# Fingerprint Classification Based on Multilayer Extreme Learning Machines
## 3 Theoretical Foundations
### Extreme Learning Machines
#### 3.2.2 Multilayer ELM  
The features of the dataset influence the learnability during both the training and the testing phases for the ELM algorithm. Consequently, feature design becomes essential, as it allows for a more effective representation of the data's inherent structure. However, this task requires expertise in the area, as well as human ingenuity, to develop suitable features. To address this challenge, AEs can be employed to train multilayer perceptron networks. An AE can be defined as an SLFN with multiple hidden layers [29]--in this sense, an unsupervised training paradigm, where the outputs and inputs are the identical. The goal of the AE is to reproduce the input signal as accurately as possible through the connections between its neurons and the biases in its hidden layers. This paper presents an ELM based on AEs that learns singular feature values to learn feature representations and classify fingerprints while reducing the penetration rate in identification databases. An AE-based multilayer M-ELM is constructed with a training architecture divided into two distinct phases: an unsupervised hierarchical feature representation phase and a supervised label classification phase (see Figure 2). In the initial phase, the ELM-AE is designed to extract sparse multilayer features from the input data using unsupervised learning. In the subsequent phase, a standard ELM is employed for final decision making. The detailed processes for each phase are presented in Algorithm 2 (unsupervised feature extraction using ELM-AE) and Algorithm 3 (supervised classifier learning phase (standard ELM)). These algorithms outline the step-by-step procedures for both phases, ensuring clarity and reproducibility of the proposed approach. Building on the supervised learning process outlined in the previous subsection, the training procedure for the unsupervised AE block in the M-ELM architecture is systematically developed. This two-phase approach ensures robust feature extraction and efficient classification. An ELM-AE can be conceptualized as a neural network comprising an input layer, a hidden layer, and an output layer (see Figure 3). It is characterized by having a number (\(n\)) of neurons in the input layer, \(d\) neurons in the hidden layer, and \(n\) neurons in the output layer and employing a nonlinear activation function (\(f(\cdot)\)) for the hidden neurons. In contrast, the neurons in the output layer are typically limited to performing summation operations. For a set of \(M\) distinct samples (\(x_{i}\in\mathbb{R}^{M\times j},i=1,\ldots,M\)), the weights (\(c\)) and biases (\(b\)) corresponding to the hidden-layer nodes are randomly generated based on a probability density function. These weights and biases are subject to the orthogonality condition, which enhances the training capacity of the neural network. The output of the hidden layer in the ELM-AE and the parameters between the input and output layers can be mathematically expressed as follows:  
\[h=f(cx+b), \tag{3}\]  
subject to  
\[c^{T}c=I, \tag{4}\]  
and  
\[b^{T}b=1, \tag{5}\]  
where \(c=[c_{1},\ldots,c_{M}]\) denotes the orthogonal random weight between the input and hidden layer, \(b=[b_{1},\ldots,b_{M}]\) represents the orthogonal random biases between the input nodes and the hidden nodes, \(I\) corresponds to the identity matrix, and \(\epsilon\) refers to the output of the connections between the last hidden layer and the output nodes. As demonstrated, the standard ELM is modified for unsupervised learning by using the input data as the output data. It is worth mentioning that the ELM-AE is designed to effectively extract input features through three distinct approaches: (1) a compressed approach, which represents entities from a higher dimensional input data space to a lower dimensional entity space; (2) a sparse approach, which is the opposite of the compressed approach; and (3) an equal-dimensional approach, where entities maintain their dimensionality in order to maximize the performance of the ELM.