N-neurons Neural Networks

Simple neural networks with constant number N of neurons per each layer (when the fixed N is actually part of the type of matrices and vectors) or even differing between layers.

The type Matrix[A, M, N] specifies that the matrix contains elements of type A, has M number of rows and N number of columns. A must have a Ring typeclass instance, as defined in spire.

The type Vector[A, N] specifies that the vectors contains elements of type A and has N number of rows. A must have a Ring typeclass instance, as defined in spire.

Each operation performed with matrices and vectors must type check, meaning one can multiply a Matrix[A, M, N] and a Matrix[A, N, P], yielding a Matrix[A, M, P], but one cannot multiply the former with a Matrix[B, N, P] or a Matrix[A, P, Q], because A differs from B, respectively, N differs from P.

N Neural Network

A neural network Network[N] is composed of M layers each of N neurons, while N+1 is the number of weights per neuron plus 1 (for the bias). For arbitrary precision arithmetic using spire.math.Real, for instance, each neuron of type Neuron[N] has a Vector[Real, N] of weights, a Real bias and an Activation function, where the latter is a Scala3 enum. This means that the Activation functions may differ with each neuron. The definition and the derivative of an Activation function must be known and given.

For a neural network, it is implemented training with backpropagation, as well as the straighter prediction, once the neurons' (biases and) weights have been trained.

For examples, see also this blog or this java-toy-neural-network project.

Neural Network

The package float builds around a generalized neural network, where the assumption of a constant number of neurons per each layer is relaxed, and thus the number of neurons may differ between layers.

Although the dimension types of the variables and values involved in the algorithms are the wildcard ?, operations on matrices and vectors are type-safe using shapes. Even assignment is safe - though not the method := by itself - because the algorithms perform reassignment only, and thus the types of matrices/vectors are asserted before assignment.

An example of a neural network with two inputs, a hidden layer with ten neurons, and one output is the following:

type N[L <: Int] = L match { case 0 => 2 case 1 => 10 case 2 => 1 }

given List[Int] = 2 :: 10 :: 1 :: Nil

Network[N, 2](loss = MSE[1](), learningRate = 3, ...)

Each type mapped by the higher-kinded type N[_] differs with the type argument: N[0] is the number of inputs, N[1] is the number of neurons in the hidden layer, while N[2] is the number of outputs. It is hence called a shape.

The implicit given_List_Int is the shape as values. Both kinds of shape (types and values) must be given. The neural network is defined as Network[N, 2](...), where N is the shape and 2 is the number of hidden layers, while the (values) shape is passed as implicit parameter. Then, 1 occuring in the argument loss = MSE[1]() is the same number of neurons in the output layer.

Note that the output layer is also a hidden layer.

Testing

Use, for instance, the following sbt command:

sbt:N Neural Networks> testOnly *double*Network*

This will run all tests with the word "Network" in package "double" (where all values, functions, or networks are based on the Double type): there is only one such suite, nnn.double.NetworkSuite.

Math

Consider a neural network with three hidden layers (the last of each is the output layer) and three neurons per each layer, like in the following table:

Input	Layer	Layer	Layer/Output
$1$	$1$	$1$
$x_1$	$a_{11} {\phi_{11} \atop \longrightarrow} h_{11}$	$a_{12} {\phi_{12} \atop \longrightarrow}h_{12}$	$a_{13} {\phi_{13} \atop \longrightarrow} y_1$
$x_2$	$a_{21} {\phi_{21} \atop \longrightarrow} h_{21}$	$a_{22} {\phi_{22} \atop \longrightarrow}h_{22}$	$a_{23} {\phi_{23} \atop \longrightarrow} y_2$
$x_3$	$a_{31} {\phi_{31} \atop \longrightarrow} h_{31}$	$a_{32} {\phi_{32} \atop \longrightarrow}h_{32}$	$a_{33} {\phi_{33} \atop \longrightarrow} y_3$

