# Beyond Stochastic Gradient Descent for Matrix Completion Based Indoor Localization
## 4 Simulation Results and Discussion
### Recovery Performance and Localization Accuracy  
For evaluating the studied solutions based on GD and its variants combined with the trilateration process, we define the following two metrics. * EDM reconstruction error using the mean square error \[MSE=\frac{||\hat{\mathbf{X}}-\mathbf{X}_{true}||_{F}^{2}}{||\mathbf{X}_{true }||_{F}^{2}},\] (39)
* Localization error \[error_{loc}=\frac{||\hat{\mathbf{C}}-\mathbf{C}||_{2}}{n-m},\] (40)  
Figure 5: Energy fraction captured by the top \(l\) singular values. As mentioned before, \(\hat{\mathbf{C}}\) is the matrix of estimated coordinates of unknown nodes and \(\mathbf{C}\) are their real coordinates. We recall that parameters \(\alpha\), \(\lambda\), \(\epsilon\), \(\mu\), \(\rho\), \(\beta_{1}\) and \(\beta_{2}\) are adjusted by simulations, selected to ensure the best result in terms of localization mean error on a validation dataset. To find the best set of these parameters, an empirical process is conducted. Thus, several simulations are required to identify the optimal value of each parameter. The parameters used in the rest of the paper are presented in Table 2. The rest of algorithms includes GD, NAG, Adagrad, RMSProp and Adadelta. To verify that trilateration guarantees better localization accuracy when more distance information is provided, we firstly apply the trilateration with observed distances only. We can easily notice that it introduces the worse localization accuracy compared to tested combinations in both noisy and noiseless environments (Figure 6b,d,f). The localization accuracy is much better when we use a complete EDM than using only the observed distances. Moreover, to apply the trilateration process, at least three detected anchors are needed. If this is not the case, the sensor node cannot be localized. This problem can be solved when using a compete EDM containing all pairwise distances. Therefore, the combination of matrix completion technique and trilateration is highly recommended. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & & **Adam** & **The Rest of Algorithms** \\ \hline \multirow{8}{*}{**Sigma shadowing = 0**} & \(\alpha\) & 0.5 & 0.5 \\ \cline{2-4}  & \(\lambda\) & 0.9 & 0.9 \\ \cline{2-4}  & \(\epsilon\) & 0.01 & 0.01 \\ \cline{2-4}  & \(\mu\) & \(-\) & 0.9 \\ \cline{2-4}  & \(\rho\) & \(-\) & 0.9 \\ \cline{2-4}  & \(\beta_{1}\) & 0.001 & \(-\) \\ \cline{2-4}  & \(\beta_{2}\) & 0.001 & \(-\) \\ \hline \multirow{8}{*}{**Sigma shadowing = 2**} & \(\alpha\) & 0.5 & 0.5 \\ \cline{2-4}  & \(\lambda\) & 0.9 & 0.9 \\ \cline{2-4}  & \(\epsilon\) & 0.1 & 0.001 \\ \cline{2-4}  & \(\mu\) & \(-\) & 0.35 \\ \cline{2-4}  & \(\rho\) & \(-\) & 0.35 \\ \cline{2-4}  & \(\beta_{1}\) & 0.001 & \(-\) \\ \cline{2-4}  & \(\alpha\) & 0.5 & 0.5 \\ \cline{2-4}  & \(\lambda\) & 0.9 & 0.9 \\ \cline{2-4}  & \(\epsilon\) & 0.1 & 0.001 \\ \cline{2-4}  & \(\mu\) & \(-\) & 0.35 \\ \cline{2-4}  & \(\rho\) & \(-\) & 0.35 \\ \cline{2-4}  & \(\beta_{1}\) & 0.001 & \(-\) \\ \cline{2-4}  & \(\beta_{2}\) & 0.001 & \(-\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Values of different used parameters. In the first set of simulations, we investigate the performances, in terms of EDM reconstruction and localization error, of different cited algorithms in a noiseless environment as shown in Figure 5(a),b. It shows the effectiveness of the location estimation of the proposed schemes for 10 simulations. Figure 5(a) illustrates the EDM reconstruction error 'MSE'. The error estimation of pairwise known distances is in the order of \(10^{-30}\). The reconstruction error varies between \(10^{-3}\) and \(10^{-1}\) for different cited  
Figure 6: Algorithms’ performances (EDM: Euclidean distance matrix). algorithms. The best EDM reconstruction rate is obtained by Adadelta which produces a localization mean error of 0.47 m. This result is close to those obtained by GD, NAG and Adam. The difference is the number of iterations required to reach the convergence of the algorithm.