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mhjensen 9d2bcb57ab Update on neural networks
2017-11-22 14:45:03 +01:00

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<!-- navigation toc: --> <li><a href="#___sec12" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;Matrix-vector notation</a></li>
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<h3 id="___sec12" class="anchor">Matrix-vector notation </h3>
We can introduce a more convenient notation for the activations in a NN.
<p>
Additionally, we can represent the biases and activations
as layer-wise column vectors \( \vec{b}_l \) and \( \vec{y}_l \), so that the \( i \)-th element of each vector
is the bias \( b_i^l \) and activation \( y_i^l \) of node \( i \) in layer \( l \) respectively.
<p>
We have that \( \mathrm{W}_l \) is a \( N_{l-1} \times N_l \) matrix, while \( \vec{b}_l \) and \( \vec{y}_l \) are \( N_l \times 1 \) column vectors.
With this notation, the sum in becomes a matrix-vector multiplication, and we can write
the equation for the activations of hidden layer 2 in
$$
\begin{equation}
\vec{y}_2 = f_2(\mathrm{W}_2 \vec{y}_{1} + \vec{b}_{2}) =
f_2\left(\left[\begin{array}{ccc}
w^2_{11} &w^2_{12} &w^2_{13} \\
w^2_{21} &w^2_{22} &w^2_{23} \\
w^2_{31} &w^2_{32} &w^2_{33} \\
\end{array} \right] \cdot
\left[\begin{array}{c}
y^1_1 \\
y^1_2 \\
y^1_3 \\
\end{array}\right] +
\left[\begin{array}{c}
b^2_1 \\
b^2_2 \\
b^2_3 \\
\end{array}\right]\right)
\tag{13}
\end{equation}
$$
and we see that the activation of node \( i \) in layer 2 is
$$
\begin{equation}
y^2_i = f_2\Bigr(w^2_{i1}y^1_1 + w^2_{i2}y^1_2 + w^2_{i3}y^1_3 + b^2_i\Bigr) =
f_2\left(\sum_{j=1}^3 w^2_{ij} y_j^1 + b^2_i\right)
\tag{14}
\end{equation}
$$
which is in accordance with. Note that
This is not just a convenient and compact notation, but also
a useful and intuitive way to think about MLPs: The output is calculated by a series of matrix-vector multiplications
and vector additions that are used as input to the activation functions. For each operation
\( \mathrm{W}_l \vec{y}_{l-1} \) we move forward one layer.
<p>
<p>
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