Persistence barcode explained

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.^[1] 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.^[2] In algebraic topology, the persistence barcodes were first introduced by Sergey Barannikov in 1994 as the "canonical forms" invariants^[2] 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.^[3]

Definition

Let

F be a fixed field. Consider a real-valued function on a chain complex f:K → 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 R the sublevel set K ^{-1} _{a = f} (( -inf ty, 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 … \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' _{i}} ) =0 .^[4] The multiset l B _{f} of the intervals [a _{i,} a _{j)} or [a _{i',} inf ty) describing the canonical form, is called the barcode, and it is the complete invariant of the filtered chain complex. M indexed over R consists of a family of F -vector spaces \{ M _{t} \} _{t} 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 .^[5] This construction is not specific to 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.^[6]

Let

I be an interval in R . Define a persistence module Q(I) via Q(I _{s)= \begin{cases} 0,} & if s\notin I;\\ F, & 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.^[7]

Then for any

R -indexed persistence module M of finite type, there exists a multiset l B _{M} of intervals such that M \cong o plus _{I \in l B _{M}} Q(I) , where the direct sum of persistence modules is carried out index-wise. The multiset l 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,^[8] 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.^[9]

References

1. Ghrist . Robert . 2007-10-26 . Barcodes: The persistent topology of data . Bulletin of the American Mathematical Society . en . 45 . 1 . 61–76 . 10.1090/S0273-0979-07-01191-3 . 0273-0979. free .
2. Framed Morse complex and its invariants . Advances in Soviet Mathematics . 1994. 93–115. 21. Sergey. Barannikov . ADVSOV . 10.1090/advsov/021/03. 9780821802373 . 125829976 .
3. Book: Carlsson . Gunnar . Zomorodian . Afra . Collins . Anne . Guibas . Leonidas . Proceedings of the 2004 Eurographics/ACM SIGGRAPH symposium on Geometry processing . Persistence barcodes for shapes . 2004-07-08 . https://dl.acm.org/doi/10.1145/1057432.1057449 . en . Nice France . ACM . 124–135 . 10.1145/1057432.1057449 . 978-3-905673-13-5. 456712 .
4. Framed Morse complex and its invariants . Advances in Soviet Mathematics . 1994. 93–115. 21. Sergey. Barannikov . ADVSOV . 10.1090/advsov/021/03. 9780821802373 . 125829976 .
5. Zomorodian . Afra . Carlsson . Gunnar . 2005 . Computing Persistent Homology . Discrete & Computational Geometry . en . 33 . 2 . 249–274 . 10.1007/s00454-004-1146-y . 0179-5376. free .
6. Crawley-Boevey . William . 2015 . Decomposition of pointwise finite-dimensional persistence modules . Journal of Algebra and Its Applications . en . 14 . 5 . 1550066 . 10.1142/S0219498815500668 . 1210.0819 . 119635797 . 0219-4988.
7. Book: Chazal . Fréderic . The structure and stability of persistence modules . de Silva . Vin . Glisse . Marc . Oudot . Steve . 2016 . 978-3-319-42545-0 . Switzerland . 960458101.
8. Botnan, Magnus, and William Crawley-Boevey. "Decomposition of persistence modules." Proceedings of the American Mathematical Society 148, no. 11 (2020): 4581-4596.
9. Webb, Cary. "Decomposition of graded modules." Proceedings of the American Mathematical Society 94, no. 4 (1985): 565-571.