Long Short-Term Memory (LSTM):

Long Short-Term Memory (LSTM) networks are a type of Recurrent Neural Network (RNN) designed to handle sequential data and overcome the limitations of traditional RNNs, such as the vanishing gradient problem. LSTMs are particularly effective for tasks involving sequences, such as time series forecasting, natural language processing, and speech recognition.

Here’s a step-by-step explanation of how LSTMs work:

Step 1: Understand the Basic Structure

An LSTM network consists of:

Input Gate: Controls how much of the new input should be added to the cell state.
Forget Gate: Decides how much of the existing cell state should be discarded.
Cell State: The internal memory of the LSTM that carries information across time steps.
Output Gate: Determines how much of the cell state should be outputted to the next layer.

Step 2: Initialize Parameters

Initialize the parameters for the LSTM, including:

Weights: For the gates and cell state transitions.
Biases: For the gates and cell state transitions.

Step 3: Forward Propagation

For each time step $t$ $t$ , compute the following:

Input Gate Calculation:
- Compute the input gate’s activation: $i_t = \sigma(W_i \cdot [h_{t-1}, x_t] + b_i)$
  - where $W_i$ $W_{i}$ is the weight matrix for the input gate, $h_{t-1}$ $h_{t - 1}$ is the previous hidden state, $x_t$ $x_{t}$ is the current input, and $b_i$ $b_{i}$ is the bias. The sigmoid function $\sigma$ $σ$ outputs values between 0 and 1.

Forget Gate Calculation:
- Compute the forget gate’s activation: $f_t = \sigma(W_f \cdot [h_{t-1}, x_t] + b_f)$
  - where $W_f$ $W_{f}$ is the weight matrix for the forget gate, and $b_f$ $b_{f}$ is the bias.
Cell State Update:
- Compute the candidate cell state: $\tilde{C}_t = \tanh(W_c \cdot [h_{t-1}, x_t] + b_c)$
  - where $W_c$ $W_{c}$ is the weight matrix for the cell state candidate, and $b_c$ $b_{c}$ is the bias.
  - Update the cell state: $C_t = f_t \cdot C_{t-1} + i_t \cdot \tilde{C}_t$
    - where $C_{t-1}$ $C_{t - 1}$ is the previous cell state.
Output Gate Calculation:
- Compute the output gate’s activation: $o_t = \sigma(W_o \cdot [h_{t-1}, x_t] + b_o)$
  - where $W_o$ $W_{o}$ is the weight matrix for the output gate, and $b_o$ $b_{o}$ is the bias.
  - Compute the hidden state: $h_t = o_t \cdot \tanh(C_t)$
    - where $\tanh(C_t)$ $tanh (C_{t})$ is the cell state after applying the tanh function.

Step 4: Update the Model

Backpropagate the errors through time (BPTT) to update the LSTM weights and biases using an optimization algorithm like Gradient Descent or Adam. The gradients are computed with respect to the loss function and the parameters of the LSTM.

Step 5: Make Predictions

Use the trained LSTM network to make predictions based on new input sequences. The predictions are derived from the final hidden state $h_t$ $h_{t}$ of the network.