 1,2, k)), (+)((+)((*)((getindex)(A, 2, 1, k), (+)((*)((getindex)(A, 3, 3, k), (getindex)(A, 4, 4, k)), (*)((*)(-1, (getindex)(A, 3, 4, k)), (getindex)(A, 4, 3, k)))), (*)((*)(-1, (getindex)(A, 2, 3, k)), (+)((*)((getindex)(A, 3, 1, k), (getindex)(A, 4, 4, k)), (*)((*)(-1, (getindex)(A, 3, 4, k)), (getindex)(A, 4, 1, k))))), (*)((getindex)(A, 2, 4, k), (+)((*)((getindex)(A, 3, 1, k), (getindex)(A, 4, 3, k)), (*)((*)(-1, (getindex)(A, 3, 3, k)), (getindex)(A, 4, 1, k))))))), (*)((getindex)(A, 1, 3, k), (+)((+)((*)((getindex)(A, 2, 1, k), (+)((*)((getindex)(A, 3,2, k), (getindex)(A, 4, 4, k)), (*)((*)(-1, (getindex)(A, 3, 4, k)), (getindex)(A, 4,2, k)))), (*)((*)(-1, (getindex)(A, 2,2, k)), (+)((*)((getindex)(A, 3, 1, k), (getindex)(A, 4, 4, k)), (*)((*)(-1, (getindex)(A, 3, 4, k)), (getindex)(A, 4, 1, k))))), (*)((getindex)(A, 2, 4, k), (+)((*)((getindex)(A, 3, 1, k), (getindex)(A, 4,2, k)), (*)((*)(-1, (getindex)(A, 3,2, k)), (getindex)(A, 4, 1, k))))))), (*)((*)(-1, (getindex)(A, 1, 4, k)), (+)((+)((*)((getindex)(A, 2, 1, k), (+)((*)((getindex)(A, 3,2, k), (getindex)(A, 4, 3, k)), (*)((*)(-1, (getindex)(A, 3, 3, k)), (getindex)(A, 4,2, k)))), (*)((*)(-1, (getindex)(A, 2,2, k)), (+)((*)((getindex)(A, 3, 1, k), (getindex)(A, 4, 3, k)), (*)((*)(-1, (getindex)(A, 3, 3, k)), (getindex)(A, 4, 1, k))))), (*)((getindex)(A, 2, 3, k), (+)((*)((getindex)(A, 3, 1, k), (getindex)(A, 4,2, k)), (*)((*)(-1, (getindex)(A, 3,2, k)), (getindex)(A, 4, 1, k))))))))
+        ˍ₋out[2, 2, k] = (/)((+)((+)((*)((getindex)(A, 1, 1, k), (+)((*)((getindex)(A, 3, 3, k), (getindex)(A, 4, 4, k)), (*)((*)(-1, (getindex)(A, 3, 4, k)), (getindex)(A, 4, 3, k)))), (*)((*)(-1, (getindex)(A, 1, 3, k)), (+)((*)((getindex)(A, 3, 1, k), (getindex)(A, 4, 4, k)), (*)((*)(-1, (getindex)(A, 3, 4, k)), (getindex)(A, 4, 1, k))))), (*)((getindex)(A, 1, 4, k), (+)((*)((getindex)(A, 3, 1, k), (getindex)(A, 4, 3, k)), (*)((*)(-1, (getindex)(A, 3, 3, k)), (getindex)(A, 4, 1, k))))), (+)((+)((+)((*)((getindex)(A, 1, 1, k), (+)((+)((*)((getindex)(A, 2,2, k), (+)((*)((getindex)(A, 3, 3, k), (getindex)(A, 4, 4, k)), (*)((*)(-1, (getindex)(A, 3, 4, k)), (getindex)(A, 4, 3, k)))), (*)((*)(-1, (getindex)(A, 2, 3, k)), (+)((*)((getindex)(A, 3,2, k), (getindex)(A, 4, 4, k)), (*)((*)(-1, (getindex)(A, 3, 4, k)), (getindex)(A, 4,2, k))))), (*)((getindex)(A, 2, 4, k), (+)((*)((getindex)(A, 3,2, k), (getindex)(A, 4, 3, k)), (*)((*)(-1, (getindex)(A, 3, 3, k)), (getindex)(A, 4,2, k)))))), (*)((*)(-1, (getindex)(A, 1,2, k)), (+)((+)((*)((getindex)(A, 2, 1, k), (+)((*)((getindex)(A, 3, 3, k), (getindex)(A, 4, 4, k)), (*)((*)(-1, (getindex)(A, 3, 4, k)), (getindex)(A, 4, 3, k)))), (*)((*)(-1, (getindex)(A, 2, 3, k)), (+)((*)((getindex)(A, 3, 1, k), (getindex)(A, 4, 4, k)), (*)((*)(-1, (getindex)(A, 3, 4, k)), (getindex)(A, 4, 1, k))))), (*)((getindex)(A, 2, 4, k), (+)((*)((getindex)(A, 3, 1, k), (getindex)(A, 4, 3, k)), (*)((*)(-1, (getindex)(A, 3, 3, k)), (getindex)(A, 4, 1, k))))))), (*)((getindex)(A, 1, 3, k), (+)((+)((*)((getindex)(A, 2, 1, k), (+)((*)((getindex)(A, 3,2, k), (getindex)(A, 4, 4, k)), (*)((*)(-1, (getindex)(A, 3, 4, k)), (getindex)(A, 4,2, k)))), (*)((*)(-1, (getindex)(A, 2,2, k)), (+)((*)((getindex)(A, 3, 1, k), (getindex)(A, 4, 4, k)), (*)((*)(-1, (getindex)(A, 3, 4, k)), (getindex)(A, 4, 1, k))))), (*)((getindex)(A, 2, 4, k), (+)((*)((getindex)(A, 3, 1, k), (getindex)(A, 4,2, k)), (*)((*)(-1, (getindex)(A, 3,2, k)), (getindex)(A, 4, 1, k))))))), (*)((*)(-1, (getindex)(A, 1, 4, k)), (+)((+)((*)((getindex)(A, 2, 1, k), (+)((*)((getindex)(A, 3,2, k), (getindex)(A, 4, 3, k)), (*)((*)(-1, (getindex)(A, 3, 3, k)), (getindex)(A, 4,2, k)))), (*)((*)(-1, (getindex)(A, 2,2, k)), (+)((*)((getindex)(A, 3, 1, k), (getindex)(A, 4, 3, k)), (*)((*)(-1, (getindex)(A, 3, 3, k)), (getindex)(A, 4, 1, k))))), (*)((getindex)(A, 2, 3, k), (+)((*)((getindex)(A, 3, 1, k), (getindex)(A, 4,2, k)), (*)((*)(-1, (getindex)(A, 3,2, k)), (getindex)(A, 4, 1, k))))))))
+        ˍ₋out[3, 2, k] = (/)((+)((+)((*)((*)(-1, (getindex)(A, 1, 1, k)), (+)((*)((getindex)(A, 3,2, k), (getindex)(A, 4, 4, k)), (*)((*)(-1, (getindex)(A, 3, 4, k)), (getindex)(A, 4,2, k)))), (*)((getindex)(A, 1,2, k), (+)((*)((getindex)(A, 3, 1, k), (getindex)(A, 4, 4, k)), (*)((*)(-1, (getindex)(A, 3, 4, k)), (getindex)(A, 4, 1, k))))), (*)((*)(-1, (getindex)(A, 1, 4, k)), (+)((*)((getindex)(A, 3, 1, k), (getindex)(A, 4,2, k)), (*)((*)(-1, (getindex)(A, 3,2, k)), (getindex)(A, 4, 1, k))))), (+)((+)((+)((*)((getindex)(A, 1, 1, k), (+)((+)((*)((getindex)(A, 2,2, k), (+)((*)((getindex)(A, 3, 3, k), (getindex)(A, 4, 4, k)), (*)((*)(-1, (getindex)(A, 3, 4, k)), (getindex)(A, 4, 3, k)))), (*)((*)(-1, (getindex)(A, 2, 3, k)), (+)((*)((getindex)(A, 3,2, k), (getindex)(A, 4, 4, k)), (*)((*)(-1, (getindex)(A, 3, 4, k)), (getindex)(A, 4,2, k))))), (*)((getindex)(A, 2, 4, k), (+)((*)((getindex)(A, 3,2, k), (getindex)(A, 4, 3, k)), (*)((*)(-1, (getindex)(A, 3, 3, k)), (getindex)(A, 4,2, k)))))), (*)((*)(-1, (getindex)(A, 1,2, k)), (+)((+)((*)((getindex)(A, 2, 1, k), (+)((*)((getindex)(A, 3, 3, k), (getindex)(A, 4, 4, k)), (*)((*)(-1, (getindex)(A, 3, 4, k)), (getindex)(A, 4, 3, k)))), (*)((*)(-1, (getindex)(A, 2, 3, k)), (+)((*)((getindex)(A, 3, 1, k), (getindex)(A, 4, 4, k)), (*)((*)(-1, (getindex)(A, 3, 4, k)), (getindex)(A, 4, 1, k))))), (*)((getindex)(A, 2, 4, k), (+)((*)((getindex)(A, 3, 1, k), (getindex)(A, 4, 3, k)), (*)((*)(-1, (getindex)(A, 3, 3, k)), (getindex)(A, 4, 1, k))))))), (*)((getindex)(A, 1, 3, k), (+)((+)((*)((getindex)(A, 2, 1, k), (+)((*)((getindex)(A, 3,2, k), (getindex)(A, 4, 4, k)), (*)((*)(-1, (getindex)(A, 3, 4, k)), (getindex)(A, 4,2, k)))), (*)((*)(-1, (getindex)(A, 2,2, k)), (+)((*)((getindex)(A, 3, 1, k), (getindex)(A, 4, 4, k)), (*)((*)(-1, (getindex)(A, 3, 4, k)), (getindex)(A, 4, 1, k))))), (*)((getindex)(A, 2, 4, k), (+)((*)((getindex)(A, 3, 1, k), (getindex)(A, 4,2, k)), (*)((*)(-1, (getindex)(A, 3,2, k)), (getindex)(A, 4, 1, k))))))), (*)((*)(-1, (getindex)(A, 1, 4, k)), (+)((+)((*)((getindex)(A, 2, 1, k), (+)((*)((getindex)(A, 3,2, k), (getindex)(A, 4, 3, k)), (*)((*)(-1, (getindex)(A, 3, 3, k)), (getindex)(A, 4,2, k)))), (*)((*)(-1, (getindex)(A, 2,2, k)), (+)((*)((getindex)(A, 3, 1, k), (getindex)(A, 4, 3, k)), (*)((*)(-1, (getindex)(A, 3, 3, k)), (getindex)(A, 4, 1, k))))), (*)((getindex)(A, 2, 3, k), (+)((*)((getindex)(A, 3, 1, k), (getindex)(A, 4,2, k)), (*)((*)(-1, (getindex)(A, 3,2, k)), (getindex)(A, 4, 1, k))))))))
+        ˍ₋out[4, 2, k] = (/)((+)((+)((*)((getindex)(A, 1, 1, k), (+)((*)((getindex)(A, 3,2, k), (getindex)(A, 4, 3, k)), (*)((*)(-1, (getindex)(A, 3, 3, k)), (getindex)(A, 4,2, k)))), (*)((*)(-1, (getindex)(A, 1,2, k)), (+)((*)((getindex)(A, 3, 1, k), (getindex)(A, 4, 3, k)), (*)((*)(-1, (getindex)(A, 3, 3, k)), (getindex)(A, 4, 1, k))))), (*)((getindex)(A, 1, 3, k), (+)((*)((getindex)(A, 3, 1, k), (getindex)(A, 4,2, k)), (*)((*)(-1, (getindex)(A, 3,2, k)), (getindex)(A, 4, 1, k))))), (+)((+)((+)((*)((getindex)(A, 1, 1, k), (+)((+)((*)((getindex)(A, 2,2, k), (+)((*)((getindex)(A, 3, 3, k), (getindex)(A, 4, 4, k)), (*)((*)(-1, (getindex)(A, 3, 4, k)), (getindex)(A, 4, 3, k)))), (*)((*)(-1, (getindex)(A, 2, 3, k)), (+)((*)((getindex)(A, 3,2, k), (getindex)(A, 4, 4, k)), (*)((*)(-1, (getindex)(A, 3, 4, k)), (getindex)(A, 4,2, k))))), (*)((getindex)(A, 2, 4, k), (+)((*)((getindex)(A, 3,2, k), (getindex)(A, 4, 3, k)), (*)((*)(-1, (getindex)(A, 3, 3, k)), (getindex)(A, 4,2, k)))))), (*)((*)(-1, (getindex)(A, 1,2, k)), (+)((+)((*)((getindex)(A, 2, 1, k), (+)((*)((getindex)(A, 3, 3, k), (getindex)(A, 4, 4, k)), (*)((*)(-1, (getindex)(A, 3, 4, k)), (getindex)(A, 4, 3, k)))), (*)((*)(-1, (getindex)(A, 2, 3, k)), (+)((*)((getindex)(A, 3, 1, k), (getindex)(A, 4, 4, k)), (*)((*)(-1, (getindex)(A, 3, 4, k)), (getindex)(A, 4, 1, k))))), (*)((getindex)(A, 2, 4, k), (+)((*)((getindex)(A, 3, 1, k), (getindex)(A, 4, 3, k)), (*)((*)(-1, (getindex)(A, 3, 3, k)), (getindex)(A, 4, 1, k))))))), (*)((getindex)(A, 1, 3, k), (+)((+)((*)((getindex)(A, 2, 1, k), (+)((*)((getindex)(A, 3,2, k), (getindex)(A, 4, 4, k)), (*)((*)(-1, (getindex)(A, 3, 4, k)), (getindex)(A, 4,2, k)))), (*)((*)(-1, (getindex)(A, 2,2, k)), (+)((*)((getindex)(A, 3, 1, k), (getindex)(A, 4, 4, k)), (*)((*)(-1, (getindex)(A, 3, 4, k)), (getindex)(A, 4, 1, k))))), (*)((getindex)(A, 2, 4, k), (+)((*)((getindex)(A, 3, 1, k), (getindex)(A, 4,2, k)), (*)((*)(-1, (getindex)(A, 3,2, k)), (getindex)(A, 4, 1, k))))))), (*)((*)(-1, (getindex)(A, 1, 4, k)), (+)((+)((*)((getindex)(A, 2, 1, k), (+)((*)((getindex)(A, 3,2, k), (getindex)(A, 4, 3, k)), (*)((*)(-1, (getindex)(A, 3, 3, k)), (getindex)(A, 4,2, k)))), (*)((*)(-1, (getindex)(A, 2,2, k)), (+)((*)((getindex)(A, 3, 1, k), (getindex)(A, 4, 3, k)), (*)((*)(-1, (getindex)(A, 3, 3, k)), (getindex)(A, 4, 1, k))))), (*)((getindex)(A, 2, 3, k), (+)((*)((getindex)(A, 3, 1, k), (getindex)(A, 4,2, k)), (*)((*)(-1, (getindex)(A, 3,2, k)), (getindex)(A, 4, 1, k))))))))
+        ˍ₋out[1, 3, k] = (/)((+)((+)((*)((getindex)(A, 1,2, k), (+)((*)((getindex)(A, 2, 3, k), (getindex)(A, 4, 4, k)), (*)((*)(-1, (getindex)(A, 2, 4, k)), (getindex)(A, 4, 3, k)))), (*)((*)(-1, (getindex)(A, 1, 3, k)), (+)((*)((getindex)(A, 2,2, k), (getindex)(A, 4, 4, k)), (*)((*)(-1, (getindex)(A, 2, 4, k)), (getindex)(A, 4,2, k))))), (*)((getindex)(A, 1, 4, k), (+)((*)((getindex)(A, 2,2, k), (getindex)(A, 4, 3, k)), (*)((*)(-1, (getindex)(A, 2, 3, k)), (getindex)(A, 4,2, k))))), (+)((+)((+)((*)((getindex)(A, 1, 1, k), (+)((+)((*)((getindex)(A, 2,2, k), (+)((*)((getindex)(A, 3, 3, k), (getindex)(A, 4, 4, k)), (*)((*)(-1, (getindex)(A, 3, 4, k)), (getindex)(A, 4, 3, k)))), (*)((*)(-1, (getindex)(A, 2, 3, k)), (+)((*)((getindex)(A, 3,2, k), (getindex)(A, 4, 4, k)), (*)((*)(-1, (getindex)(A, 3, 4, k)), (getindex)(A, 4,2, k))))), (*)((getindex)(A, 2, 4, k), (+)((*)((getindex)(A, 3,2, k), (getindex)(A, 4, 3, k)), (*)((*)(-1, (getindex)(A, 3, 3, k)), (getindex)(A, 4,2, k)))))), (*)((*)(-1, (getindex)(A, 1,2, k)), (+)((+)((*)((getindex)(A, 2, 1, k), (+)((*)((getindex)(A, 3, 3, k), (getindex)(A, 4, 4, k)), (*)((*)(-1, (getindex)(A, 3, 4, k)), (getindex)(A, 4, 3, k)))), (*)((*)(-1, (getindex)(A, 2, 3, k)), (+)((*)((getindex)(A, 3, 1, k), (getindex)(A, 4, 4, k)), (*)((*)(-1, (getindex)(A, 3, 4, k)), (getindex)(A, 4, 1, k))))), (*)((getindex)(A, 2, 4, k), (+)((*)((getindex)(A, 3, 1, k), (getindex)(A, 4, 3, k)), (*)((*)(-1, (getindex)(A, 3, 3, k)), (getindex)(A, 4, 1, k))))))), (*)((getindex)(A, 1, 3, k), (+)((+)((*)((getindex)(A, 2, 1, k), (+)((*)((getindex)(A, 3,2, k), (getindex)(A, 4, 4, k)), (*)((*)(-1, (getindex)(A, 3, 4, k)), (getindex)(A, 4,2, k)))), (*)((*)(-1, (getindex)(A, 2,2, k)), (+)((*)((getindex)(A, 3, 1, k), (getindex)(A, 4, 4, k)), (*)((*)(-1, (getindex)(A, 3, 4, k)), (getindex)(A, 4, 1, k))))), (*)((getindex)(A, 2, 4, k), (+)((*)((getindex)(A, 3, 1, k), (getindex)(A, 4,2, k)), (*)((*)(-1, (getindex)(A, 3,2, k)), (getindex)(A, 4, 1, k))))))), (*)((*)(-1, (getindex)(A, 1, 4, k)), (+)((+)((*)((getindex)(A, 2, 1, k), (+)((*)((getindex)(A, 3,2, k), (getindex)(A, 4, 3, k)), (*)((*)(-1, (getindex)(A, 3, 3, k)), (getindex)(A, 4,2, k)))), (*)((*)(-1, (getindex)(A, 2,2, k)), (+)((*)((getindex)(A, 3, 1, k), (getindex)(A, 4, 3, k)), (*)((*)(-1, (getindex)(A, 3, 3, k)), (getindex)(A, 4, 1, k))))), (*)((getindex)(A, 2, 3, k), (+)((*)((getindex)(A, 3, 1, k), (getindex)(A, 4,2, k)), (*)((*)(-1, (getindex)(A, 3,2, k)), (getindex)(A, 4, 1, k))))))))
+        ˍ₋out[2, 3, k] = (/)((+)((+)((*)((*)(-1, (getindex)(A, 1, 1, k)), (+)((*)((getindex)(A, 2, 3, k), (getindex)(A, 4, 4, k)), (*)((*)(-1, (getindex)(A, 2, 4, k)), (getindex)(A, 4, 3, k)))), (*)((getindex)(A, 1, 3, k), (+)((*)((getindex)(A, 2, 1, k), (getindex)(A, 4, 4, k)), (*)((*)(-1, (getindex)(A, 2, 4, k)), (getindex)(A, 4, 1, k))))), (*)((*)(-1, (getindex)(A, 1, 4, k)), (+)((*)((getindex)(A, 2, 1, k), (getindex)(A, 4, 3, k)), (*)((*)(-1, (getindex)(A, 2, 3, k)), (getindex)(A, 4, 1, k))))), (+)((+)((+)((*)((getindex)(A, 1, 1, k), (+)((+)((*)((getindex)(A, 2,2, k), (+)((*)((getindex)(A, 3, 3, k), (getindex)(A, 4, 4, k)), (*)((*)(-1, (getindex)(A, 3, 4, k)), (getindex)(A, 4, 3, k)))), (*)((*)(-1, (getindex)(A, 2, 3, k)), (+)((*)((getindex)(A, 3,2, k), (getindex)(A, 4, 4, k)), (*)((*)(-1, (getindex)(A, 3, 4, k)), (getindex)(A, 4,2, k))))), (*)((getindex)(A, 2, 4, k), (+)((*)((getindex)(A, 3,2, k), (getindex)(A, 4, 3, k)), (*)((*)(-1, (getindex)(A, 3, 3, k)), (getindex)(A, 4,2, k)))))), (*)((*)(-1, (getindex)(A, 1,2, k)), (+)((+)((*)((getindex)(A, 2, 1, k), (+)((*)((getindex)(A, 3, 3, k), (getindex)(A, 4, 4, k)), (*)((*)(-1, (getindex)(A, 3, 4, k)), (getindex)(A, 4, 3, k)))), (*)((*)(-1, (getindex)(A, 2, 3, k)), (+)((*)((getindex)(A, 3, 1, k), (getindex)(A, 4, 4, k)), (*)((*)(-1, (getindex)(A, 3, 4, k)), (getindex)(A, 4, 1, k))))), (*)((getindex)(A, 2, 4, k), (+)((*)((getindex)(A, 3, 1, k), (getindex)(A, 4, 3, k)), (*)((*)(-1, (getindex)(A, 3, 3, k)), (getindex)(A, 4, 1, k))))))), (*)((getindex)(A, 1, 3, k), (+)((+)((*)((getindex)(A, 2, 1, k), (+)((*)((getindex)(A, 3,2, k), (getindex)(A, 4, 4, k)), (*)((*)(-1, (getindex)(A, 3, 4, k)), (getindex)(A, 4,2, k)))), (*)((*)(-1, (getindex)(A, 2,2, k)), (+)((*)((getindex)(A, 3, 1, k), (getindex)(A, 4, 4, k)), (*)((*)(-1, (getindex)(A, 3, 4, k)), (getindex)(A, 4, 1, k))))), (*)((getindex)(A, 2, 4, k), (+)((*)((getindex)(A, 3, 1, k), (getindex)(A, 4,2, k)), (*)((*)(-1, (getindex)(A, 3,2, k)), (getindex)(A, 4, 1, k))))))), (*)((*)(-1, (getindex)(A, 1, 4, k)), (+)((+)((*)((getindex)(A, 2, 1, k), (+)((*)((getindex)(A, 3,2, k), (getindex)(A, 4, 3, k)), (*)((*)(-1, (getindex)(A, 3, 3, k)), (getindex)(A, 4,2, k)))), (*)((*)(-1, (getindex)(A, 2,2, k)), (+)((*)((getindex)(A, 3, 1, k), (getindex)(A, 4, 3, k)), (*)((*)(-1, (getindex)(A, 3, 3, k)), (getindex)(A, 4, 1, k))))), (*)((getindex)(A, 2, 3, k), (+)((*)((getindex)(A, 3, 1, k), (getindex)(A, 4,2, k)), (*)((*)(-1, (getindex)(A, 3,2, k)), (getindex)(A, 4, 1, k))))))))