The layers are fully connected, in the sense of the following nine equations:

$$\begin{align*} a_{11} = w_{11}^0 \times 1 + w_{11}^1 \times x_1 + w_{11}^2 \times x_2 + w_{11}^3 \times x_3 & & a_{12} = w_{12}^0 \times 1 + w_{12}^1 \times h_{11} + w_{12}^2 \times h_{21} + w_{12}^3 \times h_{31} & & a_{13} = w_{13}^0 \times 1 + w_{13}^1 \times h_{12} + w_{13}^2 \times h_{22} + w_{13}^3 \times h_{32} \\\ a_{21} = w_{21}^0 \times 1 + w_{21}^1 \times x_1 + w_{21}^2 \times x_2 + w_{21}^3 \times x_3 & & a_{22} = w_{22}^0 \times 1 + w_{22}^1 \times h_{11} + w_{22}^2 \times h_{21} + w_{22}^3 \times h_{31} & & a_{23} = w_{23}^0 \times 1 + w_{23}^1 \times h_{12} + w_{23}^2 \times h_{22} + w_{23}^3 \times h_{32} \\\ a_{31} = w_{31}^0 \times 1 + w_{31}^1 \times x_1 + w_{31}^2 \times x_2 + w_{31}^3 \times x_3 & & a_{32} = w_{32}^0 \times 1 + w_{32}^1 \times h_{11} + w_{32}^2 \times h_{21} + w_{32}^3 \times h_{31} & & a_{33} = w_{33}^0 \times 1 + w_{33}^1 \times h_{12} + w_{33}^2 \times h_{22} + w_{33}^3 \times h_{32} \\\ \end{align*}$$

where $w_{ij}^k$ is the weight of the $i^{th}$ neuron on the $j^{th}$ layer with respect to the $k^{th}$ output from the previous layer. We have, in matrix form:

$$\begin{align*} \begin{pmatrix} a_{11} \\\ \\\ a_{21} \\\ \\\ a_{31} \end{pmatrix} = \begin{pmatrix} w_{11}^0 & w_{11}^1 & w_{11}^2 & w_{11}^3 \\\ \\\ w_{21}^0 & w_{21}^1 & w_{21}^2 & w_{21}^3 \\\ \\\ w_{31}^0 & w_{31}^1 & w_{31}^2 & w_{31}^3 \end{pmatrix} \cdot \begin{pmatrix} 1 \\\ \\\ x_1 \\\ \\\ x_2 \\\ \\\ x_3 \end{pmatrix} & & \begin{pmatrix} a_{12} \\\ \\\ a_{22} \\\ \\\ a_{32} \end{pmatrix} = \begin{pmatrix} w_{12}^0 & w_{12}^1 & w_{12}^2 & w_{12}^3 \\\ \\\ w_{22}^0 & w_{22}^1 & w_{22}^2 & w_{22}^3 \\\ \\\ w_{32}^0 & w_{32}^1 & w_{32}^2 & w_{32}^3 \end{pmatrix} \cdot \begin{pmatrix} 1 \\\ \\\ h_{11} \\\ \\\ h_{21} \\\ \\\ h_{31} \end{pmatrix} & & \begin{pmatrix} a_{13} \\\ \\\ a_{23} \\\ \\\ a_{33} \end{pmatrix} = \begin{pmatrix} w_{13}^0 & w_{13}^1 & w_{13}^2 & w_{13}^3 \\\ \\\ w_{23}^0 & w_{23}^1 & w_{23}^2 & w_{23}^3 \\\ \\\ w_{33}^0 & w_{33}^1 & w_{33}^2 & w_{33}^3 \end{pmatrix} \cdot \begin{pmatrix} 1 \\\ \\\ h_{12} \\\ \\\ h_{22} \\\ \\\ h_{32} \end{pmatrix} \end{align*}$$

The outputs can be written more briefly using indices:

$$\begin{align*} h_{ij} = \phi_{ij}(a_{ij}) & & i = \overline{1 \dots 3}, j = \overline{1 \dots 2} \\\ y_i = \phi_{i3}(a_{i3}) & & i = \overline{1 \dots 3} \end{align*}$$

These were the equations corresponding to the forward pass. For backpropagation, we proceed backwards, from the output layer towards the input layer. Assume $L$ is the loss function; it has three known partial derivatives: $\frac{\partial{L}}{\partial{y_i}}$, where $i = \overline{1 \dots 3}$.

