What is the general equation for a recurrent neural network, particularly for the computation of a hidden state?

In the context of recurrent neural networks (RNNs), the computation of the hidden state is crucial for processing sequences of data, such as time series or natural language. The correct equation represents how the hidden state at the current time step is calculated based on the input vector from the current time step and the hidden state from the previous time step.

The equation specifies that the hidden state ( Z ) is computed as the sum of two terms: the first term, which is the current input ( X ) multiplied by the weight matrix ( W ), and the second term, which involves the hidden state from the previous time step ( H_{t-1} ) multiplied by another weight matrix. The bias term ( b ) is added to the result.

The notation also indicates that the weight matrices are transposed, which is commonly used to maintain the right dimensions for matrix multiplication. By incorporating the previous hidden state, this formulation effectively allows the network to retain information across time steps, a key characteristic that enables RNNs to handle sequential dependencies.

This equation aligns with the general structure of RNNs, capturing the essence of how they maintain context and update the hidden state based on both the current input and the previous hidden state.