+        ˍ₋out[3, 3, k] = (/)((+)((+)((*)((getindex)(A, 1, 1, k), (+)((*)((getindex)(A, 2,2, k), (getindex)(A, 4, 4, k)), (*)((*)(-1, (getindex)(A, 2, 4, k)), (getindex)(A, 4,2, k)))), (*)((*)(-1, (getindex)(A, 1,2, k)), (+)((*)((getindex)(A, 2, 1, k), (getindex)(A, 4, 4, k)), (*)((*)(-1, (getindex)(A, 2, 4, k)), (getindex)(A, 4, 1, k))))), (*)((getindex)(A, 1, 4, k), (+)((*)((getindex)(A, 2, 1, k), (getindex)(A, 4,2, k)), (*)((*)(-1, (getindex)(A, 2,2, k)), (getindex)(A, 4, 1, k))))), (+)((+)((+)((*)((getindex)(A, 1, 1, k), (+)((+)((*)((getindex)(A, 2,2, k), (+)((*)((getindex)(A, 3, 3, k), (getindex)(A, 4, 4, k)), (*)((*)(-1, (getindex)(A, 3, 4, k)), (getindex)(A, 4, 3, k)))), (*)((*)(-1, (getindex)(A, 2, 3, k)), (+)((*)((getindex)(A, 3,2, k), (getindex)(A, 4, 4, k)), (*)((*)(-1, (getindex)(A, 3, 4, k)), (getindex)(A, 4,2, k))))), (*)((getindex)(A, 2, 4, k), (+)((*)((getindex)(A, 3,2, k), (getindex)(A, 4, 3, k)), (*)((*)(-1, (getindex)(A, 3, 3, k)), (getindex)(A, 4,2, k)))))), (*)((*)(-1, (getindex)(A, 1,2, k)), (+)((+)((*)((getindex)(A, 2, 1, k), (+)((*)((getindex)(A, 3, 3, k), (getindex)(A, 4, 4, k)), (*)((*)(-1, (getindex)(A, 3, 4, k)), (getindex)(A, 4, 3, k)))), (*)((*)(-1, (getindex)(A, 2, 3, k)), (+)((*)((getindex)(A, 3, 1, k), (getindex)(A, 4, 4, k)), (*)((*)(-1, (getindex)(A, 3, 4, k)), (getindex)(A, 4, 1, k))))), (*)((getindex)(A, 2, 4, k), (+)((*)((getindex)(A, 3, 1, k), (getindex)(A, 4, 3, k)), (*)((*)(-1, (getindex)(A, 3, 3, k)), (getindex)(A, 4, 1, k))))))), (*)((getindex)(A, 1, 3, k), (+)((+)((*)((getindex)(A, 2, 1, k), (+)((*)((getindex)(A, 3,2, k), (getindex)(A, 4, 4, k)), (*)((*)(-1, (getindex)(A, 3, 4, k)), (getindex)(A, 4,2, k)))), (*)((*)(-1, (getindex)(A, 2,2, k)), (+)((*)((getindex)(A, 3, 1, k), (getindex)(A, 4, 4, k)), (*)((*)(-1, (getindex)(A, 3, 4, k)), (getindex)(A, 4, 1, k))))), (*)((getindex)(A, 2, 4, k), (+)((*)((getindex)(A, 3, 1, k), (getindex)(A, 4,2, k)), (*)((*)(-1, (getindex)(A, 3,2, k)), (getindex)(A, 4, 1, k))))))), (*)((*)(-1, (getindex)(A, 1, 4, k)), (+)((+)((*)((getindex)(A, 2, 1, k), (+)((*)((getindex)(A, 3,2, k), (getindex)(A, 4, 3, k)), (*)((*)(-1, (getindex)(A, 3, 3, k)), (getindex)(A, 4,2, k)))), (*)((*)(-1, (getindex)(A, 2,2, k)), (+)((*)((getindex)(A, 3, 1, k), (getindex)(A, 4, 3, k)), (*)((*)(-1, (getindex)(A, 3, 3, k)), (getindex)(A, 4, 1, k))))), (*)((getindex)(A, 2, 3, k), (+)((*)((getindex)(A, 3, 1, k), (getindex)(A, 4,2, k)), (*)((*)(-1, (getindex)(A, 3,2, k)), (getindex)(A, 4, 1, k))))))))
+        ˍ₋out[4, 3, k] = (/)((+)((+)((*)((*)(-1, (getindex)(A, 1, 1, k)), (+)((*)((getindex)(A, 2,2, k), (getindex)(A, 4, 3, k)), (*)((*)(-1, (getindex)(A, 2, 3, k)), (getindex)(A, 4,2, k)))), (*)((getindex)(A, 1,2, k), (+)((*)((getindex)(A, 2, 1, k), (getindex)(A, 4, 3, k)), (*)((*)(-1, (getindex)(A, 2, 3, k)), (getindex)(A, 4, 1, k))))), (*)((*)(-1, (getindex)(A, 1, 3, k)), (+)((*)((getindex)(A, 2, 1, k), (getindex)(A, 4,2, k)), (*)((*)(-1, (getindex)(A, 2,2, k)), (getindex)(A, 4, 1, k))))), (+)((+)((+)((*)((getindex)(A, 1, 1, k), (+)((+)((*)((getindex)(A, 2,2, k), (+)((*)((getindex)(A, 3, 3, k), (getindex)(A, 4, 4, k)), (*)((*)(-1, (getindex)(A, 3, 4, k)), (getindex)(A, 4, 3, k)))), (*)((*)(-1, (getindex)(A, 2, 3, k)), (+)((*)((getindex)(A, 3,2, k), (getindex)(A, 4, 4, k)), (*)((*)(-1, (getindex)(A, 3, 4, k)), (getindex)(A, 4,2, k))))), (*)((getindex)(A, 2, 4, k), (+)((*)((getindex)(A, 3,2, k), (getindex)(A, 4, 3, k)), (*)((*)(-1, (getindex)(A, 3, 3, k)), (getindex)(A, 4,2, k)))))), (*)((*)(-1, (getindex)(A, 1,2, k)), (+)((+)((*)((getindex)(A, 2, 1, k), (+)((*)((getindex)(A, 3, 3, k), (getindex)(A, 4, 4, k)), (*)((*)(-1, (getindex)(A, 3, 4, k)), (getindex)(A, 4, 3, k)))), (*)((*)(-1, (getindex)(A, 2, 3, k)), (+)((*)((getindex)(A, 3, 1, k), (getindex)(A, 4, 4, k)), (*)((*)(-1, (getindex)(A, 3, 4, k)), (getindex)(A, 4, 1, k))))), (*)((getindex)(A, 2, 4, k), (+)((*)((getindex)(A, 3, 1, k), (getindex)(A, 4, 3, k)), (*)((*)(-1, (getindex)(A, 3, 3, k)), (getindex)(A, 4, 1, k))))))), (*)((getindex)(A, 1, 3, k), (+)((+)((*)((getindex)(A, 2, 1, k), (+)((*)((getindex)(A, 3,2, k), (getindex)(A, 4, 4, k)), (*)((*)(-1, (getindex)(A, 3, 4, k)), (getindex)(A, 4,2, k)))), (*)((*)(-1, (getindex)(A, 2,2, k)), (+)((*)((getindex)(A, 3, 1, k), (getindex)(A, 4, 4, k)), (*)((*)(-1, (getindex)(A, 3, 4, k)), (getindex)(A, 4, 1, k))))), (*)((getindex)(A, 2, 4, k), (+)((*)((getindex)(A, 3, 1, k), (getindex)(A, 4,2, k)), (*)((*)(-1, (getindex)(A, 3,2, k)), (getindex)(A, 4, 1, k))))))), (*)((*)(-1, (getindex)(A, 1, 4, k)), (+)((+)((*)((getindex)(A, 2, 1, k), (+)((*)((getindex)(A, 3,2, k), (getindex)(A, 4, 3, k)), (*)((*)(-1, (getindex)(A, 3, 3, k)), (getindex)(A, 4,2, k)))), (*)((*)(-1, (getindex)(A, 2,2, k)), (+)((*)((getindex)(A, 3, 1, k), (getindex)(A, 4, 3, k)), (*)((*)(-1, (getindex)(A, 3, 3, k)), (getindex)(A, 4, 1, k))))), (*)((getindex)(A, 2, 3, k), (+)((*)((getindex)(A, 3, 1, k), (getindex)(A, 4,2, k)), (*)((*)(-1, (getindex)(A, 3,2, k)), (getindex)(A, 4, 1, k))))))))
+        ˍ₋out[1, 4, k] = (/)((+)((+)((*)((*)(-1, (getindex)(A, 1,2, k)), (+)((*)((getindex)(A, 2, 3, k), (getindex)(A, 3, 4, k)), (*)((*)(-1, (getindex)(A, 2, 4, k)), (getindex)(A, 3, 3, k)))), (*)((getindex)(A, 1, 3, k), (+)((*)((getindex)(A, 2,2, k), (getindex)(A, 3, 4, k)), (*)((*)(-1, (getindex)(A, 2, 4, k)), (getindex)(A, 3,2, k))))), (*)((*)(-1, (getindex)(A, 1, 4, k)), (+)((*)((getindex)(A, 2,2, k), (getindex)(A, 3, 3, k)), (*)((*)(-1, (getindex)(A, 2, 3, k)), (getindex)(A, 3,2, k))))), (+)((+)((+)((*)((getindex)(A, 1, 1, k), (+)((+)((*)((getindex)(A, 2,2, k), (+)((*)((getindex)(A, 3, 3, k), (getindex)(A, 4, 4, k)), (*)((*)(-1, (getindex)(A, 3, 4, k)), (getindex)(A, 4, 3, k)))), (*)((*)(-1, (getindex)(A, 2, 3, k)), (+)((*)((getindex)(A, 3,2, k), (getindex)(A, 4, 4, k)), (*)((*)(-1, (getindex)(A, 3, 4, k)), (getindex)(A, 4,2, k))))), (*)((getindex)(A, 2, 4, k), (+)((*)((getindex)(A, 3,2, k), (getindex)(A, 4, 3, k)), (*)((*)(-1, (getindex)(A, 3, 3, k)), (getindex)(A, 4,2, k)))))), (*)((*)(-1, (getindex)(A, 1,2, k)), (+)((+)((*)((getindex)(A, 2, 1, k), (+)((*)((getindex)(A, 3, 3, k), (getindex)(A, 4, 4, k)), (*)((*)(-1, (getindex)(A, 3, 4, k)), (getindex)(A, 4, 3, k)))), (*)((*)(-1, (getindex)(A, 2, 3, k)), (+)((*)((getindex)(A, 3, 1, k), (getindex)(A, 4, 4, k)), (*)((*)(-1, (getindex)(A, 3, 4, k)), (getindex)(A, 4, 1, k))))), (*)((getindex)(A, 2, 4, k), (+)((*)((getindex)(A, 3, 1, k), (getindex)(A, 4, 3, k)), (*)((*)(-1, (getindex)(A, 3, 3, k)), (getindex)(A, 4, 1, k))))))), (*)((getindex)(A, 1, 3, k), (+)((+)((*)((getindex)(A, 2, 1, k), (+)((*)((getindex)(A, 3,2, k), (getindex)(A, 4, 4, k)), (*)((*)(-1, (getindex)(A, 3, 4, k)), (getindex)(A, 4,2, k)))), (*)((*)(-1, (getindex)(A, 2,2, k)), (+)((*)((getindex)(A, 3, 1, k), (getindex)(A, 4, 4, k)), (*)((*)(-1, (getindex)(A, 3, 4, k)), (getindex)(A, 4, 1, k))))), (*)((getindex)(A, 2, 4, k), (+)((*)((getindex)(A, 3, 1, k), (getindex)(A, 4,2, k)), (*)((*)(-1, (getindex)(A, 3,2, k)), (getindex)(A, 4, 1, k))))))), (*)((*)(-1, (getindex)(A, 1, 4, k)), (+)((+)((*)((getindex)(A, 2, 1, k), (+)((*)((getindex)(A, 3,2, k), (getindex)(A, 4, 3, k)), (*)((*)(-1, (getindex)(A, 3, 3, k)), (getindex)(A, 4,2, k)))), (*)((*)(-1, (getindex)(A, 2,2, k)), (+)((*)((getindex)(A, 3, 1, k), (getindex)(A, 4, 3, k)), (*)((*)(-1, (getindex)(A, 3, 3, k)), (getindex)(A, 4, 1, k))))), (*)((getindex)(A, 2, 3, k), (+)((*)((getindex)(A, 3, 1, k), (getindex)(A, 4,2, k)), (*)((*)(-1, (getindex)(A, 3,2, k)), (getindex)(A, 4, 1, k))))))))
+        ˍ₋out[2, 4, k] = (/)((+)((+)((*)((getindex)(A, 1, 1, k), (+)((*)((getindex)(A, 2, 3, k), (getindex)(A, 3, 4, k)), (*)((*)(-1, (getindex)(A, 2, 4, k)), (getindex)(A, 3, 3, k)))), (*)((*)(-1, (getindex)(A, 1, 3, k)), (+)((*)((getindex)(A, 2, 1, k), (getindex)(A, 3, 4, k)), (*)((*)(-1, (getindex)(A, 2, 4, k)), (getindex)(A, 3, 1, k))))), (*)((getindex)(A, 1, 4, k), (+)((*)((getindex)(A, 2, 1, k), (getindex)(A, 3, 3, k)), (*)((*)(-1, (getindex)(A, 2, 3, k)), (getindex)(A, 3, 1, k))))), (+)((+)((+)((*)((getindex)(A, 1, 1, k), (+)((+)((*)((getindex)(A, 2,2, k), (+)((*)((getindex)(A, 3, 3, k), (getindex)(A, 4, 4, k)), (*)((*)(-1, (getindex)(A, 3, 4, k)), (getindex)(A, 4, 3, k)))), (*)((*)(-1, (getindex)(A, 2, 3, k)), (+)((*)((getindex)(A, 3,2, k), (getindex)(A, 4, 4, k)), (*)((*)(-1, (getindex)(A, 3, 4, k)), (getindex)(A, 4,2, k))))), (*)((getindex)(A, 2, 4, k), (+)((*)((getindex)(A, 3,2, k), (getindex)(A, 4, 3, k)), (*)((*)(-1, (getindex)(A, 3, 3, k)), (getindex)(A, 4,2, k)))))), (*)((*)(-1, (getindex)(A, 1,2, k)), (+)((+)((*)((getindex)(A, 2, 1, k), (+)((*)((getindex)(A, 3, 3, k), (getindex)(A, 4, 4, k)), (*)((*)(-1, (getindex)(A, 3, 4, k)), (getindex)(A, 4, 3, k)))), (*)((*)(-1, (getindex)(A, 2, 3, k)), (+)((*)((getindex)(A, 3, 1, k), (getindex)(A, 4, 4, k)), (*)((*)(-1, (getindex)(A, 3, 4, k)), (getindex)(A, 4, 1, k))))), (*)((getindex)(A, 2, 4, k), (+)((*)((getindex)(A, 3, 1, k), (getindex)(A, 4, 3, k)), (*)((*)(-1, (getindex)(A, 3, 3, k)), (getindex)(A, 4, 1, k))))))), (*)((getindex)(A, 1, 3, k), (+)((+)((*)((getindex)(A, 2, 1, k), (+)((*)((getindex)(A, 3,2, k), (getindex)(A, 4, 4, k)), (*)((*)(-1, (getindex)(A, 3, 4, k)), (getindex)(A, 4,2, k)))), (*)((*)(-1, (getindex)(A, 2,2, k)), (+)((*)((getindex)(A, 3, 1, k), (getindex)(A, 4, 4, k)), (*)((*)(-1, (getindex)(A, 3, 4, k)), (getindex)(A, 4, 1, k))))), (*)((getindex)(A, 2, 4, k), (+)((*)((getindex)(A, 3, 1, k), (getindex)(A, 4,2, k)), (*)((*)(-1, (getindex)(A, 3,2, k)), (getindex)(A, 4, 1, k))))))), (*)((*)(-1, (getindex)(A, 1, 4, k)), (+)((+)((*)((getindex)(A, 2, 1, k), (+)((*)((getindex)(A, 3,2, k), (getindex)(A, 4, 3, k)), (*)((*)(-1, (getindex)(A, 3, 3, k)), (getindex)(A, 4,2, k)))), (*)((*)(-1, (getindex)(A, 2,2, k)), (+)((*)((getindex)(A, 3, 1, k), (getindex)(A, 4, 3, k)), (*)((*)(-1, (getindex)(A, 3, 3, k)), (getindex)(A, 4, 1, k))))), (*)((getindex)(A, 2, 3, k), (+)((*)((getindex)(A, 3, 1, k), (getindex)(A, 4,2, k)), (*)((*)(-1, (getindex)(A, 3,2, k)), (getindex)(A, 4, 1, k))))))))
+        ˍ₋out[3, 4, k] = (/)((+)((+)((*)((*)(-1, (getindex)(A, 1, 1, k)), (+)((*)((getindex)(A, 2,2, k), (getindex)(A, 3, 4, k)), (*)((*)(-1, (getindex)(A, 2, 4, k)), (getindex)(A, 3,2, k)))), (*)((getindex)(A, 1,2, k), (+)((*)((getindex)(A, 2, 1, k), (getindex)(A, 3, 4, k)), (*)((*)(-1, (getindex)(A, 2, 4, k)), (getindex)(A, 3, 1, k))))), (*)((*)(-1, (getindex)(A, 1, 4, k)), (+)((*)((getindex)(A, 2, 1, k), (getindex)(A, 3,2, k)), (*)((*)(-1, (getindex)(A, 2,2, k)), (getindex)(A, 3, 1, k))))), (+)((+)((+)((*)((getindex)(A, 1, 1, k), (+)((+)((*)((getindex)(A, 2,2, k), (+)((*)((getindex)(A, 3, 3, k), (getindex)(A, 4, 4, k)), (*)((*)(-1, (getindex)(A, 3, 4, k)), (getindex)(A, 4, 3, k)))), (*)((*)(-1, (getindex)(A, 2, 3, k)), (+)((*)((getindex)(A, 3,2, k), (getindex)(A, 4, 4, k)), (*)((*)(-1, (getindex)(A, 3, 4, k)), (getindex)(A, 4,2, k))))), (*)((getindex)(A, 2, 4, k), (+)((*)((getindex)(A, 3,2, k), (getindex)(A, 4, 3, k)), (*)((*)(-1, (getindex)(A, 3, 3, k)), (getindex)(A, 4,2, k)))))), (*)((*)(-1, (getindex)(A, 1,2, k)), (+)((+)((*)((getindex)(A, 2, 1, k), (+)((*)((getindex)(A, 3, 3, k), (getindex)(A, 4, 4, k)), (*)((*)(-1, (getindex)(A, 3, 4, k)), (getindex)(A, 4, 3, k)))), (*)((*)(-1, (getindex)(A, 2, 3, k)), (+)((*)((getindex)(A, 3, 1, k), (getindex)(A, 4, 4, k)), (*)((*)(-1, (getindex)(A, 3, 4, k)), (getindex)(A, 4, 1, k))))), (*)((getindex)(A, 2, 4, k), (+)((*)((getindex)(A, 3, 1, k), (getindex)(A, 4, 3, k)), (*)((*)(-1, (getindex)(A, 3, 3, k)), (getindex)(A, 4, 1, k))))))), (*)((getindex)(A, 1, 3, k), (+)((+)((*)((getindex)(A, 2, 1, k), (+)((*)((getindex)(A, 3,2, k), (getindex)(A, 4, 4, k)), (*)((*)(-1, (getindex)(A, 3, 4, k)), (getindex)(A, 4,2, k)))), (*)((*)(-1, (getindex)(A, 2,2, k)), (+)((*)((getindex)(A, 3, 1, k), (getindex)(A, 4, 4, k)), (*)((*)(-1, (getindex)(A, 3, 4, k)), (getindex)(A, 4, 1, k))))), (*)((getindex)(A, 2, 4, k), (+)((*)((getindex)(A, 3, 1, k), (getindex)(A, 4,2, k)), (*)((*)(-1, (getindex)(A, 3,2, k)), (getindex)(A, 4, 1, k))))))), (*)((*)(-1, (getindex)(A, 1, 4, k)), (+)((+)((*)((getindex)(A, 2, 1, k), (+)((*)((getindex)(A, 3,2, k), (getindex)(A, 4, 3, k)), (*)((*)(-1, (getindex)(A, 3, 3, k)), (getindex)(A, 4,2, k)))), (*)((*)(-1, (getindex)(A, 2,2, k)), (+)((*)((getindex)(A, 3, 1, k), (getindex)(A, 4, 3, k)), (*)((*)(-1, (getindex)(A, 3, 3, k)), (getindex)(A, 4, 1, k))))), (*)((getindex)(A, 2, 3, k), (+)((*)((getindex)(A, 3, 1, k), (getindex)(A, 4,2, k)), (*)((*)(-1, (getindex)(A, 3,2, k)), (getindex)(A, 4, 1, k))))))))
