Non-topological persistence for data analysis and machine learning

This thesis main objective is to study possible applications of the generalisation of persistence theory introduced in [1], [2]. This generalisation extends the notion of persistence to a wider categorical setting, avoiding constructing secondary structures as topological spaces. The first field analysed is graph theory. At first, we studied which classical graph theory invariants could be used as rank function. Another aspect analysed in this thesis is the extension of the study of connectivity in graphs from a persistence viewpoint started in [1] to oriented graphs. Moreover, we studied how different orientation of the same underlying graph can change the distribution of cornerpoints in persistence diagrams, both in deterministic and random graphs. The other application field analysed is image processing. We adapted the notion of steady and ranging sets to the category of sets and used them to define activation and deactivation rules for each pixel. These notions allowed us to define a filter capable of enhancing the signal of pixels close to a border. This filter has proven to be stable under salt and pepper noise perturbation. At last, we used this filter to define a novel pooling layer for convolutional neural networks. In the experimental part, we compared the proposed layer with other state-of-the-art layers. The results show how the proposed layer outperform the other layers in term of accuracy. Moreover, by concatenating the proposed and the Max pooling, it is possible to improve accuracy further. [1] Bergomi, M.G., Ferri, M., Vertechi, P., Zuffi, L. (2020), Beyond topological persistence: Starting from networks, arXiv. [2] Bergomi, M. G., & Vertechi, P. (2020). Rank-based persistence. Theory and Applications of Categories, 35, 228-260.