Schematic view of a simple neural network with only one hidden layer. The light dots are input dimensions. The black dots are neurons with the linking weights represented as continuous lines. Learning with this network relies on Eqs. (1)–(8). X_{[1, …, i]} are the dimensions for one input vector, a_{[1, …, j]} are the activations of the hidden neurons, a_{[1, …, k]} are the activations of the output neurons, while V_ij and W_jk represent the weight matrices between the input and hidden layers, and between the hidden and output layers, respectively.