# Optimization of Signal Detection Using Deep CNN in Ultra-Massive MIMO
## 2 Materials and Methods
### Machine Learning Method  
#### 2.3.1 Extreme Learning Machine (ELM)  
This study introduces an accelerated learning method for single hidden layer feedforward networks (SLFNs) [20,21], which is designed to effectively manage networks with \(\widetilde{N}\) hidden neuron nodes. Additionally, the method is tailored to handle \(\widetilde{N}\leq\) N training samples. The focus is on significantly speeding up the learning process in SLFNs, thereby enhancing training efficiency and effectiveness. Extreme learning machine (ELM), an algorithm within the machine learning domain that employs neural networks, is known for its impressive efficiency in regression tasks and an accelerated learning rate. This was substantiated through both theoretical analysis and empirical validation. The architectural framework of ELM is detailed in Figure 3, providing a clear visual representation of its structure. n and m respectively denote the numbers of input and output data. The analysis utilizes the ELM training process with N training samples \((X_{i},t_{i})\), where \(X_{i}=[X_{i1},X_{i2},\dots,X_{in}]^{T}\)represents the input data, and \(t_{i}=[t_{i1},t_{i2},\dots,t_{im}]^{T}\) the output data. The approach to detecting patterns in SLFNs employs a mathematical model, outlined as follows:  
\[\hat{t}_{j}=\sum_{i=1}^{\tilde{N}}\beta\,o\big{(}c_{i}\cdot X_{j}+V_{i}\big{)},j =1,2,\dots,N \tag{11}\]  
\(c_{i}=[c_{i1},c_{i2},\dots,c_{in}]^{T}\) is the input weight vector that connects to the \(i\)-th hidden neuron, and \(\beta=[\beta_{1},\beta_{i2},\dots,\beta_{im}]^{T}\) refers to the output weight vector of the \(i\)-th hidden neuron node. The term \(V_{i}\) is the bias of the \(i\)-th hidden neuron node, and \(c_{i}\) indicates the activation function in SLFNs. Unlike other machine learning algorithms, ELM can randomly assign the input weight \(c_{i}\) and bias \(V_{i}\). A significant feature of ELM, contributing to its regression performance, is its verification of zero error, which allows an accurate approximation of all \(\mathbf{N}\) samples as \(i=1 Ntj-tj=0\); i.e., the \(\mathbf{N}\) equations can thus be compactly expressed as  
\[H\beta=T\;,\;\text{where}\;\beta=\begin{bmatrix}\beta_{1}\\ \vdots\\ \beta_{\tilde{N}}\end{bmatrix}^{T}\;,\;T=\begin{bmatrix}T_{1}\\ \vdots\\ T_{N}\end{bmatrix}^{T}_{N\times m} \tag{12}\]  
\[H(c_{i},\cdots,c_{N},V_{i},\cdots,V_{N},X_{i},\cdots,X_{N})=\begin{bmatrix}o( c_{1}\cdot X_{1}+V_{1})&\cdots&o(c_{\tilde{N}}\cdot X_{1}+V_{\tilde{N}})\\ \vdots&\cdots&\vdots\\ o(c_{1}\cdot X_{N}+V_{1})&\cdots&o(c_{\tilde{N}}\cdot X_{N}+V_{\tilde{N}}) \end{bmatrix}_{N\times N} \tag{13}\]  
where \(H\) is the neural network's hidden layer output matrix and \(T\) is the training data target matrix. The variable \(H\) represents the output matrix of the hidden layer in the neural network, whereas \(T\) represents the matrix that contains the training data targets. Moreover, \(\hat{\beta}\) was identified as the solution with the minimum magnitude in the least-squares approach, which is essential to the linear system of the ELM solution.  
\[\hat{\beta}=H^{p}T \tag{14}\]  
where \((*)^{p}\) is the Moore-Penrose pseudoinverse.