Fibers of point cloud persistence

TL;DR

This paper explores the geometric and algebraic structure of point cloud persistence, linking persistent homology with rigidity theory to determine when point clouds are uniquely identifiable from their persistence barcodes.

Contribution

It establishes bounds on the dimension of point cloud persistence spaces and connects rigidity properties of associated graphs to the identifiability of point clouds from their persistence.

Findings

Generic point clouds are identifiable up to isometry if their associated graph is globally rigid.

Local rigidity of the associated hypergraph ensures local identifiability from Čech persistence.

Provides combinatorial conditions linking rigidity theory with persistent homology.

Abstract

Persistent homology (PH) studies the topology of data across multiple scales by building nested collections of topological spaces called filtrations, computing homology and returning an algebraic object that can be vizualised as a barcode--a multiset of intervals. The barcode is stable and interpretable, leading to applications within mathematics and data science. We study the spaces of point clouds with the same barcode by connecting persistence with real algebraic geometry and rigidity theory. Utilizing a semi-algebraic setup of point cloud persistence, we give lower and upper bounds on its dimension and provide combinatorial conditions in terms of the local and global rigidity properties of graphs associated with point clouds and filtrations. We prove that for generic point clouds in $\mathbb{R}^d$ ($d \geq 2$), a point cloud is identifiable up to isometry from its VR persistence if the associated graph is globally rigid, and locally identifiable up to isometry from its \v{C}ech persistence if the associated hypergraph is rigid.