## 3 Proposed BICORN-RNN
### Convolutional Rearrange  
Convolution can be regarded as a weighted superposition operation of one function upon another. The most typical application is convolution neural networks (CNNs) in deep learning, which has been successfully used in object recognition, audio classification, and image processing, because of key properties: spatially shared weights and spatial pooling [35].  
RNNs pay more attention to the transmission of information between neurons at each time-step while ignoring the interconnection tendency of several points themselves, such as smoothness, volatility, periodicity, etc. Convolution is an ideal operation for extracting the indescribable local features of images, speech, and time series [36]. Therefore, it is not difficult to consider that an appropriate combination of convolution and RNNs, CO-RNNs, can enhance the ability to recollect and transmit the indispensable local features in online learning systems [37]. Actually, this \"graft\" can innovatively remove background noise and augment discrimination of data representations for classification tasks.  
Specifically, we add a convolutional layer, a pooling layer, and a full-connected layer behind the input layer of RNNs, as shown in Figure 5. We do not recommend the convolutional operation behind the hidden layer, as it would disrupt the dynamic delivery of RNNs.  
Concretely, similar to the common CNNs structure, the input example \(x(t)\) at the \(t\)-th time-step is convolved with learnable kernels \(\mathbf{K}_{j}^{c}\) (\(j=1,...,n_{\text{C}}\)) and put through a hyperbolic tangent function \(f_{c}\) to form the output feature maps \(\text{C}_{j}\) (\(j=1,...,n_{\text{C}}\)) of the convolutional layer. We use the average-pooling function \(\mathit{aver}(\cdot)\) to obtain the pooling feature maps \(\text{P}_{0j}\) (\(j=1,...,n_{\text{C}}\)). Then, \(\text{P}_{0j}\) are projected onto the full-connected layer input maps \(\text{P}_{0j}\)\({}^{\prime}\) (\(j=1,...,n_{\text{F}}\)), by the linked weight \(\mathbf{W}^{f}\in\mathbb{R}^{n_{\text{C}}\times n_{\text{F}}}\). \(\text{P}_{0j}\)\({}^{\prime}\) make convolutions with the full-connected kernel \(\mathbf{K}_{j}^{f}\) (\(j=1,...,n_{\text{F}}\)), and are then activated by hyperbolic  
Figure 5: Illustration of COnvolutional Recurrent Neural Networks (CN-RNNs) architecture and the procedures of convolution and pooling operations.  
tangent function \(f_{c}\). Thus, the concatenation of each state\(F_{j}\) (\(j=1,...,n_{\text{F}}\)) is finished, getting the new input series \(\mathbf{F}_{t}\) for CO-RNNs. The complete process is described as:  
\[\begin{array}{ll}\mathbf{C}_{j}&=f_{c}\Big{(}x(t)\otimes\mathbf{K}_{j}^{c}+ \mathbf{B}_{j}^{c}\Big{)}\\ \mathbf{Po}_{j}&=\text{down}\big{(}\mathbf{C}_{j}\otimes\text{{a}ver}(2\times 1 )\big{)}\\ F_{j}&=f_{c}\Big{(}\mathbf{Po}_{j}^{\prime}\otimes\mathbf{K}_{j}^{f}+\mathbf{B }_{j}^{f}\Big{)}\\ &=f_{c}\Big{(}\bigg{(}\sum\limits_{i=1}^{n_{\text{C}}}\mathbf{W}_{ji}^{\prime} \mathbf{Po}_{i}\bigg{)}\otimes\mathbf{K}_{j}^{f}+\mathbf{B}_{j}^{f}\Big{)}\\ \mathbf{F}_{t}&=[F_{1},F_{2},...,F_{n_{\text{F}}}]\end{array} \tag{16}\]  
where \"\(\otimes\)\" denotes the convolutional operator, \(\mathbf{B}_{j}\) is the bias, down\((\cdot)\) represents a sub-sampling function. The fixed parameters are set by K-folder cross validation. The number of kernels \(n_{\text{C}}\) and \(n_{\text{F}}\) are set to 10 and 30 manually, and the size of \(\text{a}ver(\cdot)\) plane is \(2\times 1\). Note that the random mapping described above is available and can coexist with convolutional rearranging operations in this subsection.