Functional Networks

In this letter we present functional networks. Unlike neural networks, in these networks there are no weightsassociated with the links connecting neurons, and the internal neuron functions are not fixed but learnable. These functions are not arbitrary, but subject to strong constraints to satisfy the compatibility conditions imposed by the existence of multiple links going from the last input layer to the same output units. In fact, writing the values of the output units in different forms, by considering these different links, a system of functional equations is obtained. When this system is solved, the numberof degrees of freedom of these initially multidimensional functions is considerably reduced. One example illustrates the process and shows that multidimensional functions can be reduced to functions with a single argument. To learn the resulting functions, a method based on minimizing a least squares error function is used, which, unlike the functions used in neural networks, has a single minimum.