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      where is an auxiliary variable with independent marginal , and is some vector‐valued function parameterized by . With this reparameterization trick, the variational lower bound can be estimated by sampling a batch of from :

      (8)

      where and . The selections of and are discussed in detail in Kingma and Welling [17].

6 Recurrent Neural Networks

      6.1 Introduction

      The previously introduced models have the same assumptions on the data, that is, the independence among the samples and the fixed input size. However, these assumptions may not be true in some cases, thus limiting the application of these models. For example, videos can have different lengths, and frames of the same video are not independent, and sentences of an chapter can have different lengths and are not independent.

      A RNN is another modified DNN that is used primarily to handle sequential and time series data. In a RNN, the hidden layer of each input is a function of not just the input layer but also the previous hidden layers of the inputs before it. Therefore, it addresses the issues of dependence among samples and does not have any restriction on the input size. RNNs are used primarily in natural language processing applications, such as document modeling and speech recognition.

      6.2 Architecture

As illustrated in Figure 7, a general neural network takes in input and outputs . The output of one sample will not influence the output of another sample. To capture the dependence between inputs, RNN adds a loop to connect the previous information with the current state. The graph on the left side of Figure 8 shows the structure of RNN, which has a loop connection to leverage previous information.

RNN can work with sequence data, which has input as sequence or target as sequence or both. An input sequence data can be denoted as , where each data point is a real‐valued vector. Similarly, the target sequence can be denoted as . A sample from the sequence dataset is typically a pair of one input sequence and one target sequence. The right side of Figure 8 shows the information passing process. At , network takes in a random initialed vector together with and outputs , and then at , takes in both and and outputs . This process is repeated over all data points in the input sequence.

Figure 7 Feedforward network.

Figure 8 Architecture of recurrent neural network (RNN).

      Though multiple network blocks are shown on the right side of Figure

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