Alberto Cañada Carril: Homological shape of correlations

Understanding complex data often requires looking beyond traditional metrics. This thesis introduces a novel correlation function between pairs of vectors of real numbers, obtained by extracting homological invariants from a two-dimensional pointcloud induced by the vectors. Persistent homology, a commonly used tool in Topological Data Analysis, is the main component of our analysis pipeline.

In this work, we formally define the invariant, present and justify an algorithm to compute it, and investigate its properties and interpretations. In addition, we illustrate its use through a series of examples on both synthetic and biological datasets, including protein interaction networks.