On the shape that matters – topology and geometry in data science

Abstract

The seemingly simple question, “What is the shape of things?”, gains precise mathematical meaning when examined through the lens of modern topology and geometry. This paper surveys a few methods of topological data analysis (TDA), a powerful tool for characterising and predicting the shape of a dataset. Extending beyond traditional statistics, we will present various shape descriptors offered by TDA, elucidating their computation and practical applications. Last but not least we will demonstrate the effectiveness of the presented methodology through several real-world examples.