+        ˍ₋out[4, 4, k] = (/)((+)((+)((*)((getindex)(A, 1, 1, k), (+)((*)((getindex)(A, 2,2, k), (getindex)(A, 3, 3, k)), (*)((*)(-1, (getindex)(A, 2, 3, k)), (getindex)(A, 3,2, k)))), (*)((*)(-1, (getindex)(A, 1,2, k)), (+)((*)((getindex)(A, 2, 1, k), (getindex)(A, 3, 3, k)), (*)((*)(-1, (getindex)(A, 2, 3, k)), (getindex)(A, 3, 1, k))))), (*)((getindex)(A, 1, 3, k), (+)((*)((getindex)(A, 2, 1, k), (getindex)(A, 3,2, k)), (*)((*)(-1, (getindex)(A, 2,2, k)), (getindex)(A, 3, 1, k))))), (+)((+)((+)((*)((getindex)(A, 1, 1, k), (+)((+)((*)((getindex)(A, 2,2, k), (+)((*)((getindex)(A, 3, 3, k), (getindex)(A, 4, 4, k)), (*)((*)(-1, (getindex)(A, 3, 4, k)), (getindex)(A, 4, 3, k)))), (*)((*)(-1, (getindex)(A, 2, 3, k)), (+)((*)((getindex)(A, 3,2, k), (getindex)(A, 4, 4, k)), (*)((*)(-1, (getindex)(A, 3, 4, k)), (getindex)(A, 4,2, k))))), (*)((getindex)(A, 2, 4, k), (+)((*)((getindex)(A, 3,2, k), (getindex)(A, 4, 3, k)), (*)((*)(-1, (getindex)(A, 3, 3, k)), (getindex)(A, 4,2, k)))))), (*)((*)(-1, (getindex)(A, 1,2, k)), (+)((+)((*)((getindex)(A, 2, 1, k), (+)((*)((getindex)(A, 3, 3, k), (getindex)(A, 4, 4, k)), (*)((*)(-1, (getindex)(A, 3, 4, k)), (getindex)(A, 4, 3, k)))), (*)((*)(-1, (getindex)(A, 2, 3, k)), (+)((*)((getindex)(A, 3, 1, k), (getindex)(A, 4, 4, k)), (*)((*)(-1, (getindex)(A, 3, 4, k)), (getindex)(A, 4, 1, k))))), (*)((getindex)(A, 2, 4, k), (+)((*)((getindex)(A, 3, 1, k), (getindex)(A, 4, 3, k)), (*)((*)(-1, (getindex)(A, 3, 3, k)), (getindex)(A, 4, 1, k))))))), (*)((getindex)(A, 1, 3, k), (+)((+)((*)((getindex)(A, 2, 1, k), (+)((*)((getindex)(A, 3,2, k), (getindex)(A, 4, 4, k)), (*)((*)(-1, (getindex)(A, 3, 4, k)), (getindex)(A, 4,2, k)))), (*)((*)(-1, (getindex)(A, 2,2, k)), (+)((*)((getindex)(A, 3, 1, k), (getindex)(A, 4, 4, k)), (*)((*)(-1, (getindex)(A, 3, 4, k)), (getindex)(A, 4, 1, k))))), (*)((getindex)(A, 2, 4, k), (+)((*)((getindex)(A, 3, 1, k), (getindex)(A, 4,2, k)), (*)((*)(-1, (getindex)(A, 3,2, k)), (getindex)(A, 4, 1, k))))))), (*)((*)(-1, (getindex)(A, 1, 4, k)), (+)((+)((*)((getindex)(A, 2, 1, k), (+)((*)((getindex)(A, 3,2, k), (getindex)(A, 4, 3, k)), (*)((*)(-1, (getindex)(A, 3, 3, k)), (getindex)(A, 4,2, k)))), (*)((*)(-1, (getindex)(A, 2,2, k)), (+)((*)((getindex)(A, 3, 1, k), (getindex)(A, 4, 3, k)), (*)((*)(-1, (getindex)(A, 3, 3, k)), (getindex)(A, 4, 1, k))))), (*)((getindex)(A, 2, 3, k), (+)((*)((getindex)(A, 3, 1, k), (getindex)(A, 4,2, k)), (*)((*)(-1, (getindex)(A, 3,2, k)), (getindex)(A, 4, 1, k))))))))
+        nothing
+    end
+end
+
+function tensor_inverse4(A::AbstractArray{T, 3}) where T 
+    out = similar(A)
+
+    tensor_inverse4!(out, A)
+    
+    out 
+end
+
+function tensor_inverse4!(out::AbstractArray{T, 3}, A::AbstractArray{T, 3}) where T 
+    @assert size(A, 1) == size(A, 2) == 4
+    @assert size(A) == size(out)
+
+    backend = get_backend(out)
+    inv44! = inv44_kernel!(backend)
+
+    inv44!(out, A, ndrange = size(A, 3))
+
+    nothing
+end
+
+function ChainRulesCore.rrule(::typeof(tensor_inverse4), A::AT) where {T, AT<:AbstractArray{T, 3}}
+    out = tensor_inverse4(A)
+    
+    function tensor_inverse_pullback(out_diff::AT)
+
+        NoTangent(), - tensor_transpose_tensor_mul(out, tensor_tensor_mul(out_diff, tensor_transpose(out)))
+    end 
+    out, tensor_inverse_pullback 
+end 
\ No newline at end of file
diff --git a/src/kernels/inverses/inverse_5x5.jl b/src/kernels/inverses/inverse_5x5.jl
new file mode 100644
index 000000000..70de89eb4
--- /dev/null
+++ b/src/kernels/inverses/inverse_5x5.jl
@@ -0,0 +1,61 @@
+@kernel function inv55_kernel!(ˍ₋out, A)
+    k = @index(Global)
+    @inbounds begin
+        ˍ₋out[1, 1, k] = (/)((+)((+)((+)((+)((+)((+)((+)((+)((+)((+)((+)((+)((+)((+)((+)((+)((+)((+)((+)((+)((+)((+)((+)((*)((*)((*)((getindex)(A, 2, 2, k), (getindex)(A, 3, 3, k)), (getindex)(A, 4, 4, k)), (getindex)(A, 5, 5, k)), (*)((*)((*)((*)(-1, (getindex)(A, 2, 2, k)), (getindex)(A, 3, 3, k)), (getindex)(A, 4, 5, k)), (getindex)(A, 5, 4, k))), (*)((*)((*)((*)(-1, (getindex)(A, 2, 2, k)), (getindex)(A, 3, 4, k)), (getindex)(A, 4, 3, k)), (getindex)(A, 5, 5, k))), (*)((*)((*)((getindex)(A, 2, 2, k), (getindex)(A, 3, 4, k)), (getindex)(A, 4, 5, k)), (getindex)(A, 5, 3, k))), (*)((*)((*)((getindex)(A, 2, 2, k), (getindex)(A, 3, 5, k)), (getindex)(A, 4, 3, k)), (getindex)(A, 5, 4, k))), (*)((*)((*)((*)(-1, (getindex)(A, 2, 2, k)), (getindex)(A, 3, 5, k)), (getindex)(A, 4, 4, k)), (getindex)(A, 5, 3, k))), (*)((*)((*)((*)(-1, (getindex)(A, 2, 3, k)), (getindex)(A, 3, 2, k)), (getindex)(A, 4, 4, k)), (getindex)(A, 5, 5, k))), (*)((*)((*)((getindex)(A, 2, 3, k), (getindex)(A, 3, 2, k)), (getindex)(A, 4, 5, k)), (getindex)(A, 5, 4, k))), (*)((*)((*)((getindex)(A, 2, 3, k), (getindex)(A, 3, 4, k)), (getindex)(A, 4, 2, k)), (getindex)(A, 5, 5, k))), (*)((*)((*)((*)(-1, (getindex)(A, 2, 3, k)), (getindex)(A, 3, 4, k)), (getindex)(A, 4, 5, k)), (getindex)(A, 5, 2, k))), (*)((*)((*)((*)(-1, (getindex)(A, 2, 3, k)), (getindex)(A, 3, 5, k)), (getindex)(A, 4, 2, k)), (getindex)(A, 5, 4, k))), (*)((*)((*)((getindex)(A, 2, 3, k), (getindex)(A, 3, 5, k)), (getindex)(A, 4, 4, k)), (getindex)(A, 5, 2, k))), (*)((*)((*)((getindex)(A, 2, 4, k), (getindex)(A, 3, 2, k)), (getindex)(A, 4, 3, k)), (getindex)(A, 5, 5, k))), (*)((*)((*)((*)(-1, (getindex)(A, 2, 4, k)), (getindex)(A, 3, 2, k)), (getindex)(A, 4, 5, k)), (getindex)(A, 5, 3, k))), (*)((*)((*)((*)(-1, (getindex)(A, 2, 4, k)), (getindex)(A, 3, 3, k)), (getindex)(A, 4, 2, k)), (getindex)(A, 5, 5, k))), (*)((*)((*)((getindex)(A, 2, 4, k), (getindex)(A, 3, 3, k)), (getindex)(A, 4, 5, k)), (getindex)(A, 5, 2, k))), (*)((*)((*)((getindex)(A, 2, 4, k), (getindex)(A, 3, 5, k)), (getindex)(A, 4, 2, k)), (getindex)(A, 5, 3, k))), (*)((*)((*)((*)(-1, (getindex)(A, 2, 4, k)), (getindex)(A, 3, 5, k)), (getindex)(A, 4, 3, k)), (getindex)(A, 5, 2, k))), (*)((*)((*)((*)(-1, (getindex)(A, 2, 5, k)), (getindex)(A, 3, 2, k)), (getindex)(A, 4, 3, k)), (getindex)(A, 5, 4, k))), (*)((*)((*)((getindex)(A, 2, 5, k), (getindex)(A, 3, 2, k)), (getindex)(A, 4, 4, k)), (getindex)(A, 5, 3, k))), (*)((*)((*)((getindex)(A, 2, 5, k), (getindex)(A, 3, 3, k)), (getindex)(A, 4, 2, k)), (getindex)(A, 5, 4, k))), (*)((*)((*)((*)(-1, (getindex)(A, 2, 5, k)), (getindex)(A, 3, 3, k)), (getindex)(A, 4, 4, k)), (getindex)(A, 5, 2, k))), (*)((*)((*)((*)(-1, (getindex)(A, 2, 5, k)), (getindex)(A, 3, 4, k)), (getindex)(A, 4, 2, k)), (getindex)(A, 5, 3, k))), (*)((*)((*)((getindex)(A, 2, 5, k), (getindex)(A, 3, 4, k)), (getindex)(A, 4, 3, k)), (getindex)(A, 5, 2, k))), 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(getindex)(A, 5, 1, k))))
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(getindex)(A, 5, 1, k))))
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(getindex)(A, 5, 1, k))))
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(getindex)(A, 3, 2, k)), (getindex)(A, 4, 1, k)), (getindex)(A, 5, 3, k))), (*)((*)((*)((*)((getindex)(A, 1, 4, k), (getindex)(A, 2, 5, k)), (getindex)(A, 3, 2, k)), (getindex)(A, 4, 3, k)), (getindex)(A, 5, 1, k))), (*)((*)((*)((*)((getindex)(A, 1, 4, k), (getindex)(A, 2, 5, k)), (getindex)(A, 3, 3, k)), (getindex)(A, 4, 1, k)), (getindex)(A, 5, 2, k))), (*)((*)((*)((*)((*)(-1, (getindex)(A, 1, 4, k)), (getindex)(A, 2, 5, k)), (getindex)(A, 3, 3, k)), (getindex)(A, 4, 2, k)), (getindex)(A, 5, 1, k))), (*)((*)((*)((*)((getindex)(A, 1, 5, k), (getindex)(A, 2, 1, k)), (getindex)(A, 3, 2, k)), (getindex)(A, 4, 3, k)), (getindex)(A, 5, 4, k))), (*)((*)((*)((*)((*)(-1, (getindex)(A, 1, 5, k)), (getindex)(A, 2, 1, k)), (getindex)(A, 3, 2, k)), (getindex)(A, 4, 4, k)), (getindex)(A, 5, 3, k))), (*)((*)((*)((*)((*)(-1, (getindex)(A, 1, 5, k)), (getindex)(A, 2, 1, k)), (getindex)(A, 3, 3, k)), (getindex)(A, 4, 2, k)), (getindex)(A, 5, 4, k))), (*)((*)((*)((*)((getindex)(A, 1, 5, k), (getindex)(A, 2, 1, k)), (getindex)(A, 3, 3, k)), (getindex)(A, 4, 4, k)), (getindex)(A, 5, 2, k))), (*)((*)((*)((*)((getindex)(A, 1, 5, k), (getindex)(A, 2, 1, k)), (getindex)(A, 3, 4, k)), (getindex)(A, 4, 2, k)), (getindex)(A, 5, 3, k))), (*)((*)((*)((*)((*)(-1, (getindex)(A, 1, 5, k)), (getindex)(A, 2, 1, k)), (getindex)(A, 3, 4, k)), (getindex)(A, 4, 3, k)), (getindex)(A, 5, 2, k))), (*)((*)((*)((*)((*)(-1, (getindex)(A, 1, 5, k)), (getindex)(A, 2, 2, k)), (getindex)(A, 3, 1, k)), (getindex)(A, 4, 3, k)), (getindex)(A, 5, 4, k))), (*)((*)((*)((*)((getindex)(A, 1, 5, k), (getindex)(A, 2, 2, k)), (getindex)(A, 3, 1, k)), (getindex)(A, 4, 4, k)), (getindex)(A, 5, 3, k))), (*)((*)((*)((*)((getindex)(A, 1, 5, k), (getindex)(A, 2, 2, k)), (getindex)(A, 3, 3, k)), (getindex)(A, 4, 1, k)), (getindex)(A, 5, 4, k))), (*)((*)((*)((*)((*)(-1, (getindex)(A, 1, 5, k)), (getindex)(A, 2, 2, k)), (getindex)(A, 3, 3, k)), (getindex)(A, 4, 4, k)), (getindex)(A, 5, 1, k))), (*)((*)((*)((*)((*)(-1, (getindex)(A, 1, 5, k)), (getindex)(A, 2, 2, k)), (getindex)(A, 3, 4, k)), (getindex)(A, 4, 1, k)), (getindex)(A, 5, 3, k))), (*)((*)((*)((*)((getindex)(A, 1, 5, k), (getindex)(A, 2, 2, k)), (getindex)(A, 3, 4, k)), (getindex)(A, 4, 3, k)), (getindex)(A, 5, 1, k))), (*)((*)((*)((*)((getindex)(A, 1, 5, k), (getindex)(A, 2, 3, k)), (getindex)(A, 3, 1, k)), (getindex)(A, 4, 2, k)), (getindex)(A, 5, 4, k))), (*)((*)((*)((*)((*)(-1, (getindex)(A, 1, 5, k)), (getindex)(A, 2, 3, k)), (getindex)(A, 3, 1, k)), (getindex)(A, 4, 4, k)), (getindex)(A, 5, 2, k))), (*)((*)((*)((*)((*)(-1, (getindex)(A, 1, 5, k)), (getindex)(A, 2, 3, k)), (getindex)(A, 3, 2, k)), (getindex)(A, 4, 1, k)), (getindex)(A, 5, 4, k))), (*)((*)((*)((*)((getindex)(A, 1, 5, k), (getindex)(A, 2, 3, k)), (getindex)(A, 3, 2, k)), (getindex)(A, 4, 4, k)), (getindex)(A, 5, 1, k))), (*)((*)((*)((*)((getindex)(A, 1, 5, k), (getindex)(A, 2, 3, k)), (getindex)(A, 3, 4, k)), (getindex)(A, 4, 1, k)), (getindex)(A, 5, 2, k))), (*)((*)((*)((*)((*)(-1, (getindex)(A, 1, 5, k)), (getindex)(A, 2, 3, k)), (getindex)(A, 3, 4, k)), (getindex)(A, 4, 2, k)), (getindex)(A, 5, 1, k))), (*)((*)((*)((*)((*)(-1, (getindex)(A, 1, 5, k)), (getindex)(A, 2, 4, k)), (getindex)(A, 3, 1, k)), (getindex)(A, 4, 2, k)), (getindex)(A, 5, 3, k))), (*)((*)((*)((*)((getindex)(A, 1, 5, k), (getindex)(A, 2, 4, k)), (getindex)(A, 3, 1, k)), (getindex)(A, 4, 3, k)), (getindex)(A, 5, 2, k))), (*)((*)((*)((*)((getindex)(A, 1, 5, k), (getindex)(A, 2, 4, k)), (getindex)(A, 3, 2, k)), (getindex)(A, 4, 1, k)), (getindex)(A, 5, 3, k))), (*)((*)((*)((*)((*)(-1, (getindex)(A, 1, 5, k)), (getindex)(A, 2, 4, k)), (getindex)(A, 3, 2, k)), (getindex)(A, 4, 3, k)), (getindex)(A, 5, 1, k))), (*)((*)((*)((*)((*)(-1, (getindex)(A, 1, 5, k)), (getindex)(A, 2, 4, k)), (getindex)(A, 3, 3, k)), (getindex)(A, 4, 1, k)), (getindex)(A, 5, 2, k))), (*)((*)((*)((*)((getindex)(A, 1, 5, k), (getindex)(A, 2, 4, k)), (getindex)(A, 3, 3, k)), (getindex)(A, 4, 2, k)), (getindex)(A, 5, 1, k))))
+        nothing
+    end
+end
+
+function tensor_inverse5(A::AbstractArray{T, 3}) where T 
+    out = similar(A)
+
+    tensor_inverse5!(out, A)
+    
+    out 
+end
+
+function tensor_inverse5!(out::AbstractArray{T, 3}, A::AbstractArray{T, 3}) where T 
+    @assert size(A, 1) == size(A, 2) == 5
+    @assert size(A) == size(out)
+
+    backend = get_backend(out)
+    inv55! = inv55_kernel!(backend)
+
+    inv55!(out, A, ndrange = size(A, 3))
+
+    nothing
+end
+
+function ChainRulesCore.rrule(::typeof(tensor_inverse5), A::AT) where {T, AT<:AbstractArray{T, 3}}
+    out = tensor_inverse5(A)
+    
+    function tensor_inverse_pullback(out_diff::AT)
+
+        NoTangent(), - tensor_transpose_tensor_mul(out, tensor_tensor_mul(out_diff, tensor_transpose(out)))
+    end 
+    out, tensor_inverse_pullback 
+end 

