Counting Barcodes with the same Betti Curve

TL;DR

This paper investigates the inverse problem in topological data analysis of determining the number of barcodes that yield the same Betti curve, linking it to combinatorics and juggling sequences to quantify information loss.

Contribution

It establishes a novel connection between persistent homology, Kostant partition functions, and juggling sequences, providing a new perspective on the inverse problem in TDA.

Findings

Connected the inverse problem to Kostant partition function and juggling sequences.

Proved equivalence between the inverse problem and combinatorial enumeration.

Quantified the lossy nature of the TDA pipeline from persistent homology to Betti numbers.

Abstract

This paper considers an important inverse problem in topological data analysis (TDA): How many different barcodes produce the same Betti curve? Equivalently, given a function $\beta\colon [n]=\{1<\cdots< n\} \to \mathbb{Z}_{\geq 0}$, how many different ways can we write $\beta$ as a sum of indicator functions supported on intervals in $[n]$? Our answer to this question is to connect persistent homology with the study of the Kostant partition function and the enumerative combinatorics for so-called "magic" juggling sequences studied by Ronald Graham and others. Specifically, we prove an equivalence between our inverse problem and corresponding statements in these other two settings. From an applications and statistics point of view, our work provides a quantification of how lossy the TDA pipeline is when moving from persistent homology to persistent Betti numbers.