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

Abstract: 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.