From feaf2e7b50b427a1d10683392cf2e60e0c81b3db Mon Sep 17 00:00:00 2001
From: benedict-96 <benedikt.brantner@icloud.com>
Date: Mon, 8 Apr 2024 12:16:17 +0200
Subject: [PATCH 12/33] Added cpu version for parallel computation of inverses.

---
 src/kernels/inverses/cpu_inverse.jl | 52 +++++++++++++++++++++++++++++
 1 file changed, 52 insertions(+)
 create mode 100644 src/kernels/inverses/cpu_inverse.jl

diff --git a/src/kernels/inverses/cpu_inverse.jl b/src/kernels/inverses/cpu_inverse.jl
new file mode 100644
index 000000000..128ede5b8
--- /dev/null
+++ b/src/kernels/inverses/cpu_inverse.jl
@@ -0,0 +1,52 @@
+@kernel function cpu_inverse_kernel!(B, A) 
+    k = @index(Global)
+    @views A_temp = A[:, :, k]
+    @views B_temp = B[:, :, k]
+
+    B_temp .= inv(A_temp)
+
+    nothing
+end
+
+function cpu_inverse(A::AbstractArray)
+    B = zero(A)
+    backend = KernelAbstractions.get_backend(A)
+
+    cpu_inverse! = cpu_inverse_kernel!(backend)
+    cpu_inverse!(B, A, ndrange=size(A, 3))
+
+    B 
+end
+
+@kernel function cpu_inverse_pullback_kernel!(dA, A, dB)
+    k = @index(Global)
+    @views A_temp = A[:, :, k]
+    @views dA_temp = dA[:, :, k]
+    @views dB_temp = dB[:, :, k]
+
+    copy!(dA_temp, Zygote.pullback(inv, A_temp)[2](dB_temp)[1])
+
+    nothing
+end
+
+function ChainRulesCore.rrule(::typeof(cpu_inverse), A::AbstractArray)
+    B = cpu_inverse(A)
+
+    function cpu_inverse_pullback(dB::AbstractArray)
+        dA = zero(dB)
+        backend = KernelAbstractions.get_backend(dB)
+
+        cpu_inverse_pullback! = cpu_inverse_pullback_kernel!(backend)
+        cpu_inverse_pullback!(dA, A, dB, ndrange=size(dB, 3))
+
+        return NoTangent(), dA
+    end
+
+    return B, cpu_inverse_pullback
+end
+
+function cpu_tensor_cayley(A::AbstractArray)
+    one_A = init_output(A)
+
+    tensor_tensor_mul(one_A - A, cpu_inverse(one_A + A))
+end
\ No newline at end of file

