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.

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. 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.

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