Limit Theorems for the Sum of Persistence Barcodes

2016/04/14 by Takashi Owada, Owada, Takashi · 1 citation
Computer Science · Mathematics · Neuroscience · #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Topological and Geometric Data Analysis #Tryptophan and brain disorders #math.PR

paper · pdf · doi:10.48550/arxiv.1604.04058

32 pages

arxiv created 2016/04/14 · openalex publication_date 2016/04/14 · arxiv updated 2016/04/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Topological Data Analysis (TDA) refers to an approach that uses concepts from algebraic topology to study the "shapes" of datasets. The main focus of this paper is persistent homology, a ubiquitous tool in TDA. Basing our study on this, we investigate the topological dynamics of extreme sample clouds generated by a heavy tail distribution on \mathbb R^d. In particular, we establish various limit theorems for the sum of bar lengths in the persistence barcode plot, a graphical descriptor of persistent homology. It then turns out that the growth rate of the sum of the bar lengths and the properties of the limiting processes all depend on the distance of the region of interest in \mathbb R^d from the weak core, that is, the area in which random points are placed sufficiently densely to connect with one another. If the region of interest becomes sufficiently close to the weak core, the limiting process involves a new class of Gaussian processes.

Citations

• Computing Persistent Homology
• Minimum spanning acycle and lifetime of persistent homology in the Linial-Meshulam process
• Persistence weighted Gaussian kernel for topological data analysis

Cited by

• A persistent homology approach to heart rate variability analysis with an application to sleep-wake classification

Related

• Approximating Persistent Homology for Large Datasets
• Modeling of Persistent Homology
• On the Expectation of a Persistence Diagram by the Persistence Weighted Kernel
• Stochastic Convergence of Persistence Landscapes and Silhouettes
• The density of expected persistence diagrams and its kernel based estimation