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

On The Potential of The Fractal Geometry and The CNNs Ability to Encode it