Chaos, fractals and machine learning

Abstract

The accuracy of learning a function is determined both by the underlying process that generates the sample as well as the function itself. The Lorenz butterfly, a simple weather analogy, is an example dynamical systems. Slightly more complex 6, 9 and 12, dimensional systems are also used to generate the independent variables. The non uniformly fractal distributions which are the intersection of the trajectories on a hyperplane are also used to generate variable values. As comparisons uniformly distributed (pseudo) random numbers are used as values of the independent variables. A number of functions on these hypercubes, and hyper-surfaces are defined. When the function is sampled near regions of interest and where the test set is of the same form as the learning set, both the chaotic system and fractal points have more accurate learners than the uniformly distributed ones. Using one form of distribution to learn the data, and another for testing can be particularly poor. These cross distributional results are dependent of the functional form. Aspects of machine learning relevant to fractal distributions and chaotic phenomena are developed.