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 $$\mathbb F$$ be a fixed field. Consider a real-valued function on a chain complex $$f:K \rightarrow \mathbb{R}$$ compatible with the differential, so that $$f(\sigma_i) \leq f(\tau)$$ whenever $$\partial\tau=\sum_i\sigma_i$$ in $$K$$. Then for every $$ a \in \mathbb{R}$$ the sublevel set $$K_a=f^{-1}((-\infty, a])$$ is a subcomplex of K, and the values of $$f$$ on the generators in $$K$$  define a filtration (which is in practice always finite):
 * $$ \emptyset = K_0 \subseteq K_1 \subseteq \cdots \subseteq K_n = K $$.

Then, the filtered complexes classification theorem states that for any filtered chain complex over $$\mathbb F$$, 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 $$d(e_{a_j})=e_{a_i}$$ and one-dimensional complexes with trivial differential $$d(e_{a'_i})=0$$. The multiset $$\mathcal B_f $$ of the intervals $$[a_i, a_j)$$ or $$[a_i', \infty)$$ 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 $$M$$ indexed over $$\mathbb R$$ consists of a family of $$\mathbb F$$-vector spaces $$\{ M_t \}_{t \in \mathbb R}$$ and linear maps $$\varphi_{s,t} : M_s \to M_t$$ for each $$s \leq t$$ such that $$\varphi_{s,t} \circ \varphi_{r,s} = \varphi_{r,t}$$ for all $$r \leq s \leq t$$. This construction is not specific to $$\mathbb R$$; indeed, it works identically with any totally-ordered set. 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 $$\mathbb R$$. Define a persistence module $$Q(I)$$ via $$Q(I_s)= \begin{cases} 0, & \text{if } s\notin I;\\ \mathbb F, & \text{otherwise} \end{cases}$$, where the linear maps are the identity map inside the interval. The module $$Q(I)$$ is sometimes referred to as an interval module.

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