# Harris Hawks Sparse Auto-Encoder Networks for Automatic Speech Recognition System
## Appendix A Sparse Autoencoding
### Model Fine-Tuning using Haris Hawk Optimisation  
The network has to be fine-tuned to improve the ASR system's performance. The network parameters are adjusted for reducing the error-rate classification problem. To achieve this goal, the parameters are analyzed in the N population size with an extreme number of iterations T. The output network parameters are identified according to the rabbit food searching characteristics and the objective function.  
In the population, \((X_{i};i=1,2,\ldots,N)\), the fitness value of hawks is computed with respect to the rabbit's position. Initially, the hawk's position \(X_{ra}\) is the best location. For every iteration, the rabbit's initial energy \((E_{0})\) is checked along with the jump strength \((j)\), defined in Equation (A3).  
\[\left.\begin{array}{c}E_{0}=2\;rand()\\ J=2(1-rand())\end{array}\right\}\] (A3)This computed energy and jump strength value is updated for every jump and food searching process using Equation (A4), as it is used to identify the best network parameter value.  
\[E=2E_{0}\bigg{(}1-\frac{t}{T}\bigg{)}\] (A4)  
The energy value is updated according to the prey energy value while escaping (E) on the maximum iteration T with initial energy \(E_{0}\). The \(E_{0}\) value is selected between (\(-1\), \(1\)), which determines the hawk's condition. If the value \(E_{0}\) is reduced between \(0\) to \(-1\). If \(|E|\geq 1\) (exploration phase), then it moves to a different location, and it updates continuously for selecting the effective network parameter. If \(|E|<1\), then the rabbit is in the neighborhood phase searching for the solution in the exploitation step. As said, if \(|E|\geq 1\), it is in the exploration phase; then, the location vector is updated using Equation (A5).  
\[X(t+1)=\left\{\begin{array}{c}X_{rand}(t)-r_{1}|X_{rand}(t)-2r_{2}X(t)|\;q\geq 0.5\\ (X_{rabbit}(t)-X_{m}(t))-r_{3}(LB+r_{4}(UB-LB))\;q<0.5\end{array}\right.\] (A5)  
The updating of the \(|E|\geq 1\) condition is the next iteration of the hawk's position \(X(t+1)\) updating process that is done by the rabbit position \(X_{rabbit}(t)\), the hawk's current position vector \(X(t)\), and the random numbers \(r_{1}\), \(r_{2}\), \(r_{3}\), \(r_{4}\) and q having values of (0, 1). For every iteration, the lower (LB) and upper (UB) boundary of the searching region is considered with the current population \(X_{rand}(t)\) and the position \(X_{m}\) of hawks. Suppose \(|E|<1\) (\(r\geq 0.5\)_and_\(|E|\geq 0.5\)), then it goes to the exploitation phase, and the energy factor is updated using Equation (A6).  
\[X(t+1)=\Delta X(t)-E|JX_{rabbit}(t)-X(t)|\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\;\;  
Here, random values between (0, 1) are selected for \(u\) and \(v\) and 1.5 is the constant value for \(\beta\). At last, the \(|E|<1\) (\(r<0.5\)_and_\(|E|<0.5\)), then the updating process is done by using Equation (A12).  
\[X(t+1)=\left\{\begin{array}{l}Y\;\text{if}\;F(Y)<F(X(t))\\ Z\;\text{if}\;F(Z)<F(X(t))\end{array}\right.\] (A12)  
Here, \(Y\) and \(Z\) are computed as follows,  
\[Y=X_{rabbit}(t)-E|JX_{rabbit}(t)-X_{m}(t)|\] (A13)  
\[Z=Y+S*LF(D)\] (A14)  
\[X_{m}(t)=\frac{1}{N}\sum_{i=1}^{N}X_{i}(t)\] (A15)  
According to this process, the network parameters are updated continuously, which reduces the recognition issues and the existing research problem. Based on the encoder network performance, the convolute network identifies the speech by examining the acoustic, lexicon, and language model effectively.