## 5 Discussion  
The main goal of an autoencoder is to learn a compressed representation of the input and then reconstruct it within a reasonable computation time. This study proposes a robust sparse optimization-based method for training ELM-AE networks. With the control parameters specified in the results section, we see that the training times for G-FISTA and LC-FISTA are less than for FISTA. Table 9 shows these important results in terms of percentages. For the CIFAR10, UMist, MNIST, NORB, SCars, and Caltech256 databases, the average computation times for G-FISTA are 44.48%, 14.42%, 35.87%, 42.97%, 48.42%, and 39.43% faster than for FISTA, respectively. The LC-FISTA algorithm achieved speedups of 42.41%, 18.71%, 27.70%, 47.93%, 47.32%, and 43.90%. In addition, the indicators recorded in Tables 4-8 are coherent, and show comparable performance for all databases. The proposed training scheme is an important automatic learning method since it can help make accurate decisions in shorter periods. Consequently, incorporating fast optimization techniques into the autoencoder architecture can help improve the performance of current models, and particularly the first-order methods of the shrinkage-thresholding class.  
The proximal step of FISTA requires the main computational effort, and this also holds true for G-FISTA and LC-FISTA. The additional computation required in G-FISTA and LC-FISTA increases the convergence speed, but the execution cost is higher. However, this problem can be controlled, since G-FISTA and LC-FISTA converge with fewer iterations.  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Algorithm**} & \multicolumn{5}{c}{**RMSE**} & \multicolumn{5}{c}{**Training Time (Minutes)**} \\ \cline{2-9}  & **UMist** & **NORB** & **SCars** & **Caltech256** & **UMist** & **NORB** & **SCars** & **Caltech256** \\ \hline FISTA & 0.0211 & 0.0001 & 0.0537 & 0.0494 & 49 & 21 & 44 & 139 \\ G-FISTA & 0.0225 & 0.0001 & 0.0537 & 0.0494 & 40 & 12 & 27 & 85 \\ LC-FISTA & 0.0220 & 0.0001 & 0.0537 & 0.0494 & 37 & 11 & 26 & 70 \\ \hline \hline \end{tabular}
\end{table}
Table 8: The performance of each algorithm for the \(\lambda\) chosen in Table 3.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{\(L\)} & \multirow{2}{*}{**Algorithm**} & \multicolumn{5}{c}{**Training Time (Seg)**} \\ \cline{3-8}  & & **CIFAR10** & **UMist** & **MNIST** & **NORB** & **SCars** & **Caltech256** \\ \hline
100 & G-FISTA & 43.84 & 17.80 & 34.31 & 45.85 & 50.99 & 43.82 \\  & LC-FISTA & 41.10 & 19.86 & 25.52 & 49.97 & 46.07 & 48.07 \\
500 & G-FISTA & 44.34 & 10.07 & 37.43 & 45.26 & 49.34 & 37.83 \\  & LC-FISTA & 43.02 & 16.90 & 29.89 & 49.87 & 48.91 & 42.75 \\
1000 & G-FISTA & 45.25 & 9.93 & — & 41.95 & 47.95 & 40.49 \\  & LC-FISTA & 43.10 & 15.81 & — & 47.19 & 48.92 & 44.09 \\
1500 & G-FISTA & — & 16.26 & — & 40.63 & 47.18 & 38.14 \\  & LC-FISTA & — & 19.57 & — & 45.86 & 47.41 & 43.32 \\
2000 & G-FISTA & — & 18.05 & — & 41.14 & 46.63 & 36.88 \\  & LC-FISTA & — & 21.37 & — & 46.76 & 45.28 & 41.26 \\ \hline Av. time & G-FISTA & 44.48\% & 14.42\% & 35.87\% & 42.97\% & 48.42\% & 39.43\% \\ Av. time & LC-FISTA & 42.41\% & 18.71\% & 27.70\% & 47.93\% & 47.32\% & 43.90\% \\ \hline \hline \end{tabular} Note: The formula \(\%=(1-A_{1}/A_{2})\times 100\) compares the computation time of algorithms \(A_{1}\) and \(A_{2}\).  
\end{table}
Table 9: Average speeds of the G-FISTA and LC-FISTA algorithms compared with FISTA for training of the ELM-AE.  
In the experimental evaluation described in Section 4, we raised the complexity of the problem by increasing the number of neurons in the hidden layer to get a clearer idea of the performance when solving applications involving higher dimensional datasets. Based on this evaluation criterion, we see that the proposed method can be configured to solve large-scale problems, since the vector operations involved in the ELM-AE architecture and the proposed mechanism are block-separable.