Persistence barcode

In topological data analysis, a persistence barcode, sometimes shortened to barcode, is an algebraic invariant associated with a filtered chain complex or a persistence module that characterizes the stability of topological features throughout a growing family of spaces. 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. In algebraic topology, the persistence barcodes were first introduced by Sergey Barannikov in 1994 as the "canonical forms" invariants 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.

Definition

Let $𝔽$ be a fixed field. Consider a real-valued function on a chain complex $f:K\to ℝ$ compatible with the differential, so that $f({\sigma }_i)\le f(\tau )$ whenever $\partial \tau =\sum _i{\sigma }_i$ in $K$. Then for every $a\in ℝ$ the sublevel set $K_a=f^{-1}((-\mathrm{\infty },a])$ is a subcomplex of K, and the values of $f$ on the generators in $K$ define a filtration (which is in practice always finite):

$\mathrm{\varnothing }=K_0\subseteq K_1\subseteq \cdots \subseteq K_n=K$.

Then, the filtered complexes classification theorem states that for any filtered chain complex over $𝔽$, there exists a linear transformation that preserves the filtration and brings the filtered complex into so called canonical form, a canonically defined direct sum of filtered complexes of two types: two-dimensional complexes with trivial homology $d(e_{a_j})=e_{a_i}$ and one-dimensional complexes with trivial differential $d(e_{a{\text{'}}_i})=0$. The multiset ${ℬ}_f$ of the intervals $[a_i,a_j)$ or $[{a_i}^{\prime },\mathrm{\infty })$ describing the canonical form, is called the barcode, and it is the complete invariant of the filtered chain complex.

The concept of a persistence module is intimately linked to the notion of a filtered chain complex. A persistence module $M$ indexed over $ℝ$ consists of a family of $𝔽$-vector spaces $\{M_t{\}}_{t\in ℝ}$ and linear maps ${\phi }_{s,t}:M_s\to M_t$ for each $s\le t$ such that ${\phi }_{s,t}\circ {\phi }_{r,s}={\phi }_{r,t}$ for all $r\le s\le t$. This construction is not specific to $ℝ$; 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 $M$ 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.

Let $I$ be an interval in $ℝ$. Define a persistence module $Q(I)$ via $Q(I_s)=\{\begin{array}{cc}0,& \text{if }s\notin I;\\ 𝔽,& \text{otherwise}\end{array}$, where the linear maps are the identity map inside the interval. The module $Q(I)$ is sometimes referred to as an interval module.

Then for any $ℝ$-indexed persistence module $M$ of finite type, there exists a multiset ${ℬ}_M$ of intervals such that $M\cong \underset{I\in {ℬ}_M}{⨁}Q(I)$, where the direct sum of persistence modules is carried out index-wise. The multiset ${ℬ}_M$ is called the barcode of $M$, and it is unique up to a reordering of the intervals.

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, 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.