Persistence Barcode

In topological data analysis, a persistence barcode, sometimes shortened to barcode, is an algebraic invariant of a persistence module that characterizes the stability of topological features throughout a growing family of spaces.^[1] Formally, a persistence barcode consists of a multiset of intervals in the extended real line, where the length of each interval corresponds to the lifetime of a topological feature in a filtration, usually built on a point cloud, a graph, a function, or, more generally, a simplicial complex or a chain complex. Generally, longer intervals in a barcode correspond to more robust features, whereas shorter intervals are more likely to be noise in the data. A persistence barcode is a complete invariant that captures all the topological information in a filtration.^[2] In algebraic topology, the persistence barcodes were first introduced by Sergey Barannikov in 1994 as the "canonical forms" invariants^[2] consisting of a multiset of line segments with ends on two parallel lines, and later, in geometry processing, by Gunnar Carlsson et al. in 2004.^[3]

Definition

Let [math]\displaystyle{ \mathbb F }[/math] be a fixed field. Then a persistence module [math]\displaystyle{ M }[/math] indexed over [math]\displaystyle{ \mathbb R }[/math] consists of a family of [math]\displaystyle{ \mathbb F }[/math]-vector spaces [math]\displaystyle{ \{ M_t \}_{t \in \mathbb R} }[/math] and linear maps [math]\displaystyle{ \varphi_{s,t} : M_s \to M_t }[/math] for each [math]\displaystyle{ s \leq t }[/math] such that [math]\displaystyle{ \varphi_{s,t} \circ \varphi_{r,s} = \varphi_{r,t} }[/math] for all [math]\displaystyle{ r \leq s \leq t }[/math].^[4] This construction is not specific to [math]\displaystyle{ \mathbb R }[/math]; indeed, it works identically with any totally-ordered set.

A series of four nested simplicial complexes and the 0-dimensional persistence barcode of the resulting filtration.

A persistence module [math]\displaystyle{ M }[/math] is said to be of finite type if it contains a finite number of unique finite-dimensional vector spaces. The latter condition is sometimes referred to as pointwise finite-dimensional.^[5]

Let [math]\displaystyle{ I }[/math] be an interval in [math]\displaystyle{ \mathbb R }[/math]. Define a persistence module [math]\displaystyle{ Q(I) }[/math] via [math]\displaystyle{ Q(I_s)= \begin{cases} 0, & \text{if } s\notin I;\\ \mathbb F, & \text{otherwise} \end{cases} }[/math], where the linear maps are the identity map inside the interval. The module [math]\displaystyle{ Q(I) }[/math] is sometimes referred to as an interval module.^[6]

Then for any [math]\displaystyle{ \mathbb R }[/math]-indexed persistence module [math]\displaystyle{ M }[/math] of finite type, there exists a multiset [math]\displaystyle{ \mathcal B_M }[/math] of intervals such that [math]\displaystyle{ M \cong \bigoplus_{I \in \mathcal B_M}Q(I) }[/math], where the direct sum of persistence modules is carried out index-wise. The multiset [math]\displaystyle{ \mathcal B_M }[/math] is called the barcode of [math]\displaystyle{ M }[/math], and it is unique up to a reordering of the intervals.^[3]

This result was extended to the case of pointwise finite-dimensional persistence modules indexed over an arbitrary totally-ordered set by William Crawley-Boevey and Magnus Botnan in 2020,^[7] building upon known results from the structure theorem for finitely generated modules over a PID, as well as the work of Cary Webb for the case of the integers.^[8]

References

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