more boring typos

2018-10-19 06:07:18 +02:00
parent a067cf3e04
commit 092fdecd9b
18 changed files with 384 additions and 384 deletions
@@ -370,40 +370,40 @@ The feedforward step is similar to as for the neural netowork, but now consideri

 <p>
 The \( i \)-th neuron at layer \( l \) recieves the result
-\( \vec{x}_j^{(l-1),\text{hidden} } \) from the \( j \)-th neuron at layer
+\( \hat{x}_j^{(l-1),\mathrm{hidden} } \) from the \( j \)-th neuron at layer
 \( l-1 \). The \( i \)-th neuron at layer \( l \) weights all of the elements in
-\( \vec{x}_j^{(l-1),\text{hidden} } \) with a weight vector \( \vec
-w_{i,j}^{(l), \ \text{hidden} } \) with as many weigths as there are
-elements in$\vec{x}_j^{(l-1),\text{hidden} }$, and adds a bias
-\( b_i^{(l), \ \text{hidden} } \):
+\( \hat{x}_j^{(l-1),\mathrm{hidden} } \) with a weight vector \( \vec
+w_{i,j}^{(l), \ \mathrm{hidden} } \) with as many weigths as there are
+elements in$\hat{x}_j^{(l-1),\mathrm{hidden} }$, and adds a bias
+\( b_i^{(l), \ \mathrm{hidden} } \):

 $$
 \begin{aligned}
-z_{i,j}^{(l),\ \text{hidden}} &= b_i^{(l), \ \text{hidden}} + \big(\vec{w}_{i}^{(l), \ \text{hidden}}\big)^T\vec{x}_j^{(l-1),\text{hidden} } \\
+z_{i,j}^{(l),\ \mathrm{hidden}} &= b_i^{(l), \ \mathrm{hidden}} + \big(\hat{w}_{i}^{(l), \ \mathrm{hidden}}\big)^T\hat{x}_j^{(l-1),\mathrm{hidden} } \\
 &= 
 \begin{pmatrix}
-b_i^{(l), \ \text{hidden}}  & \big(\vec{w}_{i}^{(l), \ \text{hidden}}\big)^T
+b_i^{(l), \ \mathrm{hidden}}  & \big(\hat{w}_{i}^{(l), \ \mathrm{hidden}}\big)^T
 \end{pmatrix}
 \begin{pmatrix}
 1  \\
-\vec{x}_j^{(l-1),\text{hidden} }
+\hat{x}_j^{(l-1),\mathrm{hidden} }
 \end{pmatrix} 
 \end{aligned}
 $$

 <p>
-The output from the \( i \)-th neuron at the hidden layer \( l \) becomes a vector \( \vec{z}_{i}^{(l),\ \text{hidden}} \):
+The output from the \( i \)-th neuron at the hidden layer \( l \) becomes a vector \( \hat{z}_{i}^{(l),\ \mathrm{hidden}} \):

 $$
 \begin{aligned}
-\vec{z}_{i}^{(l),\ \text{hidden}} &= \Big(  b_i^{(l), \ \text{hidden}} + \big(\vec{w}_{i}^{(l), \ \text{hidden}}\big)^T\vec{x}_1^{(l-1),\text{hidden} }, \ \dots \ , \  b_i^{(l), \ \text{hidden}} + \big(\vec{w}_{i}^{(l), \ \text{hidden}}\big)^T\vec{x}_{N_{hidden}^{(l-1)}}^{(l-1),\text{hidden} } \Big) \\
+\hat{z}_{i}^{(l),\ \mathrm{hidden}} &= \Big(  b_i^{(l), \ \mathrm{hidden}} + \big(\hat{w}_{i}^{(l), \ \mathrm{hidden}}\big)^T\hat{x}_1^{(l-1),\mathrm{hidden} }, \ \dots \ , \  b_i^{(l), \ \mathrm{hidden}} + \big(\hat{w}_{i}^{(l), \ \mathrm{hidden}}\big)^T\hat{x}_{N_{hidden}^{(l-1)}}^{(l-1),\mathrm{hidden} } \Big) \\
 &= 
 \begin{pmatrix}
-b_i^{(l), \ \text{hidden}} & \big(\vec{w}_{i}^{(l), \ \text{hidden}}\big)^T
+b_i^{(l), \ \mathrm{hidden}} & \big(\hat{w}_{i}^{(l), \ \mathrm{hidden}}\big)^T
 \end{pmatrix}
 \begin{pmatrix}
 1 & 1 & \dots & 1 \\
-\vec{x}_{1}^{(l-1),\text{hidden} } & \vec{x}_{2}^{(l-1),\text{hidden} } & \dots & \vec{x}_{N_{hidden}^{(l-1)}}^{(l-1),\text{hidden} }
+\hat{x}_{1}^{(l-1),\mathrm{hidden} } & \hat{x}_{2}^{(l-1),\mathrm{hidden} } & \dots & \hat{x}_{N_{hidden}^{(l-1)}}^{(l-1),\mathrm{hidden} }
 \end{pmatrix}
 \end{aligned}
 $$