From cabaa9908692d3e0b749a694af605296b2164796 Mon Sep 17 00:00:00 2001
From: benedict-96 <benedikt.brantner@icloud.com>
Date: Mon, 8 Apr 2024 12:17:24 +0200
Subject: [PATCH 13/33] Test is not making sense any more.
 VolumePreservingAttention has diverged too far from MultiHeadAttention.

---
 .../apply_multi_head_attention.jl             | 26 -------------------
 1 file changed, 26 deletions(-)
 delete mode 100644 test/attention_layer/apply_multi_head_attention.jl

diff --git a/test/attention_layer/apply_multi_head_attention.jl b/test/attention_layer/apply_multi_head_attention.jl
deleted file mode 100644
index 967411b48..000000000
--- a/test/attention_layer/apply_multi_head_attention.jl
+++ /dev/null
@@ -1,26 +0,0 @@
-using GeometricMachineLearning, Test 
-using NNlib: softmax
-using LinearAlgebra: I
-import Random 
-
-Random.seed!(1234)
-
-function compare_attention_to_mha(N, batch_size=10, T=Float32)
-    model₁ = MultiHeadAttention(N, 1, add_connection=false)
-    model₂ = Attention(N, softmax, add_connection=false)
-    model₃ = MultiHeadAttention(N, 1, add_connection=true)
-    model₄ = Attention(N, softmax, add_connection=true)
-
-    ps₂ = initialparameters(CPU(), T, model₂)
-    ps₁ = (PQ=(head_1=ps₂.PQ,), PK=(head_1=ps₂.PK,), PV=(head_1=typeof(ps₂.PK)(I(N)),))
-
-    mat = rand(T, N, N)
-    ten = rand(T, N, N, batch_size)
-    @test isapprox(model₁(mat, ps₁), model₂(mat, ps₂))
-    @test isapprox(model₁(ten, ps₁), model₂(ten, ps₂))
-
-    @test isapprox(model₃(mat, ps₁), model₄(mat, ps₂))
-    @test isapprox(model₃(ten, ps₁), model₄(ten, ps₂))
-end
-
-compare_attention_to_mha(10)
\ No newline at end of file

From da04c41a81fc59ec1bda245845b0c180d625eb7e Mon Sep 17 00:00:00 2001
From: benedict-96 <benedikt.brantner@icloud.com>
Date: Mon, 8 Apr 2024 12:21:13 +0200
Subject: [PATCH 14/33] Performing Cayley transform in parallel based on the
 the parallel inverses.

---
 src/kernels/inverses/tensor_cayley.jl | 24 ++++++++++++++++++++++++
 1 file changed, 24 insertions(+)
 create mode 100644 src/kernels/inverses/tensor_cayley.jl

diff --git a/src/kernels/inverses/tensor_cayley.jl b/src/kernels/inverses/tensor_cayley.jl
new file mode 100644
index 000000000..ab8ce1d4c
--- /dev/null
+++ b/src/kernels/inverses/tensor_cayley.jl
@@ -0,0 +1,24 @@
+function cpu_tensor_cayley(A::AbstractArray{T, 3}) where T 
+    one_tensor = init_output(A)
+    tensor_tensor_mul(one_tensor - A, cpu_inverse(one_tensor + A))
+end
+
+function tensor_cayley2(A::AbstractArray{T, 3}) where T 
+    one_tensor = init_output(A)
+    tensor_tensor_mul(one_tensor - A, tensor_inverse2(one_tensor + A))
+end
+
+function tensor_cayley3(A::AbstractArray{T, 3}) where T 
+    one_tensor = init_output(A)
+    tensor_tensor_mul(one_tensor - A, tensor_inverse3(one_tensor + A))
+end
+
+function tensor_cayley4(A::AbstractArray{T, 3}) where T 
+    one_tensor = init_output(A)
+    tensor_tensor_mul(one_tensor - A, tensor_inverse4(one_tensor + A))
+end
+
+function tensor_cayley5(A::AbstractArray{T, 3}) where T 
+    one_tensor = init_output(A)
+    tensor_tensor_mul(one_tensor - A, tensor_inverse5(one_tensor + A))
+end
\ No newline at end of file

From f4159f3cb26b1017cf0804f0c57f7364953eda4c Mon Sep 17 00:00:00 2001
From: benedict-96 <benedikt.brantner@icloud.com>
Date: Mon, 8 Apr 2024 12:22:04 +0200
Subject: [PATCH 15/33] Updated some of the functions associated with
 SkewSymMatrix.

---
 src/arrays/skew_symmetric.jl | 50 ++++++++++++++++++++++++------------
 1 file changed, 34 insertions(+), 16 deletions(-)

diff --git a/src/arrays/skew_symmetric.jl b/src/arrays/skew_symmetric.jl
index 058345ab3..118daa6a1 100644
--- a/src/arrays/skew_symmetric.jl
+++ b/src/arrays/skew_symmetric.jl
@@ -8,7 +8,6 @@ The first index is the row index, the second one the column index.
 
 The struct two fields: `S` and `n`. The first stores all the entries of the matrix in a sparse fashion (in a vector) and the second is the dimension ``n`` for ``A\in\mathbb{R}^{n\times{}n}``.
 """
-
 mutable struct SkewSymMatrix{T, AT <: AbstractVector{T}} <: AbstractMatrix{T}
     S::AT
     n::Int
@@ -20,17 +19,8 @@ mutable struct SkewSymMatrix{T, AT <: AbstractVector{T}} <: AbstractMatrix{T}
     function SkewSymMatrix(S::AbstractMatrix{T}) where {T}
         n = size(S, 1)
         @assert size(S, 2) == n
-        S_vec = zeros(T, n*(n-1)÷2)
-        # make the input skew-symmetric if it isn't already
-        S = T(.5)*(S - S')
-        # this is disgusting and should be removed! Here because indexing for GPUs not supported.
-        S_cpu = Matrix{T}(S)
-        # map the sub-diagonal elements to a vector 
-        for i in 2:n
-            S_vec[((i-1)*(i-2)÷2+1):(i*(i-1)÷2)] = S_cpu[i,1:(i-1)]
-        end
-        S_vec₂ = Base.typename(typeof(S)).wrapper{eltype(S), 1}(S_vec)
-        new{T,typeof(S_vec₂)}(S_vec₂, n)
+        S_vec = map_to_Skew(S)
+        new{T,typeof(S_vec)}(S_vec, n)
     end
 end 
 
@@ -39,9 +29,9 @@ function Base.getindex(A::SkewSymMatrix, i::Int, j::Int)
         return zero(eltype(A))
     end
     if i > j
-        return A.S[(i-2)*(i-1)÷2+j]
+        return A.S[(i-2) * (i-1) ÷ 2 + j]
     end
-    return - A.S[(j-2)*(j-1)÷2+i]
+    return - A.S[ (j-2) * (j-1) ÷ 2 + i] 
 end
 
 
@@ -164,7 +154,7 @@ function Base.:*(B::AbstractMatrix{T}, A::SkewSymMatrix{T}) where T
 end
 
 function Base.:*(A::SkewSymMatrix, b::AbstractVector{T}) where T
-    A*reshape(b, size(b), 1)
+    A*reshape(b, length(b), 1)
 end
 
 function Base.one(A::SkewSymMatrix{T}) where T
@@ -202,4 +192,32 @@ end
 
 function Base.copy(A::SkewSymMatrix)
     SkewSymMatrix(copy(A.S), A.n)
-end
\ No newline at end of file
+end
+
+@kernel function assign_Skew_val_kernel!(S, A_skew, i)
+    j = @index(Global)
+    S[((i - 2) * (i - 1) ÷ 2 + j)] = A_skew[i, j]
+end
+
+function map_to_Skew(A::AbstractMatrix{T}) where T
+    n = size(A, 1)
+    @assert size(A, 2) == n 
+    A_skew = T(.5)*(A - A')
+    backend = KernelAbstractions.get_backend(A)
+    S = KernelAbstractions.zeros(backend, T, n * (n - 1) ÷ 2)
+    assign_Skew_val! = assign_Skew_val_kernel!(backend)
+    for i in 2:n
+        assign_Skew_val!(S, A_skew, i, ndrange = (i - 1))
+    end
+    S
+end
+
+function Base.copyto!(A::SkewSymMatrix, B::SkewSymMatrix)
+    A.S .= B.S
+    nothing
+end
+
+# define routines for generalizing ChainRulesCore to SkewSymMatrix 
+ChainRulesCore.ProjectTo(A::SkewSymMatrix) = ProjectTo{SkewSymMatrix}(; skew_sym = ProjectTo(A.S))
+(project::ProjectTo{SkewSymMatrix})(dA::AbstractMatrix) = SkewSymMatrix(project.skew_sym(map_to_Skew(dA)), size(dA, 2))
+(project::ProjectTo{SkewSymMatrix})(dA::SkewSymMatrix) = SkewSymMatrix(project.skew_sym(dA.S), dA.n)
\ No newline at end of file

From 904239309881a50a6eea4a3f6327e7c89619f528 Mon Sep 17 00:00:00 2001
From: benedict-96 <benedikt.brantner@icloud.com>
Date: Mon, 8 Apr 2024 12:22:20 +0200
Subject: [PATCH 16/33] Updated docs.:

---
 src/arrays/symmetric.jl | 8 --------
 1 file changed, 8 deletions(-)

