## 3 The Radial Basis Function Network  
The main objective of the EIT inverse problem is to find the conductivity by measuring the current and voltage on the surface. Under the assumption that there exists a mapping from the boundary data to the conductivity distribution \(\sigma\), the target is to either find this mapping or approximate it well enough so that, given the boundary data, we can predict the conductivity distribution inside the material with reasonably good accuracy. We use rbf networks to approximate this mapping. Radial basis function networks are capable of universal approximation [33], and since we are trying to approximate a function, it is a suitable choice. Usually, rbf networks feed forward with input, output, and single hidden layers. The activation functions in the hidden layer are rbfs \(\phi_{j}(||\mathbf{x}-\mathbf{c}_{j}||)\) where \(\mathbf{c}_{j}\) is some fixed center vector for the \(j\)-th rbf, and \(\mathbf{x}\) is the input vector, which contains the input data. If the hidden layer has \(k\) nodes, there are \(k\) activation functions, so \(j\) runs from 1 to \(k\). The norm taken of \(\mathbf{x}-\mathbf{c}_{j}\) is the \(\ell_{2}\)-norm. Therefore, \(\phi\) is a function of the distance between the data point and the center point in a d-dimensional Euclidean space, where d is the input vector's dimension. The most common choice of an rbf, and the one that we employ, is the Gaussian function, mainly  
\[\phi_{j}=e^{-||\mathbf{x}-\mathbf{c}_{j}||^{2}/2b_{j}^{2}} \tag{13}\]  
where \(c_{j}\) and \(b_{j}\) are clearly the mean and the standard deviation for \(\phi_{j}\). We determined \(c_{j}\) using \(k\) means clustering of the data set for a hidden layer of size \(k\) and determined \(b_{j}\) using  
\[b_{j}=\frac{d_{j}}{\sqrt{2k}} \tag{14}\]  
where \(k\) is the number of hidden neurons and \(d_{j}=max(||c_{j}-c_{i}||)\), \(i\
eq j\). It is also common to use a Gaussian function of the same width. In such a case, all the \(b_{j}\) would equal a nonzero real number \(b\). We try both of these choices and compare their performance. The basic structure of an rbf network is shown in Figure 1. The input vector has \(n\) components, and the output vector has \(m\) components. The hidden layer has \(k\) nodes. In that case, the components of the output are given by  
\[y_{i}=\sum_{j}w_{ij}\phi_{j}(\mathbf{x}). \tag{15}\]  
This can be written more compactly in matrix form as  
\[\mathbf{y}=W\Phi(\mathbf{x}) \tag{16}\]  
where \(W\) is an \(m\times k\) matrix with \(w_{ij}\) as its \(ij\)-th element. Equation (16) is for a single input and a single output vector. If we have a training set of size \(N\), there will be \(N\) input vectors. So, a training set size of \(N\) would have  
\[Y=W\Phi(X) \tag{17}\]  
where each column in \(Y\), \(X\), and \(\Phi(X)\) correspond to a single training datum and would individually satisfy Equation (16). In the case of the EIT inverse problem, \(\mathbf{x}\) would be the surface voltage measurements, while \(\mathbf{y}\) would be the conductivity distribution (on some mesh). We assume that there is some ideal mapping \(f:X\to Y\), and we wish to approximate that target function as closely as possible using the weight matrix \(W\) and the rbfs \(\phi_{1}\) to \(\phi_{k}\). We define the loss function by  
\[E=\sum_{j=1}^{n}\|\tilde{y}^{(j)}-y^{(j)}\|_{2}^{2} \tag{18}\]  
where \(n\) is the training size, \(\tilde{y}^{(j)}\) is the output vector from the \(j\)-th training data, \(y^{(j)}\) is obtained from Equation (16) using the input vector from the \(j\)-th training data. We use three explicit regularization schemes to add a term \(\lambda\|W\|\) to the loss function in the optimization problem. They are Tikhonov, Lasso, and Elastic Net regularization, respectively.