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. 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 be a fixed field. Then a persistence module indexed over consists of a family of -vector spaces and linear maps for each such that for all . This construction is not specific to ; indeed, it works identically with any totally-ordered set.
A persistence module 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 be an interval in . Define a persistence module via , where the linear maps are the identity map inside the interval. The module is sometimes referred to as an interval module.
Then for any -indexed persistence module of finite type, there exists a multiset of intervals such that , where the direct sum of persistence modules is carried out index-wise. The multiset is called the barcode of , 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.