From these, we start with the derivatives of $L$ with respect to the weights in the last (output) layer ($w_{i3}^k$, where $i = \overline{1 \dots 3}$ and $k = \overline{0 \dots 3}$). Using the chain rule the following equations hold:

$$\begin{align} \frac{\partial{L}}{\partial{w_{13}^0}} = \frac{\partial{L}}{\partial{y_1}} \times \phi_{13}'(a_{13}) \times 1 & & \frac{\partial{L}}{\partial{w_{13}^1}} = \frac{\partial{L}}{\partial{y_1}} \times \phi_{13}'(a_{13}) \times h_{12} & & \frac{\partial{L}}{\partial{w_{13}^2}} = \frac{\partial{L}}{\partial{y_1}} \times \phi_{13}'(a_{13}) \times h_{22} & & \frac{\partial{L}}{\partial{w_{13}^3}} = \frac{\partial{L}}{\partial{y_1}} \times \phi_{13}'(a_{13}) \times h_{32} & (1) \end{align}$$ $$\begin{align} \frac{\partial{L}}{\partial{w_{23}^0}} = \frac{\partial{L}}{\partial{y_2}} \times \phi_{23}'(a_{23}) \times 1 & & \frac{\partial{L}}{\partial{w_{23}^1}} = \frac{\partial{L}}{\partial{y_2}} \times \phi_{23}'(a_{23}) \times h_{12} & & \frac{\partial{L}}{\partial{w_{23}^2}} = \frac{\partial{L}}{\partial{y_2}} \times \phi_{23}'(a_{23}) \times h_{22} & & \frac{\partial{L}}{\partial{w_{23}^3}} = \frac{\partial{L}}{\partial{y_2}} \times \phi_{23}'(a_{23}) \times h_{32} & (2) \end{align}$$ $$\begin{align} \frac{\partial{L}}{\partial{w_{33}^0}} = \frac{\partial{L}}{\partial{y_3}} \times \phi_{33}'(a_{33}) \times 1 & & \frac{\partial{L}}{\partial{w_{33}^1}} = \frac{\partial{L}}{\partial{y_3}} \times \phi_{33}'(a_{33}) \times h_{12} & & \frac{\partial{L}}{\partial{w_{33}^2}} = \frac{\partial{L}}{\partial{y_3}} \times \phi_{33}'(a_{33}) \times h_{22} & & \frac{\partial{L}}{\partial{w_{33}^3}} = \frac{\partial{L}}{\partial{y_3}} \times \phi_{33}'(a_{33}) \times h_{32} & (3) \end{align}$$

We note that $\frac{\partial{L}}{\partial{y_i}} \times \phi_{i3}'(a_{i3})$, where $i = \overline{1 \dots 3}$, occur repeatedly: we may thus introduce the following matrix named $\delta$ (using the Hadamard product $\odot$):

$$\delta = \begin{pmatrix} \frac{\partial{L}}{\partial{y_1}} \\\ \\\ \frac{\partial{L}}{\partial{y_2}} \\\ \\\ \frac{\partial{L}}{\partial{y_3}} \end{pmatrix} \odot \begin{pmatrix} \phi_{13}'(a_{13}) \\\ \\\ \phi_{23}'(a_{23}) \\\ \\\ \phi_{33}'(a_{33}) \end{pmatrix}$$

Then, the previous equations become (under the notation $\nabla^{(3)}$ - the partial derivatives of $L$ with respect to the weights on the $3^{rd}$ layer):

$$\nabla^{(3)} = \begin{pmatrix} \frac{\partial{L}}{\partial{w_{13}^0}} & \frac{\partial{L}}{\partial{w_{13}^1}} & \frac{\partial{L}}{\partial{w_{13}^2}} & \frac{\partial{L}}{\partial{w_{13}^3}} \\\ \\\ \frac{\partial{L}}{\partial{w_{23}^0}} & \frac{\partial{L}}{\partial{w_{23}^1}} & \frac{\partial{L}}{\partial{w_{23}^2}} & \frac{\partial{L}}{\partial{w_{23}^3}} \\\ \\\ \frac{\partial{L}}{\partial{w_{33}^0}} & \frac{\partial{L}}{\partial{w_{33}^1}} & \frac{\partial{L}}{\partial{w_{33}^2}} & \frac{\partial{L}}{\partial{w_{33}^3}} \end{pmatrix} = \delta \cdot \begin{pmatrix} 1 & h_{12} & h_{22} & h_{32} \end{pmatrix}$$