diff --git a/src/arrays/symmetric.jl b/src/arrays/symmetric.jl
index 0fcd7b506..688aed306 100644
--- a/src/arrays/symmetric.jl
+++ b/src/arrays/symmetric.jl
@@ -14,14 +14,6 @@ Internally the `struct` saves a vector $S$ of size $n(n+1)\div2$. The conversion
 ```
 
 So ``S`` stores a string of vectors taken from $A$: $S = [\tilde{a}_1, \tilde{a}_2, \ldots, \tilde{a}_n]$ with $\tilde{a}_i = [[A]_{i1},[A]_{i2},\ldots,[A]_{ii}]$.
-
-TODO: 
-- [x] Overload Adjoint operation for SymmetricMatrix!! (Aᵀ = A)
-- [ ] implement matrix and vector products (to also work on GPU)
-- [x] implement zero initialization (for optimizer)
-- [ ] perform some tests (also with Zygote)
-- [x] update the constructor (to work better for GPU)
-- [ ] implement multiplication with a tensor
 """
 mutable struct SymmetricMatrix{T, AT <: AbstractVector{T}} <: AbstractMatrix{T}
     S::AT

From be7dc9bafa277763c30514aa4553a10605caaca0 Mon Sep 17 00:00:00 2001
From: benedict-96 <benedikt.brantner@icloud.com>
Date: Mon, 8 Apr 2024 12:23:11 +0200
Subject: [PATCH 17/33] Various tests for parallel inverses and Cayley
 transforms.

---
 test/kernels/tensor_cayley.jl  |  47 +++++++++++++
 test/kernels/tensor_inverse.jl | 117 +++++++++++++++++++++++++++++++++
 2 files changed, 164 insertions(+)
 create mode 100644 test/kernels/tensor_cayley.jl
 create mode 100644 test/kernels/tensor_inverse.jl

diff --git a/test/kernels/tensor_cayley.jl b/test/kernels/tensor_cayley.jl
new file mode 100644
index 000000000..79f37f5ca
--- /dev/null
+++ b/test/kernels/tensor_cayley.jl
@@ -0,0 +1,47 @@
+using GeometricMachineLearning: tensor_cayley2, tensor_cayley3, tensor_cayley4, tensor_cayley5
+using GeometricMachineLearning: tensor_transpose
+using Test
+import Random 
+
+Random.seed!(123)
+
+function check_orthonormal_property(B::Array)
+    for i in axes(B, 3)
+        B_temp = @view B[:, :, i]
+        @test B_temp * B_temp' ≈ one(B_temp)
+    end
+end
+
+function test_tensor_cayley2(T::Type=Float32, third_dim::Int=10)
+    A = rand(2, 2, third_dim)
+    B = tensor_cayley2(A - tensor_transpose(A))
+    check_orthonormal_property(B)
+end
+
+function test_tensor_cayley3(T::Type=Float32, third_dim::Int=10)
+    A = rand(3, 3, third_dim)
+    B = tensor_cayley3(A - tensor_transpose(A))
+    check_orthonormal_property(B)
+end
+
+function test_tensor_cayley4(T::Type=Float32, third_dim::Int=10)
+    A = rand(4, 4, third_dim)
+    B = tensor_cayley4(A - tensor_transpose(A))
+    check_orthonormal_property(B)
+end
+
+function test_tensor_cayley5(T::Type=Float32, third_dim::Int=10)
+    A = rand(5, 5, third_dim)
+    B = tensor_cayley5(A - tensor_transpose(A))
+    check_orthonormal_property(B)
+end
+
+function check_all(T::Type, third_dim::Int=10)
+    test_tensor_cayley2(T, third_dim)
+    test_tensor_cayley3(T, third_dim)
+    test_tensor_cayley4(T, third_dim)
+    test_tensor_cayley5(T, third_dim)
+end
+
+check_all(Float32)
+check_all(Float64)
\ No newline at end of file
diff --git a/test/kernels/tensor_inverse.jl b/test/kernels/tensor_inverse.jl
new file mode 100644
index 000000000..252da0b4d
--- /dev/null
+++ b/test/kernels/tensor_inverse.jl
@@ -0,0 +1,117 @@
+using GeometricMachineLearning: tensor_inverse2, tensor_inverse3, tensor_inverse4, tensor_inverse5
+using Test
+import Zygote
+
+function test55_inverse(k::Int = 10)
+    A = rand(5, 5, k)
+
+    A_inv = tensor_inverse5(A)
+    for i = 1:k
+        @test inv(A[:, :, i]) ≈ A_inv[:, :, i]
+    end
+end
+
+test55_inverse()
+
+function test55_inverse_pullback(k::Int = 10)
+    A = rand(5, 5, k)
+
+    pullback_total = Zygote.pullback(tensor_inverse5, A)
+
+    out_diff = rand(5, 5, k)
+
+    for i = 1:k 
+        pullback_k = Zygote.pullback(inv, A[:, :, k])
+        
+        @test pullback_total[1][:, :, k] ≈ pullback_k[1]
+        @test pullback_total[2](out_diff)[1][:, :, k] ≈ pullback_k[2](out_diff[:, :, k])[1]
+    end
+end
+
+test55_inverse_pullback()
+
+function test44_inverse(k::Int = 10)
+    A = rand(4, 4, k)
+
+    A_inv = tensor_inverse4(A)
+    for i = 1:k
+        @test inv(A[:, :, i]) ≈ A_inv[:, :, i]
+    end
+end
+
+test44_inverse()
+
+function test44_inverse_pullback(k::Int = 10)
+    A = rand(4, 4, k)
+
+    pullback_total = Zygote.pullback(tensor_inverse4, A)
+
+    out_diff = rand(4, 4, k)
+
+    for i = 1:k 
+        pullback_k = Zygote.pullback(inv, A[:, :, k])
+        
+        @test pullback_total[1][:, :, k] ≈ pullback_k[1]
+        @test pullback_total[2](out_diff)[1][:, :, k] ≈ pullback_k[2](out_diff[:, :, k])[1]
+    end
+end
+
+test44_inverse_pullback()
+
+function test33_inverse(k::Int = 10)
+    A = rand(3, 3, k)
+
+    A_inv = tensor_inverse3(A)
+    for i = 1:k
+        @test inv(A[:, :, i]) ≈ A_inv[:, :, i]
+    end
+end
+
+test33_inverse()
+
+
+function test33_inverse_pullback(k::Int = 10)
+    A = rand(3, 3, k)
+
+    pullback_total = Zygote.pullback(tensor_inverse3, A)
+
+    out_diff = rand(3, 3, k)
+
+    for i = 1:k 
+        pullback_k = Zygote.pullback(inv, A[:, :, k])
+        
+        @test pullback_total[1][:, :, k] ≈ pullback_k[1]
+        @test pullback_total[2](out_diff)[1][:, :, k] ≈ pullback_k[2](out_diff[:, :, k])[1]
+    end
+end
+
+test33_inverse_pullback()
+
+function test22_inverse(k::Int = 10)
+    A = rand(2, 2, k)
+
+    A_inv = tensor_inverse2(A)
+    for i = 1:k
+        @test inv(A[:, :, i]) ≈ A_inv[:, :, i]
+    end
+end
+
+test22_inverse()
+
+
+function test22_inverse_pullback(k::Int = 10)
+    A = rand(2, 2, k)
+
+    pullback_total = Zygote.pullback(tensor_inverse2, A)
+
+    out_diff = rand(2, 2, k)
+
+    for i = 1:k 
+        pullback_k = Zygote.pullback(inv, A[:, :, k])
+        
+        @test pullback_total[1][:, :, k] ≈ pullback_k[1]
+        @test pullback_total[2](out_diff)[1][:, :, k] ≈ pullback_k[2](out_diff[:, :, k])[1]
+    end
+end
+
+test22_inverse_pullback()
\ No newline at end of file

From 45702ad2a8329d986d88aba0febb3b3d8f2e6b18 Mon Sep 17 00:00:00 2001
From: benedict-96 <benedikt.brantner@icloud.com>
Date: Mon, 8 Apr 2024 12:24:13 +0200
Subject: [PATCH 18/33] Tensor_mat_skew_sym assign is needed for the attention
 mechanism where we compute correlations with regular weighting.

---
 .../inverses/tensor_mat_skew_sym_assign.jl    | 51 +++++++++++++++++
 .../tensor_mat_skew_sym_assign.jl             | 55 +++++++++++++++++++
 2 files changed, 106 insertions(+)
 create mode 100644 src/kernels/inverses/tensor_mat_skew_sym_assign.jl
 create mode 100644 src/kernels/kernel_ad_routines/tensor_mat_skew_sym_assign.jl

diff --git a/src/kernels/inverses/tensor_mat_skew_sym_assign.jl b/src/kernels/inverses/tensor_mat_skew_sym_assign.jl
new file mode 100644
index 000000000..b86c0cc78
--- /dev/null
+++ b/src/kernels/inverses/tensor_mat_skew_sym_assign.jl
@@ -0,0 +1,51 @@
+@doc raw"""
+A kernel that computes the weighted scalar products of all combinations of vectors in the matrix `Z` except where the two vectors are the same and writes the result into a *tensor of skew symmetric matrices* `C`. 
+"""
+@kernel function tensor_mat_skew_sym_assign_kernel!(C::AbstractArray{T, 3}, Z::AbstractArray{T, 3}, A::AbstractMatrix{T}) where T
+    i, j, k = @index(Global, NTuple)
+
+    temp = zero(T)
+
+    if i > j
+        for l in axes(Z, 1)
+            for m in axes(Z, 1)
+                temp += Z[l, i, k] * A[l, m] * Z[m, j, k]
+            end
+        end
+    end
+
+    C[i, j, k] = temp
+
+    nothing
+end
+
+function tensor_mat_skew_sym_assign!(C::AbstractArray{T, 3}, Z::AbstractArray{T, 3}, A::AbstractMatrix{T}) where {T}
+    backend = KernelAbstractions.get_backend(Z)
+
+    tensor_mat_skew_sym_assign_k! = tensor_mat_skew_sym_assign_kernel!(backend)
+
+    tensor_mat_skew_sym_assign_k!(C, Z, A, ndrange=size(C))
+end
+
+@doc raw"""
+Takes as input: 
+- `Z::AbstractArray{T, 3}`: A tensor that stores a bunch of time series. 
+- `A::AbstractMatrix`: A matrix that is used to perform various scalar products. 
+
+For one of these time series the function performs the following computation: 
+
+```math
+    (z^{(i)}, z^{(j)}) \mapsto (z^{(i)})^TAz^{(j)} \text{ for } i > j.
+```
+
+The result of this are ``n(n-2)\div2`` scalar products. These scalar products are written into a lower-triangular matrix and the final output of the function is a tensor of these lower-triangular matrices. 
+"""
+function tensor_mat_skew_sym_assign(Z::AT, A::AbstractMatrix{T}) where {T, AT <: AbstractArray{T, 3}}
+    backend = KernelAbstractions.get_backend(Z)
+
+    C = KernelAbstractions.zeros(backend, T, size(Z, 2), size(Z, 2), size(Z, 3))
+
+    tensor_mat_skew_sym_assign!(C, Z, A)
+
+    C
+end
\ No newline at end of file
diff --git a/src/kernels/kernel_ad_routines/tensor_mat_skew_sym_assign.jl b/src/kernels/kernel_ad_routines/tensor_mat_skew_sym_assign.jl
new file mode 100644
index 000000000..e24541ad0
--- /dev/null
+++ b/src/kernels/kernel_ad_routines/tensor_mat_skew_sym_assign.jl
@@ -0,0 +1,55 @@
+@kernel function dz_kernel!(dZ::AT, Z::AT, A::AbstractMatrix{T}, dB::AT, sys_dim::Int, seq_length::Int) where {T, AT <: AbstractArray{T, 3}}
+    k, n, h = @index(Global, NTuple)
+
+    temp = zero(T)
+    for m = 1:sys_dim
+        for j = 1:(n - 1)
+            temp += A[k, m] * Z[m, j, h] * dB[n, j, h]
+        end
+        for j = (n + 1):seq_length
+            temp += A[m, k] * Z[m, j, h] * dB[j, n, h]
+        end
+    end
+
+    dZ[k, n, h] = temp
+
+    nothing
+end
+
+@kernel function da_kernel!(dA::AT, Z::AT, ::BT, dB::AT, ::Int, seq_length::Int) where {T, AT <: AbstractArray{T, 3}, BT <: AbstractMatrix{T}}
+    m, n, h = @index(Global, NTuple)
+
+    temp = zero(T)
+    for j = 1:seq_length
+        for i = (j+1):seq_length
+            temp += Z[m, i, h] * Z[n, j, h] * dB[i, j, h]
+        end
+    end
+
+    dA[m, n, h] = temp
+
+    nothing
+end
+
+function ChainRulesCore.rrule(::typeof(tensor_mat_skew_sym_assign), Z::AbstractArray{T, 3}, A::AbstractArray{T, 2}) where T
+    @assert size(A, 1) == size(Z, 1) 
+    B = tensor_mat_skew_sym_assign(Z, A)
+    function tensor_mat_skew_sym_assign_pullback(dB::AbstractArray{T, 3})
+        f̄ = NoTangent()
+        backend = KernelAbstractions.get_backend(dB)
+        dz! = dz_kernel!(backend)
+        da! = da_kernel!(backend)
+
+        dZ = zero(Z)
+        dA = KernelAbstractions.zeros(backend, T, size(A)..., size(dB, 3))
+        
+        sys_dim, seq_length, _ = size(Z)
+        
+        dz!(dZ, Z, A, dB, sys_dim, seq_length, ndrange = size(dZ))
+        da!(dA, Z, A, dB, sys_dim, seq_length, ndrange = size(dA))
+
+        return f̄, dZ, reshape(sum(dA, dims = 3), size(A)) 
+    end
+
+    return B, tensor_mat_skew_sym_assign_pullback
+end
\ No newline at end of file

From ebbe058d217adc58e8a5a61600b889e3e7d0e8b8 Mon Sep 17 00:00:00 2001
From: benedict-96 <benedikt.brantner@icloud.com>
Date: Mon, 8 Apr 2024 12:24:46 +0200
Subject: [PATCH 19/33] Attention -> VolumePreservingAttention.

---
 src/layers/volume_preserving_attention.jl | 97 +++++++++++++++++++++++
 1 file changed, 97 insertions(+)
 create mode 100644 src/layers/volume_preserving_attention.jl

diff --git a/src/layers/volume_preserving_attention.jl b/src/layers/volume_preserving_attention.jl
new file mode 100644
index 000000000..dfd6ccb35
--- /dev/null
+++ b/src/layers/volume_preserving_attention.jl
@@ -0,0 +1,97 @@
+@doc raw"""
+# Volume-preserving attention (single head attention)
+
+Drawbacks: 
+- the super fast activation is only implemented for sequence lengths of 2, 3, 4 and 5.
+- other sequence lengths only work on CPU for now (lu decomposition has to be implemented to work for tensors in parallel).
+
+## Constructor 
+
+The constructor is called with: 
+- `dim::Int`: The system dimension 
+- `seq_length::Int`: The sequence length to be considered. The default is zero, i.e. arbitrary sequence lengths; this works for all sequence lengths but doesn't apply the super-fast activation. 
+- `skew_sym::Bool` (keyword argument): specifies if we the weight matrix is skew symmetric or arbitrary (default is false).
+
+## Functor 
+Applying a layer of type `VolumePreservingAttention` does the following: 
+- First we perform the operation ``X \mapsto X^T A X =: C``, where ``X\in\mathbb{R}^{N\times\mathtt{seq\_length}}`` is a vector containing time series data and ``A`` is the skew symmetric matrix associated with the layer. 
+- In a second step we compute the Cayley transform of ``C``; ``\Lambda = \mathrm{Cayley}(C)``.
+- The output of the layer is then ``X\Lambda``.
+"""
+struct VolumePreservingAttention{M, N, ScalarProductType, SL} <: AbstractExplicitLayer{M, N}
+
+    function VolumePreservingAttention(dim::Int, seq_length::Int = 0; skew_sym::Bool=false)
+        if skew_sym
+            return new{dim, dim, :skew_sym, seq_length}()
+        else
+            return new{dim, dim, :arbitrary, seq_length}()
+        end
+    end
+end
+
+function orthonormal_activation_cayley(::VolumePreservingAttention, A::AbstractArray{T, 3}) where T 
+    cpu_tensor_cayley(A)
+end
+
+function orthonormal_activation_cayley(::VolumePreservingAttention{M, M, ScalarProductType, 2}, A::AbstractArray{T, 3}) where {T, M, ScalarProductType} 
+    tensor_cayley2(A)
+end
+
+function orthonormal_activation_cayley(::VolumePreservingAttention{M, M, ScalarProductType, 3}, A::AbstractArray{T, 3}) where {T, M, ScalarProductType} 
+    tensor_cayley3(A)
+end
+
+function orthonormal_activation_cayley(::VolumePreservingAttention{M, M, ScalarProductType, 4}, A::AbstractArray{T, 3}) where {T, M, ScalarProductType} 
+    tensor_cayley4(A)
+end
+
