Think back to the feedforward networks we discussed earlier (or that you might know from introductory material). Each input sample is processed independently. If you feed the network the word "hot" and then the word "dog", the network has no inherent mechanism to know that "dog" followed "hot". It treats them as separate events. This is a major limitation when dealing with sequences, where order and context are fundamental. Sentences, stock prices, or musical notes derive meaning from their position relative to others.

This is where the hidden state ( $h_t$ ) enters the picture. It is the central concept that allows Recurrent Neural Networks to overcome the memory limitations of feedforward networks. You can think of the hidden state as the network's memory. At each time step $t$ , the RNN doesn't just process the current input $x_t$ ; it also incorporates information from the previous hidden state, $h_{t-1}$ .

Recall the core calculation for the hidden state:

$h_t = f(W_{hh}h_{t-1} + W_{xh}x_t + b_h)$

Notice the important term $W_{hh}h_{t-1}$ . This directly injects the information summarized in the previous hidden state $h_{t-1}$ (transformed by the weight matrix $W_{hh}$ ) into the calculation of the current hidden state $h_t$ . Because $h_{t-1}$ itself was calculated using $h_{t-2}$ , and $h_{t-2}$ using $h_{t-3}$ , and so on, the current hidden state $h_t$ becomes a function of the entire preceding sequence of inputs ( $x_0, x_1, ..., x_t$ ).

In essence, the hidden state acts as a running summary or a compressed representation of everything the network has "seen" up to the current time step. It carries context forward, allowing the network's output at time $t$ , $y_t$ , to be influenced not just by the current input $x_t$ , but also by the inputs that came before it.

The hidden state $h$ acts as the connection between time steps. Information from the input $x_t$ and the previous hidden state $h_{t-1}$ merge to form the current hidden state $h_t$ , which then influences the output $y_t$ and is passed on to the next time step $t+1$ . Weight matrices ( $W_{xh}, W_{hh}, W_{hy}$ ) govern these transformations.

It's important to understand that this memory isn't perfect or infinite. The hidden state is typically a fixed-size vector. As the network processes longer sequences, summarizing all relevant past information into this fixed-size representation becomes challenging. Information from the distant past might get diluted or overwritten by more recent inputs. This limitation is a significant factor in the development of more advanced architectures like LSTMs and GRUs, which we will explore later.

However, for many tasks involving short-to-medium term dependencies, the hidden state mechanism of a simple RNN provides the necessary memory that feedforward networks lack. It's the critical component that allows RNNs to learn patterns and relationships that unfold over time within sequential data. Without the hidden state propagating information across steps, an RNN would effectively collapse back into a standard feedforward network, losing its ability to model sequences.

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