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 ${\displaystyle \mathbb {F} }$ be a fixed field. Then a persistence module ${\displaystyle M}$ indexed over ${\displaystyle \mathbb {R} }$ consists of a family of ${\displaystyle \mathbb {F} }$-vector spaces ${\displaystyle \{M_{t}\}_{t\in \mathbb {R} }}$ and linear maps ${\displaystyle \varphi _{s,t}:M_{s}\to M_{t}}$ for each ${\displaystyle s\leq t}$ such that ${\displaystyle \varphi _{s,t}\circ \varphi _{r,s}=\varphi _{r,t}}$ for all ${\displaystyle r\leq s\leq t}$.^[4] This construction is not specific to ${\displaystyle \mathbb {R} }$; 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 ${\displaystyle M}$ 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 ${\displaystyle I}$ be an interval in ${\displaystyle \mathbb {R} }$. Define a persistence module ${\displaystyle Q(I)}$ via ${\displaystyle Q(I_{s})={\begin{cases}0,&{\text{if }}s\notin I;\\\mathbb {F} ,&{\text{otherwise}}\end{cases}}}$, where the linear maps are the identity map inside the interval. The module ${\displaystyle Q(I)}$ is sometimes referred to as an interval module.^[6]

Then for any ${\displaystyle \mathbb {R} }$-indexed persistence module ${\displaystyle M}$ of finite type, there exists a multiset ${\displaystyle {\mathcal {B}}_{M}}$ of intervals such that ${\displaystyle M\cong \bigoplus _{I\in {\mathcal {B}}_{M}}Q(I)}$, where the direct sum of persistence modules is carried out index-wise. The multiset ${\displaystyle {\mathcal {B}}_{M}}$ is called the barcode of ${\displaystyle M}$, 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