Persistence barcode

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

${\displaystyle \emptyset =K_{0}\subseteq K_{1}\subseteq \cdots \subseteq K_{n}=K}$.

Then, the filtered complexes classification theorem states that for any filtered chain complex over ${\displaystyle \mathbb {F} }$, 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 ${\displaystyle d(e_{a_{j}})=e_{a_{i}}}$ and one-dimensional complexes with trivial differential ${\displaystyle d(e_{a'_{i}})=0}$.^[2] The multiset ${\displaystyle {\mathcal {B}}_{f}}$ of the intervals ${\displaystyle [a_{i},a_{j})}$ or ${\displaystyle [a_{i}',\infty )}$ 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 ${\displaystyle M}$ 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 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]