How to calculate the Hausdorff dimension using fractal structures

摘要

In this paper, we provide the first known overall algorithm to calculate the Hausdorff dimension of any compact Euclidean subset. This novel approach is based on both a new discrete model of fractal dimension for a fractal structure which considers finite coverings and a theoretical result that the authors contributed previously in [14]. This new procedure combines fractal techniques with tools from Machine Learning Theory. In particular, we use a support vector machine to decide the value of the Hausdorff dimension. In addition to that, we artificially generate a wide collection of examples that allows us to train our algorithm and to test its performance by external proof. Some analyses about the accuracy of this approach are also provided.