## 3 Proposed BICORN-RNN
### Random Mapping Input Space
Let \(\mathbf{W}_{d_{k}}^{\text{HI}}\) be the \(d_{k}-\text{th}\) row vector of \(\mathbf{W}^{\text{HI}}\) in Equation (1). Defining \(w_{d_{k}}\) such that \(\mathbf{W}_{d_{k}}^{\text{HI}}\)\(w_{d_{k}}=1\), then we have,  
\[ \begin{array}{ll}d_{kk}^{\prime}(t)&=f\Big{(}\mathbf{W}_{d_{k}}^{\text{HI}}( x(t)+w_{d_{k}})+\alpha\Big{)}\\ &=f\Big{(}\mathbf{W}_{d_{k}}^{\text{HI}}.x(t)+\mathbf{W}_{d_{k}}^{\text{HI}}.w _{d_{k}}+\alpha\Big{)}\\ &=f\Big{(}\mathbf{W}_{d_{k}}^{\text{HI}}.x(t)+1+\alpha\Big{)}\\ &\geq f\Big{(}\mathbf{W}_{d_{k}}^{\text{HI}}.x(t)+\alpha\Big{)}=h_{d_{k}}(t) \end{array}  \tag{11}\]  
\[y^{\prime}_{d_{k}}(t)=g\big{(}h^{\prime}_{d_{k}}(t)\big{)}\geq g\big{(}h_{d_{ k}}(t)\big{)}=y_{d_{k}}(t) \tag{12}\]  
where \(h_{d_{k}}(t)\),\(y_{d_{k}}(t)\) refer to the \(d_{k}-\text{th}\) element of the vector \(h(t)\), \(y(t)\) respectively. \(\alpha\) is the fixed term in Equation (1), regarded as a constant. In the TSC task, the loss function is often chosen as Cross-Entropy in Equation (8), and then,  
\[-\ln y^{\prime}_{d_{k}}(t)\leq-\ln y_{d_{k}}(t) \tag{13}\]  
which leads to \(L(\xi^{\prime},\theta)\leq L(\xi,\theta)\). It is important to note that learning the projection matrix may have a pure-perfect result in training sets, but not, regretfully, in testing sets, because of the overly confident strategy. On the premise of predictions \(y(t\text{-}m)\), \(m=1,\ldots,t-1\) in previous time-steps, learning-updating the elements of \(\mathbf{P}_{i}\) may suffer from over-fitting in all possibilities, which provided bad data or even a negative contribution in classification performance in the testing process. A conventional approach to avoid this is adding a regularization term in the loss function to constrain dynamic tuning. However, this would lead to additional parameters at each time-step, i.e., it will increase the computational complexity exponentially. Here, we might set the parameter matrix \(\mathbf{P}_{i}\) randomly without training, surprisingly, which is an agile and effective approach to solve both the overfitting and increasing computational complexity. It is worth explaining that we aim to train an excellent generalization RNNs with the support of previous prediction information, rather than focusing on decorating data in a supervised way. Furthermore, this modification will not raise the dimension of the input space, and avoid the curse of dimensionality. In fact, there is a large amount of computation in the training phase of RN-RNNs. The purpose of indexing from 1 to \(t-1\) is to force the retention of the information at earlier time-steps, highlighting its learning ability and reducing its dependency on long sequences. For more application scenarios, the sequence information at a very early time-step is often useless or redundant. Considering the limitation of computation and the demand of specific applications, we can modify Equation (10) to Equation (14) or Equation (15), as shown in following formulae:  
\[x^{\prime}(t)=\sigma\Bigg{(}x(t)+\sum_{i=t-N-1}^{t-1}\beta_{i}\mathbf{P}_{i}y (i)\Bigg{)} \tag{14}\]  
\[x^{\prime}(t)=\sigma\Bigg{(}x(t)+\sum_{i=1,1+M,1+2M,...}^{t-1}\beta_{i} \mathbf{P}_{i}y(i)\Bigg{)} \tag{15}\]Equation (10) indicates that the predictive information from the 1-th to (_t_-1)-th time-steps needs to be maintained and transmitted, to evoke the classification decision at the _t_-th time-step. However, the modified formulae have more pertinence in indexes with different meanings:  
Equation (14) indicates that only the prediction values from the (_t_-\(N\)-1)-th to (_t_-1)-th time-steps, (_N_+1)-points sequence, are retained, because the value other than \(N\)-steps is too distant, its importance is greatly reduced, and it will not be involved in the classification decision at the _t_-th time-step. Equation (15) indicates that the prediction values from the 1-th to (_t_-1)-th time-steps are sampled at equal intervals \(M\).