+function orthonormal_activation_cayley(::VolumePreservingAttention{M, M, ScalarProductType, 5}, A::AbstractArray{T, 3}) where {T, M, ScalarProductType} 
+    tensor_cayley5(A)
+end
+
+# function orthonormal_activation_cayley(::VolumePreservingAttention{M, M, 6}, A::AbstractArray{T, 3}) where {T, M} 
+#     tensor_cayley6(A)
+# end
+
+# function orthonormal_activation_cayley(A::AbstractMatrix{T}) where T 
+#     reshape(orthonormal_activation_cayley(reshape(A, size(A)..., 1)), size(A)...)
+# end
+
+function parameterlength(::VolumePreservingAttention{M, M, :skew_sym}) where {M}
+    M * (M-1) ÷ 2
+end
+
+function parameterlength(::VolumePreservingAttention{M, M, :arbitrary}) where {M}
+    M ^2
+end
+
+function initialparameters(backend::KernelAbstractions.Backend, T::Type, d::VolumePreservingAttention{M, M, :skew_sym}; rng::AbstractRNG=Random.default_rng(), initializer!::AbstractNeuralNetworks.AbstractInitializer=GlorotUniform()) where {M}
+    V = KernelAbstractions.allocate(backend, T, parameterlength(d))
+    initializer!(rng, V)
+    (A = SkewSymMatrix(V, M), )
+end
+
+function initialparameters(backend::KernelAbstractions.Backend, T::Type, ::VolumePreservingAttention{M, M, :arbitrary}; rng::AbstractRNG=Random.default_rng(), initializer!::AbstractNeuralNetworks.AbstractInitializer=GlorotUniform()) where {M}
+    A = KernelAbstractions.allocate(backend, T, M, M)
+    initializer!(rng, A)
+    (A = A, )
+end
+
+function (d::VolumePreservingAttention{M, M, :skew_sym})(x::AbstractArray{T, 3}, ps::NamedTuple) where {T, M}
+    tensor_tensor_mul(  x, 
+                        orthonormal_activation_cayley(d, 
+                            tensor_transpose_tensor_mul(x, mat_tensor_mul(ps.A, x))
+                        )
+    )
+end
+
+function (d::VolumePreservingAttention{M, M, :arbitrary})(x::AbstractArray{T, 3}, ps::NamedTuple) where {T, M}
+    x_interim = tensor_mat_skew_sym_assign(x, ps.A) / T(√M)
+    tensor_tensor_mul(  x, 
+                        orthonormal_activation_cayley(d, x_interim - tensor_transpose(x_interim))
+                        )
+end
+
+function (d::VolumePreservingAttention)(x::AbstractMatrix, ps::NamedTuple)
+    reshape(d(reshape(x, size(x)..., 1), ps), size(x)...)
+end
\ No newline at end of file

From 66e563c6429d0871638a46836998629c9204a57c Mon Sep 17 00:00:00 2001
From: benedict-96 <benedikt.brantner@icloud.com>
Date: Mon, 8 Apr 2024 12:25:39 +0200
Subject: [PATCH 20/33] Now testing the two options for the volume-preserving
 attention layers.

---
 test/attention_layer/attention_setup.jl | 50 +++++++++----------------
 1 file changed, 18 insertions(+), 32 deletions(-)

diff --git a/test/attention_layer/attention_setup.jl b/test/attention_layer/attention_setup.jl
index 2a8d01e7d..03ea246ab 100644
--- a/test/attention_layer/attention_setup.jl
+++ b/test/attention_layer/attention_setup.jl
@@ -1,47 +1,33 @@
 using GeometricMachineLearning, Test
-using GeometricMachineLearning: upper_triangular_asymmetrize
-using GeometricMachineLearning: orthonormal_activation
 using LinearAlgebra: det
 import Random 
 
 Random.seed!(1234)
 
-function attention_tests(N, T=Float32)
-    model₁ = Attention(N, Stiefel=false)
-    model₂ = Attention(N, Stiefel=true)
-    model₃ = Attention(N, orthonormal_activation, Stiefel=false)
-    # same as model₁, but with the add connection
-    model₄ = Attention(N, Stiefel=false, add_connection=true)
+function volume_preserving_attention_tests(N, T=Float32)
+    model₁ = VolumePreservingAttention(N, skew_sym = false)
+    model₂ = VolumePreservingAttention(N, skew_sym = true)
 
     ps₁ = initialparameters(CPU(), T, model₁)
     ps₂ = initialparameters(CPU(), T, model₂)
-    ps₃ = initialparameters(CPU(), T, model₃)
-    @test typeof(ps₂.PQ) <: StiefelManifold 
-    @test typeof(ps₂.PK) <: StiefelManifold 
+    @test typeof(ps₁.A) <: AbstractMatrix{T} 
+    @test typeof(ps₂.A) <: SkewSymMatrix{T} 
 
-    A = randn(N, N)
+    # check if the layers are volume preserving
+    A = randn(T, N, N)
     det₁ = det(A)
     det₂ = det(model₁(A, ps₁))
     det₃ = det(model₂(A, ps₂))
-    det₄ = det(model₃(A, ps₃))
-    @test isapprox(det₁, det₂)
-    @test isapprox(det₂, det₃)
-    @test isapprox(det₃, det₄)
-
-    @test isapprox(model₁(A, ps₁), model₄(A, ps₁)-A)
-
-    A = reshape(rand(SkewSymMatrix{T}, N), N, N, 1)
-    @test isapprox(A, upper_triangular_asymmetrize(A))
-
-    model = Chain(model₁, model₂)
-    ps = (ps₁, ps₂)
-    dx₁ = (PQ=rand(T, N, N), PK=rand(T, N, N))
-    dx₂ = (PQ=rand(StiefelLieAlgHorMatrix{T}, N, N), PK=rand(StiefelLieAlgHorMatrix{T}, N, N))
-    dx = (dx₁, dx₂)
-    o = Optimizer(AdamOptimizer(), ps)
-    optimization_step!(o, model, ps, dx)
-    @test typeof(ps₂.PQ) <: StiefelManifold 
-    @test typeof(ps₂.PQ) <: StiefelManifold 
+    @test det₁ ≈ det₂
+    @test det₂ ≈ det₃
 end
 
-attention_tests(10)
\ No newline at end of file
+# this checks the cpu version
+volume_preserving_attention_tests(10)
+
+# this checks the "gpu versions"
+volume_preserving_attention_tests(2)
+volume_preserving_attention_tests(3)
+volume_preserving_attention_tests(4)
+volume_preserving_attention_tests(5)
+volume_preserving_attention_tests(6)
\ No newline at end of file

From 3c550cb2481c8c41488aff2afc707b6f98b0995e Mon Sep 17 00:00:00 2001
From: benedict-96 <benedikt.brantner@icloud.com>
Date: Mon, 8 Apr 2024 12:32:50 +0200
Subject: [PATCH 21/33] Removed old routines connected with previous parallel
 inverse computation and added new one (tensor_mat_skew_sym_assign).

---
 test/custom_ad_rules/kernel_pullbacks.jl | 6 ++----
 1 file changed, 2 insertions(+), 4 deletions(-)

diff --git a/test/custom_ad_rules/kernel_pullbacks.jl b/test/custom_ad_rules/kernel_pullbacks.jl
index 34dd7ad4b..c987ee802 100644
--- a/test/custom_ad_rules/kernel_pullbacks.jl
+++ b/test/custom_ad_rules/kernel_pullbacks.jl
@@ -1,4 +1,4 @@
-using GeometricMachineLearning: tensor_mat_mul, mat_tensor_mul, tensor_tensor_mul, tensor_transpose_tensor_mul, assign_q_and_p, tensor_transpose, assign_matrix, assign_tensor, assign_output_estimate, vec_tensor_mul, upper_triangular_asymmetrize
+using GeometricMachineLearning: tensor_mat_mul, mat_tensor_mul, tensor_tensor_mul, tensor_transpose_tensor_mul, assign_q_and_p, tensor_transpose, assign_output_estimate, vec_tensor_mul, tensor_mat_skew_sym_assign
 using ChainRulesTestUtils
 using Printf
 import Random 
@@ -17,11 +17,9 @@ function main(first_dim, second_dim, third_dim, third_tensor_dim)
     test_rrule(tensor_tensor_mul, rand(first_dim, second_dim, third_tensor_dim), rand(second_dim, third_dim, third_tensor_dim))
     test_rrule(tensor_transpose_tensor_mul, rand(second_dim, first_dim, third_tensor_dim), rand(second_dim, third_dim, third_tensor_dim))
     test_rrule(tensor_transpose, rand(first_dim, second_dim, third_tensor_dim))
-    test_rrule(assign_matrix, rand(first_dim, second_dim, third_tensor_dim), third_tensor_dim)
-    test_rrule(assign_tensor, rand(first_dim, second_dim), third_tensor_dim, 1)
     test_rrule(assign_output_estimate, rand(first_dim, second_dim, third_tensor_dim), 1)
     test_rrule(vec_tensor_mul, rand(first_dim), rand(first_dim, second_dim, third_tensor_dim))
-    test_rrule(upper_triangular_asymmetrize, rand(first_dim, first_dim, third_tensor_dim))
+    test_rrule(tensor_mat_skew_sym_assign, rand(first_dim, second_dim, third_tensor_dim), rand(first_dim, first_dim), check_thunked_output_tangent = false)
 end
 
 const dim_range = 10

From 8a63299987822c0f1bcf1d8e0900460107d33095 Mon Sep 17 00:00:00 2001
From: benedict-96 <benedikt.brantner@icloud.com>
Date: Mon, 8 Apr 2024 12:33:15 +0200
Subject: [PATCH 22/33] Updated to VolumePreservingAttention.

---
 test/parameterlength/check_parameterlengths.jl | 8 ++++----
 1 file changed, 4 insertions(+), 4 deletions(-)

diff --git a/test/parameterlength/check_parameterlengths.jl b/test/parameterlength/check_parameterlengths.jl
index 0673b73e1..350d06aca 100644
--- a/test/parameterlength/check_parameterlengths.jl
+++ b/test/parameterlength/check_parameterlengths.jl
@@ -11,8 +11,8 @@ function check_parameterlength(N, n)
     model₃₂ = Chain(model₃, model₂)
     model₅ = MultiHeadAttention(N, N÷n, Stiefel=false)
     model₆ = MultiHeadAttention(N, N÷n, Stiefel=true)
-    model₇ = Attention(N, Stiefel=false)
-    model₈ = Attention(N, Stiefel=true)
+    model₇ = VolumePreservingAttention(N, skew_sym = true)
+    model₈ = VolumePreservingAttention(N, skew_sym = false)
 
     @test parameterlength(model₁) == N*N+N
     @test parameterlength(model₂) == N*N 
@@ -21,8 +21,8 @@ function check_parameterlength(N, n)
     @test parameterlength(model₃₂) == parameterlength(model₃) + parameterlength(model₂)
     @test parameterlength(model₅) == 3*N*N
     @test parameterlength(model₆) == 3*N÷n*Int(n*(N-(n+1)/2))
-    @test parameterlength(model₇) == 2*N*N
-    @test parameterlength(model₈) == 2*Int(N*(N-(N+1)/2))
+    @test parameterlength(model₇) == N * (N - 1) ÷ 2
+    @test parameterlength(model₈) == N * N
 end
 
 check_parameterlength(10,5)
\ No newline at end of file

From 4b4084ad0865d8a15f921cf2a4ad4ccff929a3d2 Mon Sep 17 00:00:00 2001
From: benedict-96 <benedikt.brantner@icloud.com>
Date: Mon, 8 Apr 2024 12:33:37 +0200
Subject: [PATCH 23/33] Added parallel inverse/Cayley tests.

---
 test/runtests.jl | 6 ++++--
 1 file changed, 4 insertions(+), 2 deletions(-)

diff --git a/test/runtests.jl b/test/runtests.jl
index 4eecc8f5d..e57477207 100644
--- a/test/runtests.jl
+++ b/test/runtests.jl
@@ -25,7 +25,6 @@ using SafeTestsets
 @safetestset "Check if the optimization_step! changes the parameters of the transformer       " begin include("transformer_related/transformer_optimizer.jl") end
 
 @safetestset "Attention layer #1                                                              " begin include("attention_layer/attention_setup.jl") end
-@safetestset "(MultiHead)Attention                                                            " begin include("attention_layer/apply_multi_head_attention.jl") end
 @safetestset "Classification layer                                                            " begin include("layers/classification.jl") end
 @safetestset "Optimizer #1                                                                    " begin include("optimizers/utils/global_sections.jl") end
 @safetestset "Optimizer #2                                                                    " begin include("optimizers/utils/optimization_step.jl") end
@@ -45,4 +44,7 @@ using SafeTestsets
 @safetestset "Test mnist_utils.                                                               " begin include("data_loader/mnist_utils.jl") end
 @safetestset "Test the data loader in combination with optimization_step!                     " begin include("data_loader/data_loader_optimization_step.jl") end
 @safetestset "Optimizer functor with data loader for Adam                                     " begin include("data_loader/optimizer_functor_with_adam.jl") end
-@safetestset "Test data loader for a tensor (q and p data)                                    " begin include("data_loader/draw_batch_for_tensor_test.jl") end
\ No newline at end of file
+@safetestset "Test data loader for a tensor (q and p data)                                    " begin include("data_loader/draw_batch_for_tensor_test.jl") end
+
+@safetestset "Test parallel inverses                                                          " begin include("kernels/tensor_inverse.jl") end
+@safetestset "Test parallel Cayley                                                            " begin include("kernels/tensor_cayley.jl") end
\ No newline at end of file

From 2d67808ba4e129d06dd1ceeabad32d49a1e0dae9 Mon Sep 17 00:00:00 2001
From: benedict-96 <benedikt.brantner@icloud.com>
Date: Mon, 8 Apr 2024 12:34:27 +0200
Subject: [PATCH 24/33] Added parallel inverse/Cayly routines; removed
 Attention, added VolumePreservingAttention.

---
 src/GeometricMachineLearning.jl | 13 ++++++++++---
 1 file changed, 10 insertions(+), 3 deletions(-)

diff --git a/src/GeometricMachineLearning.jl b/src/GeometricMachineLearning.jl
index bea5100e8..82cacdc66 100644
--- a/src/GeometricMachineLearning.jl
+++ b/src/GeometricMachineLearning.jl
@@ -56,7 +56,13 @@ module GeometricMachineLearning
     include("kernels/mat_tensor_mul.jl")
     include("kernels/tensor_transpose.jl")
     include("kernels/exponentials/tensor_exponential.jl")
-    include("kernels/inverses/inverse_kernel.jl")
+    include("kernels/inverses/cpu_inverse.jl")
+    include("kernels/inverses/inverse_2x2.jl")
+    include("kernels/inverses/inverse_3x3.jl")
+    include("kernels/inverses/inverse_4x4.jl")
+    include("kernels/inverses/inverse_5x5.jl")
+    include("kernels/inverses/tensor_cayley.jl")
+    include("kernels/inverses/tensor_mat_skew_sym_assign.jl")
     include("kernels/vec_tensor_mul.jl")
 
     include("kernels/kernel_ad_routines/assign_q_and_p.jl")
@@ -65,6 +71,7 @@ module GeometricMachineLearning
     include("kernels/kernel_ad_routines/tensor_tensor_mul.jl")
     include("kernels/kernel_ad_routines/tensor_transpose_tensor_mul.jl")
     include("kernels/kernel_ad_routines/tensor_transpose.jl")
+    include("kernels/kernel_ad_routines/tensor_mat_skew_sym_assign.jl")
     include("kernels/kernel_ad_routines/vec_tensor_mul.jl")
     # export tensor_mat_mul
 
@@ -133,7 +140,7 @@ module GeometricMachineLearning
     include("layers/stiefel_layer.jl")
     include("layers/grassmann_layer.jl")
     include("layers/multi_head_attention.jl")
-    include("layers/attention_layer.jl")
+    include("layers/volume_preserving_attention.jl")
     include("layers/transformer.jl")
