2.-Statistics/rnn.md

@@ -49,30 +49,30 @@ An **LSTM** is a type of RNN designed to mitigate the vanishing gradient problem
 The LSTM cell updates are governed by the following equations:
 
 1. **Forget Gate**: Decides what information to forget:
-   $$
-   f_t = \sigma(W_f \cdot [h_{t-1}, x_t] + b_f)
-   $$
+$$
+f_t = \sigma(W_f \cdot [h_{t-1}, x_t] + b_f)
+$$
 
 2. **Input Gate**: Decides which new information to store:
-   $$
-   i_t = \sigma(W_i \cdot [h_{t-1}, x_t] + b_i)
-   $$
-   $$
-   \tilde{C}_t = \tanh(W_C \cdot [h_{t-1}, x_t] + b_C)
-   $$
+$$
+i_t = \sigma(W_i \cdot [h_{t-1}, x_t] + b_i)
+$$
+$$
+\tilde{C}_t = \tanh(W_C \cdot [h_{t-1}, x_t] + b_C)
+$$
 
 3. **Cell State Update**:
-   $$
-   C_t = f_t \cdot C_{t-1} + i_t \cdot \tilde{C}_t
-   $$
+$$
+C_t = f_t \cdot C_{t-1} + i_t \cdot \tilde{C}_t
+$$
 
 4. **Output Gate**: Decides the new hidden state:
-   $$
-   o_t = \sigma(W_o \cdot [h_{t-1}, x_t] + b_o)
-   $$
-   $$
-   h_t = o_t \cdot \tanh(C_t)
-   $$
+$$
+o_t = \sigma(W_o \cdot [h_{t-1}, x_t] + b_o)
+$$
+$$
+h_t = o_t \cdot \tanh(C_t)
+$$
 
 Where:
 - **$f_t$, $i_t$, $o_t$** are the forget, input, and output gates.
@@ -103,14 +103,14 @@ A **Gated Recurrent Unit (GRU)** is a simpler alternative to LSTMs. It has fewer
 The GRU uses two gates to control the flow of information:
 
 1. **Update Gate**: Decides how much past information to keep:
-   $$
-   z_t = \sigma(W_z \cdot [h_{t-1}, x_t] + b_z)
-   $$
+$$
+z_t = \sigma(W_z \cdot [h_{t-1}, x_t] + b_z)
+$$
 
 2. **Reset Gate**: Decides how much of the previous hidden state to forget:
-   $$
-   r_t = \sigma(W_r \cdot [h_{t-1}, x_t] + b_r)
-   $$
+$$
+r_t = \sigma(W_r \cdot [h_{t-1}, x_t] + b_r)
+$$
 
 The new hidden state is calculated as:
 $$