nLab stability of persistence diagrams

Contents

Context

Homological algebra

homological algebra

(also nonabelian homological algebra)

Introduction

Context

Basic definitions

Stable homotopy theory notions

Constructions

Lemmas

diagram chasing

Homology theories

Theorems

Representation theory

Contents

Idea

The central theorem in persistent homology states how the data retained in persistence diagrams/barcodes is stable under small deformations of the initial data.

Specifically, the original form of the stability theorem (Cohen, Steiner, Edelsbrunner & Harer 2007) applies to persistence modules V(f) V(f)_\bullet given by the connected components of the sub-level sets of a continuous function XfX \xrightarrow{f} \mathbb{R} on some data set XX

V(f) lH 0(f 1((,l])) V(f)_l \;\coloneqq\; H_0\Big( f^{-1}\big( (-\infty, l] \big) \Big)

(equipped with the evident inclusions) and states that as the function ff is deformed to another continuous function gg, the bottleneck distance d Bd_B (CSEH07, p. 3)

d B(X,Y)infXγYsupxXxγ(x) d_B \big( X ,\, Y \big) \;\coloneqq\; \underset{ X \underoverset{\sim}{\gamma}{\to} Y }{inf} \;\; \underset{x \in X}{sup} \; \Vert x - \gamma(x) \Vert_\infty

between the corresponding persistence diagrams PDgr(V(f) )PDgr\big( V(f)_\bullet \big) is bounded by the supremum norm of the difference between the two functions:

d B(PDgr(V(f) ),PDgr(V(f) ))fg . d_B \Big( PDgr\big( V(f)_\bullet \big) ,\, PDgr\big( V(f)_\bullet \big) \Big) \;\leq\; \Vert f - g \Vert_\infty \,.

Various generalizations of this stability result exist, notably the algebraic stability theorem (CCGGO09).

From Botnan & Lesnick 18, p. 2:

The algebraic stability theorem is perhaps the central theorem in the theory of persistent homology; it provides the core mathematical justification for the use of persistent homology in the study of noisy data. The theorem is used, in one form or another, in nearly all available results on the approximation, inference, and estimation of persistent homology.

References

The stability theorem originates in:

The algebraic stability theorem:

Further developments:

Generalization to zigzag persistence modules:

Refinement to persistent homotopy:

Version for persistent cohomotopy:

Last revised on May 23, 2022 at 08:37:43. See the history of this page for a list of all contributions to it.