     include("layers/psd_like_layer.jl")
     include("layers/classification.jl")
@@ -142,7 +149,7 @@ module GeometricMachineLearning
     export StiefelLayer, GrassmannLayer, ManifoldLayer
     export PSDLayer
     export MultiHeadAttention
-    export Attention
+    export VolumePreservingAttention
     export ResNet
     export Transformer
     export Classification

From 4636d0b8ca1dab139ef810153a1252b090d9a10c Mon Sep 17 00:00:00 2001
From: benedict-96 <benedikt.brantner@icloud.com>
Date: Mon, 8 Apr 2024 12:51:42 +0200
Subject: [PATCH 25/33] Got rid of section that was toggled.

---
 src/layers/volume_preserving_attention.jl | 4 ----
 1 file changed, 4 deletions(-)

diff --git a/src/layers/volume_preserving_attention.jl b/src/layers/volume_preserving_attention.jl
index dfd6ccb35..483121c6f 100644
--- a/src/layers/volume_preserving_attention.jl
+++ b/src/layers/volume_preserving_attention.jl
@@ -49,10 +49,6 @@ function orthonormal_activation_cayley(::VolumePreservingAttention{M, M, ScalarP
     tensor_cayley5(A)
 end
 
-# function orthonormal_activation_cayley(::VolumePreservingAttention{M, M, 6}, A::AbstractArray{T, 3}) where {T, M} 
-#     tensor_cayley6(A)
-# end
-
 # function orthonormal_activation_cayley(A::AbstractMatrix{T}) where T 
 #     reshape(orthonormal_activation_cayley(reshape(A, size(A)..., 1)), size(A)...)
 # end

From 21339ea2ad91b4c60d3a83a1476adbdd91a3d9db Mon Sep 17 00:00:00 2001
From: benedict-96 <benedikt.brantner@icloud.com>
Date: Tue, 9 Apr 2024 09:53:55 +0200
Subject: [PATCH 26/33] Specified resolution when calling make png
 (docs/src/tikz/).

---
 docs/src/tikz/Makefile | 4 +++-
 1 file changed, 3 insertions(+), 1 deletion(-)

diff --git a/docs/src/tikz/Makefile b/docs/src/tikz/Makefile
index 77300f1d4..f0585c15b 100644
--- a/docs/src/tikz/Makefile
+++ b/docs/src/tikz/Makefile
@@ -1,5 +1,5 @@
 all: pdf 
-	$(MAKE) png 
+	$(MAKE) png   res1=150 res2=200 res3=100 res4=180
 	$(MAKE) logo 
 	$(MAKE) clean 
 
@@ -46,6 +46,8 @@ png:
 	pdftocairo 	-png -r $(res3) -transp -singlefile  structs_visualization.pdf						structs_visualization
 	pdftocairo 	-png -r $(res3) -transp -singlefile  structs_visualization_dark.pdf					structs_visualization_dark
 	pdftocairo 	-png -r $(res1) -transp -singlefile	 logo.pdf										logo 
+	pdftocairo 	-png -r 500	    -transp -singlefile	 logo_with_name.pdf								logo_with_name 
+	pdftocairo 	-png -r 500	    -transp -singlefile	 logo_with_name_dark.pdf						logo_with_name_dark 
 	pdftocairo  -png -r $(res1) -transp -singlefile  symplectic_autoencoder.pdf						symplectic_autoencoder
 	pdftocairo  -png -r $(res1) -transp -singlefile  symplectic_autoencoder_dark.pdf				symplectic_autoencoder_dark
 	pdftocairo  -png -r $(res1) -transp -singlefile  solution_manifold_2.pdf						solution_manifold_2

From 7c29629286f0c97a8802092af2626af5809474bb Mon Sep 17 00:00:00 2001
From: benedict-96 <benedikt.brantner@icloud.com>
Date: Tue, 9 Apr 2024 10:10:16 +0200
Subject: [PATCH 27/33] Removed skew_symmetric and vp stuff (not yet in this
 pr).

---
 docs/src/tikz/Makefile | 4 ----
 1 file changed, 4 deletions(-)

diff --git a/docs/src/tikz/Makefile b/docs/src/tikz/Makefile
index f0585c15b..6ec62f34a 100644
--- a/docs/src/tikz/Makefile
+++ b/docs/src/tikz/Makefile
@@ -57,10 +57,6 @@ png:
 	pdftocairo  -png -r $(res1) -transp -singlefile  tensor_dark.pdf 								tensor_dark
 	pdftocairo  -png -r $(res4) -transp -singlefile  tensor_sampling.pdf 							tensor_sampling
 	pdftocairo  -png -r $(res4) -transp -singlefile  tensor_sampling_dark.pdf 						tensor_sampling_dark
-	pdftocairo  -png -r $(res4) -transp -singlefile  skew_sym_visualization.pdf 					skew_sym_visualization
-	pdftocairo  -png -r $(res4) -transp -singlefile  skew_sym_visualization_dark.pdf 				skew_sym_visualization_dark
-	pdftocairo  -png -r $(res1) -transp -singlefile  vp_feedforward.pdf 							vp_feedforward
-	pdftocairo  -png -r $(res1) -transp -singlefile  vp_feedforward_dark.pdf 						vp_feedforward_dark
 
 logo:
 	cp logo_with_name.png 		../assets/logo.png

From a0c5b772276972aedb6075f38d396f72d1472d24 Mon Sep 17 00:00:00 2001
From: benedict-96 <benedikt.brantner@icloud.com>
Date: Tue, 9 Apr 2024 10:32:11 +0200
Subject: [PATCH 28/33] Adjusted to new BrenierTwoFluid interface.

---
 docs/src/tutorials/grassmann_layer.md | 4 ++--
 1 file changed, 2 insertions(+), 2 deletions(-)

diff --git a/docs/src/tutorials/grassmann_layer.md b/docs/src/tutorials/grassmann_layer.md
index c0f5fb5d8..6059b4931 100644
--- a/docs/src/tutorials/grassmann_layer.md
+++ b/docs/src/tutorials/grassmann_layer.md
@@ -48,7 +48,7 @@ const ε = 0.1                 # entropic regularization. √ε is a length.  #
 const q = 1.0                 # annealing parameter                       # hide
 const Δ = 1.0                 # characteristic domain size                # hide
 const s = ε                   # current scale: no annealing -> equals ε   # hide
-const tol = 1e-4              # marginal condition tolerance              # hide 
+const tol = 1e-6              # marginal condition tolerance              # hide 
 const crit_it = 20            # acceleration inference                    # hide
 const p_η = 2
 
@@ -58,7 +58,7 @@ function compute_wasserstein_gradient(ensemble1::AT, ensemble2::AT) where AT<:Ab
     V = SinkhornVariable(copy(ensemble1'), ones(number_of_particles1) / number_of_particles1)
     W = SinkhornVariable(copy(ensemble2'), ones(number_of_particles2) / number_of_particles2)
     params = SinkhornParameters(; ε=ε,q=1.0,Δ=1.0,s=s,tol=tol,crit_it=crit_it,p_η=p_η,sym=false,acc=true) # hide
-    S = SinkhornDivergence(V, W, c, params, true)
+    S = SinkhornDivergence(V, W, c, params; islog = true)
     initialize_potentials!(S)
     compute!(S)
     value(S), x_gradient!(S, ∇c)'

From 7048bfdbf6abaf4aa66f99af3c07151912b26c1a Mon Sep 17 00:00:00 2001
From: benedict-96 <benedikt.brantner@icloud.com>
Date: Tue, 9 Apr 2024 10:51:49 +0200
Subject: [PATCH 29/33] Not testing 5x5 inverse now (too expensive).

---
 test/kernels/tensor_cayley.jl  | 2 +-
 test/kernels/tensor_inverse.jl | 4 ++--
 2 files changed, 3 insertions(+), 3 deletions(-)

diff --git a/test/kernels/tensor_cayley.jl b/test/kernels/tensor_cayley.jl
index 79f37f5ca..d671776af 100644
--- a/test/kernels/tensor_cayley.jl
+++ b/test/kernels/tensor_cayley.jl
@@ -40,7 +40,7 @@ function check_all(T::Type, third_dim::Int=10)
     test_tensor_cayley2(T, third_dim)
     test_tensor_cayley3(T, third_dim)
     test_tensor_cayley4(T, third_dim)
-    test_tensor_cayley5(T, third_dim)
+    # test_tensor_cayley5(T, third_dim)
 end
 
 check_all(Float32)
diff --git a/test/kernels/tensor_inverse.jl b/test/kernels/tensor_inverse.jl
index 252da0b4d..f130126ab 100644
--- a/test/kernels/tensor_inverse.jl
+++ b/test/kernels/tensor_inverse.jl
@@ -11,7 +11,7 @@ function test55_inverse(k::Int = 10)
     end
 end
 
-test55_inverse()
+# test55_inverse()
 
 function test55_inverse_pullback(k::Int = 10)
     A = rand(5, 5, k)
@@ -28,7 +28,7 @@ function test55_inverse_pullback(k::Int = 10)
     end
 end
 
-test55_inverse_pullback()
+# test55_inverse_pullback()
 
 function test44_inverse(k::Int = 10)
     A = rand(4, 4, k)

From 5281c47fce237353adc19f45552aea63398fc939 Mon Sep 17 00:00:00 2001
From: benedict-96 <benedikt.brantner@icloud.com>
Date: Tue, 9 Apr 2024 10:53:44 +0200
Subject: [PATCH 30/33] Adjusted to new interface.

---
 docs/src/tutorials/grassmann_layer.md | 4 ++--
 1 file changed, 2 insertions(+), 2 deletions(-)

diff --git a/docs/src/tutorials/grassmann_layer.md b/docs/src/tutorials/grassmann_layer.md
index c0f5fb5d8..6059b4931 100644
--- a/docs/src/tutorials/grassmann_layer.md
+++ b/docs/src/tutorials/grassmann_layer.md
@@ -48,7 +48,7 @@ const ε = 0.1                 # entropic regularization. √ε is a length.  #
 const q = 1.0                 # annealing parameter                       # hide
 const Δ = 1.0                 # characteristic domain size                # hide
 const s = ε                   # current scale: no annealing -> equals ε   # hide
-const tol = 1e-4              # marginal condition tolerance              # hide 
+const tol = 1e-6              # marginal condition tolerance              # hide 
 const crit_it = 20            # acceleration inference                    # hide
 const p_η = 2
 
@@ -58,7 +58,7 @@ function compute_wasserstein_gradient(ensemble1::AT, ensemble2::AT) where AT<:Ab
     V = SinkhornVariable(copy(ensemble1'), ones(number_of_particles1) / number_of_particles1)
     W = SinkhornVariable(copy(ensemble2'), ones(number_of_particles2) / number_of_particles2)
     params = SinkhornParameters(; ε=ε,q=1.0,Δ=1.0,s=s,tol=tol,crit_it=crit_it,p_η=p_η,sym=false,acc=true) # hide
-    S = SinkhornDivergence(V, W, c, params, true)
+    S = SinkhornDivergence(V, W, c, params; islog = true)
     initialize_potentials!(S)
     compute!(S)
     value(S), x_gradient!(S, ∇c)'

From 41d686b4005e685d35ce752243fa80ff8230e22e Mon Sep 17 00:00:00 2001
From: benedict-96 <benedikt.brantner@icloud.com>
Date: Tue, 9 Apr 2024 10:55:04 +0200
Subject: [PATCH 31/33] Updated/fixed workflow.

---
 docs/src/tikz/Makefile | 6 ++++--
 1 file changed, 4 insertions(+), 2 deletions(-)

diff --git a/docs/src/tikz/Makefile b/docs/src/tikz/Makefile
index 77300f1d4..2bbf8a533 100644
--- a/docs/src/tikz/Makefile
+++ b/docs/src/tikz/Makefile
@@ -1,5 +1,5 @@
 all: pdf 
-	$(MAKE) png 
+	$(MAKE) png   res1=150 res2=200 res3=100 res4=180
 	$(MAKE) logo 
 	$(MAKE) clean 
 
@@ -45,7 +45,9 @@ png:
 	pdftocairo	-png -r $(res2) -transp -singlefile	 sympnet_architecture_dark.pdf					sympnet_architecture_dark
 	pdftocairo 	-png -r $(res3) -transp -singlefile  structs_visualization.pdf						structs_visualization
 	pdftocairo 	-png -r $(res3) -transp -singlefile  structs_visualization_dark.pdf					structs_visualization_dark
-	pdftocairo 	-png -r $(res1) -transp -singlefile	 logo.pdf										logo 
+	pdftocairo 	-png -r $(res1) -transp -singlefile	 logo.pdf										logo
+	pdftocairo  -png -r 500     -transp -singlefile  logo_with_name.pdf                             logo_with_name
+	pdftocairo  -png -r 500     -transp -singlefile  logo_with_name_dark.pdf                        logo_with_name_dark
 	pdftocairo  -png -r $(res1) -transp -singlefile  symplectic_autoencoder.pdf						symplectic_autoencoder
 	pdftocairo  -png -r $(res1) -transp -singlefile  symplectic_autoencoder_dark.pdf				symplectic_autoencoder_dark
 	pdftocairo  -png -r $(res1) -transp -singlefile  solution_manifold_2.pdf						solution_manifold_2

From ab3f63008a05ff116c8c42e773bbd70532d4a200 Mon Sep 17 00:00:00 2001
From: benedict-96 <benedikt.brantner@icloud.com>
Date: Tue, 9 Apr 2024 11:09:13 +0200
Subject: [PATCH 32/33] Removed files that are part of different pr
 (vp_feedforward).

---
 docs/src/tikz/Makefile | 4 ----
 1 file changed, 4 deletions(-)

diff --git a/docs/src/tikz/Makefile b/docs/src/tikz/Makefile
index 2bbf8a533..81b05c408 100644
--- a/docs/src/tikz/Makefile
+++ b/docs/src/tikz/Makefile
@@ -57,10 +57,6 @@ png:
 	pdftocairo  -png -r $(res1) -transp -singlefile  tensor_dark.pdf 								tensor_dark
 	pdftocairo  -png -r $(res4) -transp -singlefile  tensor_sampling.pdf 							tensor_sampling
 	pdftocairo  -png -r $(res4) -transp -singlefile  tensor_sampling_dark.pdf 						tensor_sampling_dark
-	pdftocairo  -png -r $(res4) -transp -singlefile  skew_sym_visualization.pdf 					skew_sym_visualization
-	pdftocairo  -png -r $(res4) -transp -singlefile  skew_sym_visualization_dark.pdf 				skew_sym_visualization_dark
-	pdftocairo  -png -r $(res1) -transp -singlefile  vp_feedforward.pdf 							vp_feedforward
-	pdftocairo  -png -r $(res1) -transp -singlefile  vp_feedforward_dark.pdf 						vp_feedforward_dark
 
 logo:
 	cp logo_with_name.png 		../assets/logo.png

From 88bf05fb462a1455cd6aa4e5e703a7afc9db14ad Mon Sep 17 00:00:00 2001
From: Michael Kraus <michael.kraus@ipp.mpg.de>
Date: Tue, 9 Apr 2024 11:19:22 +0200
Subject: [PATCH 33/33] Add pre-push git hook.

---
 .githooks/pre-push | 19 +++++++++++++++++++
 README.md          |  9 +++++++++
 2 files changed, 28 insertions(+)
 create mode 100755 .githooks/pre-push

diff --git a/.githooks/pre-push b/.githooks/pre-push
new file mode 100755
index 000000000..0e3dc747d
--- /dev/null
+++ b/.githooks/pre-push
@@ -0,0 +1,19 @@
+# pre-push git hook that runs all tests before pushing
+
+red='\033[0;31m'
+green='\033[0;32m'
+no_color='\033[0m'
+
+reponame=$(basename `git rev-parse --show-toplevel`)
+
+
+echo "\nRunning pre-push hook\n"
+echo "Testing $reponame"
+julia --project=@.  -e "using Pkg; Pkg.test(\"GeometricMachineLearning\")"
+
+if [[ $? -ne 0 ]]; then
+        echo "\n${red}ERROR - Tests must pass before push!\n${no_color}"
+  exit 1
+fi
+
+echo "\n${green}Git hook was SUCCESSFUL!${no_color}\n"
diff --git a/README.md b/README.md
index 9b6dd535c..ac40d6b99 100644
--- a/README.md
+++ b/README.md
@@ -63,3 +63,12 @@ More examples like this can be found in the docs.
 - Brantner B. Generalizing Adam To Manifolds For Efficiently Training Transformers. arXiv preprint arXiv:2305.16901, 2023.
 - Brantner B., Kraus M. Symplectic Autoencoders for Model Reduction of Hamiltonian Systems. arXiv preprint arXiv:2312.10004, 2023.
 - Brantner B., Romemont G., Kraus M., Li Z. Structure-Preserving Transformers for Learning Parametrized Hamiltonian Systems. arXiv preprint arXiv:2312.11166, 2023.
+
+
+## Development
+
+We are using git hooks, e.g., to enforce that all tests pass before pushing.
+In order to activate these hooks, the following command must be executed once:
+```
+git config core.hooksPath .githooks
+```