Let us now write the twelve partial derivatives of $L$ with respect to the weights on the $2^{nd}$ layer, each equation being a sum of three terms:

$$\begin{align} \frac{\partial{L}}{\partial{w_{12}^0}} & = \frac{\partial{L}}{\partial{y_1}} \times \phi_{13}'(a_{13}) \times w_{13}^1 \times \phi_{12}'(a12) \times 1 & + \frac{\partial{L}}{\partial{y_2}} \times \phi_{23}'(a_{23}) \times w_{23}^1 \times \phi_{12}'(a12) \times 1 & + \frac{\partial{L}}{\partial{y_3}} \times \phi_{33}'(a_{33}) \times w_{33}^1 \times \phi_{12}'(a12) \times 1 & (4) \\\ \frac{\partial{L}}{\partial{w_{12}^1}} & = \frac{\partial{L}}{\partial{y_1}} \times \phi_{13}'(a_{13}) \times w_{13}^1 \times \phi_{12}'(a12) \times h_{11} & + \frac{\partial{L}}{\partial{y_2}} \times \phi_{23}'(a_{23}) \times w_{23}^1 \times \phi_{12}'(a12) \times h_{11} & + \frac{\partial{L}}{\partial{y_3}} \times \phi_{33}'(a_{33}) \times w_{33}^1 \times \phi_{12}'(a12) \times h_{11} & (5) \end{align}$$ $$\begin{align} \frac{\partial{L}}{\partial{w_{12}^2}} & = \frac{\partial{L}}{\partial{y_1}} \times \phi_{13}'(a_{13}) \times w_{13}^1 \times \phi_{12}'(a12) \times h_{21} & + \frac{\partial{L}}{\partial{y_2}} \times \phi_{23}'(a_{23}) \times w_{23}^1 \times \phi_{12}'(a12) \times h_{21} & + \frac{\partial{L}}{\partial{y_3}} \times \phi_{33}'(a_{33}) \times w_{33}^1 \times \phi_{12}'(a12) \times h_{21} & (6) \\\ \frac{\partial{L}}{\partial{w_{12}^3}} & = \frac{\partial{L}}{\partial{y_1}} \times \phi_{13}'(a_{13}) \times w_{13}^1 \times \phi_{12}'(a12) \times h_{31} & + \frac{\partial{L}}{\partial{y_2}} \times \phi_{23}'(a_{23}) \times w_{23}^1 \times \phi_{12}'(a12) \times h_{31} & + \frac{\partial{L}}{\partial{y_3}} \times \phi_{33}'(a_{33}) \times w_{33}^1 \times \phi_{12}'(a12) \times h_{31} & (7) \end{align}$$ $$\begin{align} \frac{\partial{L}}{\partial{w_{22}^0}} & = \frac{\partial{L}}{\partial{y_1}} \times \phi_{13}'(a_{13}) \times w_{13}^2 \times \phi_{22}'(a22) \times 1 & + \frac{\partial{L}}{\partial{y_2}} \times \phi_{23}'(a_{23}) \times w_{23}^2 \times \phi_{22}'(a22) \times 1 & + \frac{\partial{L}}{\partial{y_3}} \times \phi_{33}'(a_{33}) \times w_{33}^2 \times \phi_{22}'(a22) \times 1 & (8) \\\ \frac{\partial{L}}{\partial{w_{22}^1}} & = \frac{\partial{L}}{\partial{y_1}} \times \phi_{13}'(a_{13}) \times w_{13}^2 \times \phi_{22}'(a22) \times h_{11} & + \frac{\partial{L}}{\partial{y_2}} \times \phi_{23}'(a_{23}) \times w_{23}^2 \times \phi_{22}'(a22) \times h_{11} & + \frac{\partial{L}}{\partial{y_3}} \times \phi_{33}'(a_{33}) \times