# Utilizing Constructed Neural Networks for Autism Screening
## 2 Background Information
### MLP Description  
Artificial neural networks [56] are a type of machine learning tool that has been successfully applied to various in different scientific fields, including chemistry [57; 58; 59], economics [60; 61], and medicine [62; 63]. A neural network is commonly defined as a function \(N\left(\overset{\rightarrow}{x},\overset{\rightarrow}{w}\right)\) where \(\overset{\rightarrow}{x}\) is the input vector of patterns and \(\overset{\rightarrow}{w}\) is the so-called weight vector of the parameters that needs to be estimated. The estimation of this vector can be achieved by minimizing the error function defined as:  
\[E\left(N\left(\overset{\rightarrow}{x},\overset{\rightarrow}{w}\right) \right)=\sum_{i=1}^{M}\left(N\left(\overset{\rightarrow}{x}_{i},\overset{ \rightarrow}{w}\right)-t_{i}\right)^{2} \tag{1}\]  
The training data for the neural network are defined by the set of pairs \(\left(\overset{\rightarrow}{x}_{i},t_{i}\right)\). To minimize the error function, various methods such as the back propagation method [64] or the RPROP method [65] have been used, along with global optimization methods like quasi-Newton methods [66], genetic algorithms [67; 68], particle swarm optimization [69], and more. For the experiments in this study, a radial basis function (RBF) neural network [70] and two optimization methods were used: the limited-memory Broyden-Fletcher-Goldfarb-Shanno (BFGS) [71] method and a BFGS variant of Powell [72]. RBF neural networks are a specialized type of neural network with three layers: an input layer that receives data, a hidden layer using radial basis functions (typically Gaussian) for transforming inputs based on their distance to a central point, and an output layer that synthesizes the hidden layer's outputs to make predictions. RBF networks excel in localized input processing, quick learning, and complex function modeling, and they can approximate any continuous function with sufficient neurons. The RBF neural network excels in machine learning tasks like classification, regression, and clustering [70]. Particularly effective in high-dimensional spaces, RBF networks are known for their capability to tackle complex patterns efficiently. Their key advantages over other neural architectures include their rapid training and testing phases and the precise approximation of continuous functions [73]. BFGS is an iterative optimization technique primarily used in various domains [74], including machine learning [75], mainly aimed at estimating the inverse of the Hessian matrix (matrix of second-order partial derivatives). With the help of an estimate, we can determine the search direction to minimize the objective function. Furthermore, this method enhances its estimation with each iteration by using gradient information for correction. BFGS is a widely used approach in machine learning to optimize neural network weights because of its excellent convergence capabilities. It is beneficial for solving high-dimensional problems. However, if the initial estimate is not chosen carefully, the BFGS algorithm may end up with a suboptimal solution for objective functions that are not convex. On the other hand, convex optimization problems provide both quick and reliable solutions. The neural minimizer method [76], previously based on RBF neural networks [77], has now been updated with an artificial neural network that is trained using a local minimization technique called limited-memory BFGS (L-BFGS). This technique is relatively inexpensive regarding calculations and storage space [71]. This modified neural minimizer method has been widely used in optimization problems like image reconstruction [78] and seismic waveform tomography [79]. Proposed modifications have been suggested that would utilize contemporary parallel computing systems [80]. Sometimes, the large number of parameters in an artificial neural network makes it impossible to use an RBF neural network in a global optimization method for minimizing the error.