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This is a very basic question about the functionality of RNN / LSTM. My question is about the RNN outputs. Is there a single output for only the last element in the sequence, or is there an output for every element of the sequence?

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There is an output for every node, referred as hidden state in vanilla RNN (LSTM nodes has 2 outputs). In certain cases you only care about the output of the last node so you just avoid previous ones. But each hidden state is still used: Hidden states of the nodes transfered to subsequent nodes thus they are used to construct hidden state (output) of the last node in the sequence.

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