# nLab stability of persistence diagrams

Contents

### Context

#### Homological algebra

homological algebra

Introduction

diagram chasing

### Theorems

#### Representation theory

representation theory

geometric 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 $V(f)_\bullet$ given by the connected components of the sub-level sets of a continuous function $X \xrightarrow{f} \mathbb{R}$ on some data set $X$

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

(equipped with the evident inclusions) and states that as the function $f$ is deformed to another continuous function $g$, the bottleneck distance $d_B$ (CSEH07, p. 3)

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

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

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

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

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.

The stability theorem originates in:

The algebraic stability theorem:

Further developments:

Generalization to zigzag persistence modules:

Refinement to persistent homotopy:

Version for persistent cohomotopy: