  The fractal dimension provides a statistical index of object complexity by
studying how the pattern changes with the measuring scale. Although useful in
several classification tasks, the fractal dimension is under-explored in deep
learning applications. In this work, we investigate the features that are
learned by deep models and we study whether these deep networks are able to
encode features as complex and high-level as the fractal dimensions.
Specifically, we conduct a correlation analysis experiment to show that deep
networks are not able to extract such a feature in none of their layers. We
combine our analytical study with a human evaluation to investigate the
differences between deep learning networks and models that operate on the
fractal feature solely. Moreover, we show the effectiveness of fractal features
in applications where the object structure is crucial for the classification
task. We empirically show that training a shallow network on fractal features
achieves performance comparable, even superior in specific cases, to that of
deep networks trained on raw data while requiring less computational resources.
Fractals improved the accuracy of the classification by 30% on average while
requiring up to 84% less time to train. We couple our empirical study with a
complexity analysis of the computational cost of extracting the proposed
fractal features, and we study its limitation.