w_{33}^2 \times \phi_{22}'(a22) \times h_{11} & (9) \end{align}$$ $$\begin{align} \frac{\partial{L}}{\partial{w_{22}^2}} & = \frac{\partial{L}}{\partial{y_1}} \times \phi_{13}'(a_{13}) \times w_{13}^2 \times \phi_{22}'(a22) \times h_{21} & + \frac{\partial{L}}{\partial{y_2}} \times \phi_{23}'(a_{23}) \times w_{23}^2 \times \phi_{22}'(a22) \times h_{21} & + \frac{\partial{L}}{\partial{y_3}} \times \phi_{33}'(a_{33}) \times w_{33}^2 \times \phi_{22}'(a22) \times h_{21} & (10) \\\ \frac{\partial{L}}{\partial{w_{22}^3}} & = \frac{\partial{L}}{\partial{y_1}} \times \phi_{13}'(a_{13}) \times w_{13}^2 \times \phi_{22}'(a22) \times h_{31} & + \frac{\partial{L}}{\partial{y_2}} \times \phi_{23}'(a_{23}) \times w_{23}^2 \times \phi_{22}'(a22) \times h_{31} & + \frac{\partial{L}}{\partial{y_3}} \times \phi_{33}'(a_{33}) \times w_{33}^2 \times \phi_{22}'(a22) \times h_{31} & (11) \end{align}$$ $$\begin{align} \frac{\partial{L}}{\partial{w_{32}^0}} & = \frac{\partial{L}}{\partial{y_1}} \times \phi_{13}'(a_{13}) \times w_{13}^3 \times \phi_{32}'(a32) \times 1 & + \frac{\partial{L}}{\partial{y_2}} \times \phi_{23}'(a_{23}) \times w_{23}^3 \times \phi_{32}'(a32) \times 1 & + \frac{\partial{L}}{\partial{y_3}} \times \phi_{33}'(a_{33}) \times w_{33}^3 \times \phi_{32}'(a32) \times 1 & (12) \\\ \frac{\partial{L}}{\partial{w_{32}^1}} & = \frac{\partial{L}}{\partial{y_1}} \times \phi_{13}'(a_{13}) \times w_{13}^3 \times \phi_{32}'(a32) \times h_{11} & + \frac{\partial{L}}{\partial{y_2}} \times \phi_{23}'(a_{23}) \times w_{23}^3 \times \phi_{32}'(a32) \times h_{11} & + \frac{\partial{L}}{\partial{y_3}} \times \phi_{33}'(a_{33}) \times w_{33}^3 \times \phi_{32}'(a32) \times h_{11} & (13) \end{align}$$ $$\begin{align} \frac{\partial{L}}{\partial{w_{32}^2}} & = \frac{\partial{L}}{\partial{y_1}} \times \phi_{13}'(a_{13}) \times w_{13}^3 \times \phi_{32}'(a32) \times h_{21} & + \frac{\partial{L}}{\partial{y_2}} \times \phi_{23}'(a_{23}) \times w_{23}^3 \times \phi_{32}'(a32) \times h_{21} & + \frac{\partial{L}}{\partial{y_3}} \times \phi_{33}'(a_{33}) \times w_{33}^3 \times \phi_{32}'(a32) \times h_{21} & (14) \\\ \frac{\partial{L}}{\partial{w_{32}^3}} & = \frac{\partial{L}}{\partial{y_1}} \times \phi_{13}'(a_{13}) \times w_{13}^3 \times \phi_{32}'(a32) \times h_{31} & + \frac{\partial{L}}{\partial{y_2}} \times \phi_{23}'(a_{23}) \times w_{23}^3 \times \phi_{32}'(a32) \times h_{31} & + \frac{\partial{L}}{\partial{y_3}} \times \phi_{33}'(a_{33}) \times w_{33}^3 \times \phi_{32}'(a32) \times h_{31} & (15) \end{align}$$

