Applications of Persistent Homology: Data Classification and Intrinsic Dimension of Manifold

The main aim of the phd was to apply new techniques and mathematical tools to Data Analysis, mainly to the analysis of foundry data. The partner of the project, that is Enginsoft, a company in Padova, provided data acquired and processed by one of its own software: SmartProdActive. After a detailed analysis of this tool and, we began our mathematical study, keeping in mind the task to address: solving a classification problem on highly imbalanced datasets. First we focused our attention on kernels, allowed us to build a connection between the approximation theory and the Machine Learning framework, where kernels are widely used with the so-called kernel trick. In this context we have introduced and described a suitable modification of KNN using topological information, called TDA Local. Then, we have moved to another context: classification of Persistent Dagrams using SVM with Persistence Kernels. Finally, we investigated the use of Persistent Homology in the context of Intrinsic Dimension of Manifold. All topics have been enriched with a lot of numerical tests and the related Python code are available in the GitHub repository.