Functional Data Representation with Merge Trees

In this paper we face the problem of representation of functional data with the tools of algebraic topology. We represent functions by means of merge trees, which, like the more commonly used persistence diagrams, are invariant under homeomorphic reparametrizations of the functions they represent, thus allowing for a statistical analysis which is indifferent to functional misalignment. We consider a recently defined metric for merge trees and we prove some theoretical results related to its specific implementation when merge trees represent functions, establishing also a class of consistent estimators with convergence rates. To showcase the good properties of our topological approach to functional data analysis, we test it on the Aneurisk65 dataset replicating, from our different perspective, the supervised classification analysis which contributed to make this dataset a benchmark for methods dealing with misaligned functional data. In the Appendix we provide an extensive comparison between merge trees and persistence diagrams, highlighting similarities and differences, which can guide the analyst in choosing between the two representations.