Let us now observe what is the product of the transpose of the $3^{rd}$ layer's weights matrix, and delta:

$${W^{(3)}}^T \cdot \delta = \begin{pmatrix} w_{13}^0 & w_{23}^0 & w_{33}^0 \\\ \\\ w_{13}^1 & w_{23}^1 & w_{33}^1 \\\ \\\ w_{13}^2 & w_{23}^2 & w_{33}^2 \\\ \\\ w_{13}^3 & w_{23}^3 & w_{33}^3 \\\ \end{pmatrix} \cdot \begin{pmatrix} \frac{\partial{L}}{\partial{y_1}} \times \phi_{13}'(a_{13}) \\\ \\\ \frac{\partial{L}}{\partial{y_2}} \times \phi_{23}'(a_{23}) \\\ \\\ \frac{\partial{L}}{\partial{y_3}} \times \phi_{33}'(a_{33}) \end{pmatrix} =$$ $$= \begin{pmatrix} \frac{\partial{L}}{\partial{y_1}} \times \phi_{13}'(a_{13}) \times w_{13}^0 + \frac{\partial{L}}{\partial{y_2}} \times \phi_{23}'(a_{23}) \times w_{23}^0 + \frac{\partial{L}}{\partial{y_3}} \times \phi_{33}'(a_{33}) \times w_{33}^0 \\\ \\\ \frac{\partial{L}}{\partial{y_1}} \times \phi_{13}'(a_{13}) \times w_{13}^1 + \frac{\partial{L}}{\partial{y_2}} \times \phi_{23}'(a_{23}) \times w_{23}^1 + \frac{\partial{L}}{\partial{y_3}} \times \phi_{33}'(a_{33}) \times w_{33}^1 \\\ \\\ \frac{\partial{L}}{\partial{y_1}} \times \phi_{13}'(a_{13}) \times w_{13}^2 + \frac{\partial{L}}{\partial{y_2}} \times \phi_{23}'(a_{23}) \times w_{23}^2 + \frac{\partial{L}}{\partial{y_3}} \times \phi_{33}'(a_{33}) \times w_{33}^2 \\\ \\\ \frac{\partial{L}}{\partial{y_1}} \times \phi_{13}'(a_{13}) \times w_{13}^3 + \frac{\partial{L}}{\partial{y_2}} \times \phi_{23}'(a_{23}) \times w_{23}^3 + \frac{\partial{L}}{\partial{y_3}} \times \phi_{33}'(a_{33}) \times w_{33}^3 \\\ \end{pmatrix}$$

Let us further drop the first row (because it is not used) in the above matrix (denoting this transient matrix by ${\left( {W^{(3)}}^T \cdot \delta \right)}^*$):

$${\left( {W^{(3)}}^T \cdot \delta \right)}^* = \begin{pmatrix} \frac{\partial{L}}{\partial{y_1}} \times \phi_{13}'(a_{13}) \times w_{13}^1 + \frac{\partial{L}}{\partial{y_2}} \times \phi_{23}'(a_{23}) \times w_{23}^1 + \frac{\partial{L}}{\partial{y_3}} \times \phi_{33}'(a_{33}) \times w_{33}^1 \\\ \\\ \frac{\partial{L}}{\partial{y_1}} \times \phi_{13}'(a_{13}) \times w_{13}^2 + \frac{\partial{L}}{\partial{y_2}} \times \phi_{23}'(a_{23}) \times w_{23}^2 + \frac{\partial{L}}{\partial{y_3}} \times \phi_{33}'(a_{33}) \times w_{33}^2 \\\ \\\ \frac{\partial{L}}{\partial{y_1}} \times \phi_{13}'(a_{13}) \times w_{13}^3 + \frac{\partial{L}}{\partial{y_2}} \times \phi_{23}'(a_{23}) \times w_{23}^3 + \frac{\partial{L}}{\partial{y_3}} \times \phi_{33}'(a_{33}) \times w_{33}^3 \\\ \end{pmatrix}$$

and apply the following Hadamard product:

