Multilayer perceptrons and fractals

In this article, a mathematical relationship between the gradient descent technique and contractive maps is examined. This relationship is based upon the observation that the convergence of the gradient descent technique can be proved using results in fractal theory - more specifically, results concerning contractive maps - as opposed to results based on Taylor's series. This proof, involving the eigenvalues of the Hessian matrix of the gradient descent technique's objective function, is presented. A simple example is given in which steps from the aforementioned proof are used to find conditions under which a specific multilayer perceptron is guaranteed to converge. Since the gradient descent technique is used in multilayer perceptrons, and contractive maps give rise to fractals, a theoretical relationship is thus established between multilayer perceptrons and fractals.