Statistical Analysis of Persistence Intensity Functions

Abstract

Persistence diagrams are two-dimensional plots that summarize the topological features of functions and are an important part of topological data analysis. A problem that has received much attention is how deal with sets of persistence diagrams. How do we summarize them, average them or cluster them? One approach -- the persistence intensity function -- was introduced informally by Edelsbrunner, Ivanov, and Karasev (2012). Here we provide a modification and formalization of this approach. Using the persistence intensity function, we can visualize multiple diagrams, perform clustering and conduct two-sample tests.

10 pages, 5 figures

References in corpus (3)

• Persistence Images: A Stable Vector Representation of Persistent Homology
• Robust Topological Inference: Distance To a Measure and Kernel Distance
• A Stable Multi-Scale Kernel for Topological Machine Learning

Cited by in corpus (16)

• Is the Observable Universe Consistent with the Cosmological Principle?
• Persistence Images: A Stable Vector Representation of Persistent Homology
• Persistent Homology of Complex Networks for Dynamic State Detection
• The persistence landscape and some of its properties
• Topological Feature Vectors for Chatter Detection in Turning Processes
• Approximating Continuous Functions on Persistence Diagrams Using Template Functions
• The density of expected persistence diagrams and its kernel based estimation
• Noise robustness of persistent homology on greyscale images, across filtrations and signatures
• Stabilizing the unstable output of persistent homology computations
• Bootstrapping Persistent Betti Numbers and Other Stabilizing Statistics
• Differentiating small-scale subhalo distributions in CDM and WDM models using persistent homology
• Using Persistent Homology Topological Features to Characterize Medical Images: Case Studies on Lung and Brain Cancers
• A computationally efficient framework for vector representation of persistence diagrams
• The self-similar evolution of stationary point processes via persistent homology
• Persistent Homology and Graphs Representation Learning
• On the estimation of persistence intensity functions and linear representations of persistence diagrams