$${\left( {W^{(3)}}^T \cdot \delta \right)}^* \odot \begin{pmatrix} \phi_{12}'(a_{12}) \\\ \\\ \phi_{22}'(a_{22}) \\\ \\\ \phi_{32}'(a_{32}) \end{pmatrix} = \begin{pmatrix} \frac{\partial{L}}{\partial{y_1}} \times \phi_{13}'(a_{13}) \times w_{13}^1 + \frac{\partial{L}}{\partial{y_2}} \times \phi_{23}'(a_{23}) \times w_{23}^1 + \frac{\partial{L}}{\partial{y_3}} \times \phi_{33}'(a_{33}) \times w_{33}^1 \\\ \\\ \frac{\partial{L}}{\partial{y_1}} \times \phi_{13}'(a_{13}) \times w_{13}^2 + \frac{\partial{L}}{\partial{y_2}} \times \phi_{23}'(a_{23}) \times w_{23}^2 + \frac{\partial{L}}{\partial{y_3}} \times \phi_{33}'(a_{33}) \times w_{33}^2 \\\ \\\ \frac{\partial{L}}{\partial{y_1}} \times \phi_{13}'(a_{13}) \times w_{13}^3 + \frac{\partial{L}}{\partial{y_2}} \times \phi_{23}'(a_{23}) \times w_{23}^3 + \frac{\partial{L}}{\partial{y_3}} \times \phi_{33}'(a_{33}) \times w_{33}^3 \\\ \end{pmatrix} \odot \begin{pmatrix} \phi_{12}'(a_{12}) \\\ \\\ \phi_{22}'(a_{22}) \\\ \\\ \phi_{32}'(a_{32}) \end{pmatrix}$$

We obtain the following result, again reassigned to $\delta$:

$$\delta = \begin{pmatrix} \frac{\partial{L}}{\partial{y_1}} \times \phi_{13}'(a_{13}) \times w_{13}^1 \times \phi_{12}'(a_{12}) + \frac{\partial{L}}{\partial{y_2}} \times \phi_{23}'(a_{23}) \times w_{23}^1 \times \phi_{12}'(a_{12}) + \frac{\partial{L}}{\partial{y_3}} \times \phi_{33}'(a_{33}) \times w_{33}^1 \times \phi_{12}'(a_{12}) \\\ \\\ \frac{\partial{L}}{\partial{y_1}} \times \phi_{13}'(a_{13}) \times w_{13}^2 \times \phi_{22}'(a_{22}) + \frac{\partial{L}}{\partial{y_2}} \times \phi_{23}'(a_{23}) \times w_{23}^2 \times \phi_{22}'(a_{22}) + \frac{\partial{L}}{\partial{y_3}} \times \phi_{33}'(a_{33}) \times w_{33}^2 \times \phi_{22}'(a_{22}) \\\ \\\ \frac{\partial{L}}{\partial{y_1}} \times \phi_{13}'(a_{13}) \times w_{13}^3 \times \phi_{32}'(a_{32}) + \frac{\partial{L}}{\partial{y_2}} \times \phi_{23}'(a_{23}) \times w_{23}^3 \times \phi_{32}'(a_{32}) + \frac{\partial{L}}{\partial{y_3}} \times \phi_{33}'(a_{33}) \times w_{33}^3 \times \phi_{32}'(a_{32}) \\\ \end{pmatrix}$$

The first row of this matrix corresponds to equations $(4)-(7)$, the second row to equations $(8)-(11)$, and the third row to equations $(12)-(15)$.

Then, the equations $(4)-(15)$ become (under the notation $\nabla^{(2)}$ - the partial derivatives of $L$ with respect to the weights on the $2^{nd}$ layer):

$$\nabla^{(2)} = \begin{pmatrix} \frac{\partial{L}}{\partial{w_{12}^0}} & \frac{\partial{L}}{\partial{w_{12}^1}} & \frac{\partial{L}}{\partial{w_{12}^2}} & \frac{\partial{L}}{\partial{w_{12}^3}} \\\ \\\ \frac{\partial{L}}{\partial{w_{22}^0}} & \frac{\partial{L}}{\partial{w_{22}^1}} & \frac{\partial{L}}{\partial{w_{22}^2}} & \frac{\partial{L}}{\partial{w_{22}^3}} \\\ \\\ \frac{\partial{L}}{\partial{w_{32}^0}} & \frac{\partial{L}}{\partial{w_{32}^1}} & \frac{\partial{L}}{\partial{w_{32}^2}} & \frac{\partial{L}}{\partial{w_{32}^3}} \end{pmatrix} = \delta \cdot \begin{pmatrix} 1 & h_{11} & h_{21} & h_{31} \end{pmatrix}$$

Math (cont'd)