# Fingerprint Classification Based on Multilayer Extreme Learning Machines
## 5 Results and Discussion
### Hyperparameter Optimization  
For this study, the sigmoid function (\(g(x)=1/[1-exp(x)]\)) is considered as the activation function, since it guarantees the universal approximation of the SLFNs, making it suitable for any ELM algorithm dealing with nonlinear problems. In addition, the regularization parameter (C) is omitted across all ELM versions for simplicity. It should be noted that the inclusion of the regularization parameter (C) in the model serves to prevent overfitting by balancing the training error and the norm of the output weights. Hyperparameter optimization for the different ELM models is first performed heuristically with the aim of maximizing their performance while minimizing complexity, thereby avoiding overtraining. This process considers various feature extractors (Capelli02, Hong08, and Liu10) and fingerprint qualities (Default, HQNoPert, and VQAndPert). Figures 5-7 illustrate the performance in terms of the number of hidden neurons in the original ELM (single hidden layer), showing results from both the training and validation stages. The training results demonstrate the successful convergence of single-hidden-layer networks trained with the ELM algorithm while highlighting the issue of overfitting that occurs as network complexity (i.e., the number of hidden neurons) increases. The validation results, however, are more indicative of the generalization capacity of the model, since they reflect performance on  
Figure 4: Samples concerning fingerprint image quality: (**a**) default; (**b**) HQNoPert; (**c**) VQAndPert. unseen data. For each descriptor, there is a number of hidden neurons that maximizes the generalization ability of the ELM. This number is only slightly dependent on the fingerprint quality. Nevertheless, for robustness and generalization purposes (so that the results are independent of the fingerprint quality), the following numbers of hidden neurons are adopted: 1000, 2000, and 750 for the Capelli02, Hong08, and Lui10 descriptors, respectively. The accuracies achieved by these numbers of neurons are highlighted with markers. Note that the number of hidden neurons is constrained to 8000 samples, since, theoretically, it has been shown that to achieve the best performance of an ELM, the number of hidden neurons must be less than the number of samples. The following subsection discusses the results regarding fingerprint quality and descriptors in a particular and general manner. Figure 5: The accuracy in the training and validation phases in terms of the number of hidden neurons of the original ELM, considering Capelli02 as a descriptor. Figure 6: The accuracy vs. the number of hidden neurons of the original ELM in the training and validation phases when the descriptor corresponds to Hong08. The results regarding the optimization of two- and three-hidden-layer ELM models (M-ELM) are presented below in Figures 8-13. In Figures 8-10, performance is displayed as contour plots, showing the relationship between the number of neurons in the first and second layers. An irregular zone can be observed where the accuracy reaches its highest value, which is slightly dependent on the quality and, more noticeably, on the descriptor. To avoid overfitting, the optimization yields the following configuration: both the first and second hidden layers contain 1000 neurons (\(N1=N2=1000\)). Increasing the complexity of the ELM (e.g., adding more neurons in each hidden layer) does not necessarily maximize performance, as evidenced by the significant decrease in accuracy when \(N1=N2=7000\). Furthermore, the optimal performance is achieved when the first hidden layer contains more neurons than the second. This is because one of the functions of unsupervised learning in ELM autoencoders is to derive meaningful feature representations at the input level. Thus, simplifying the network architecture in deeper layers is logical to maintain optimal performance.