Basi omologiche in analisi topologica dei dati

In recent years topological data analysis (TDA) has gained more and more momentum in the field of data science and machine learning. In particular, persistent homology, the main tool used to perform TDA, has become widely spread and even though efficient algorithms to perform it have been developed and have found applications in several domains, the problem of efficiently determining minimal $k$-homology bases in persistent homology is still open. We provide a comparative analysis of some current state of the art academic research papers on topological data analysis and persistent homology, with particular focus minimal persistent homology bases. Our main contribution consist in a work of re-definitions of the concepts of persistent homology basis, interval basis and homological scaffold. Thanks to this re-definitions, we have been able to compare and combine such concepts and this allowed us to obtain some interesting results: in the document we proved that is not always possible to define an interval basis that induces at the same time a minimal persistent homology basis, we showed it is possible to well define interval bases and persistent homology bases on the same and thus to compare and try to combine the two. Finally, we showed that is possible to well define an homological scaffold of an interval basis and we hint on possible future research on it.