Probability measures on the space of persistence diagrams. (English) Zbl 1247.68310

Mileyko, Yuriy; Mukherjee, Sayan; Harer, John

Inverse Probl. 27, No. 12, Article ID 124007, 22 p. (2011).

This interesting paper is devoted to the study of persistent homology from a probabilistic point of view. The authors prove that if we assume that the set \(D_p\) of persistence diagrams with finite degree-\(p\) total persistence is endowed with a regular enough probability measure, then we can define a Fréchet expectation on it (Theorems 24 and 28). This result is based on the proof that the space of persistence diagrams endowed with the Wasserstein metric is complete (Theorem 6). Separability of \(D_p\) is also proven. Finally, the authors show how a measure on point samples allows to obtain a measure on persistence diagrams, opening a way to applications of their results in statistical inference. A clarifying example ends the paper.

Reviewer: Patrizio Frosini (Bologna)

MSC:

68U05	Computer graphics; computational geometry (digital and algorithmic aspects)
55N35	Other homology theories in algebraic topology
65D18	Numerical aspects of computer graphics, image analysis, and computational geometry
68Q87	Probability in computer science (algorithm analysis, random structures, phase transitions, etc.)
60D05	Geometric probability and stochastic geometry
60B05	Probability measures on topological spaces

Keywords:

persistent homology; Fréchet variance; Fréchet expectation; Wasserstein metric

Software:

EMD