Persistence barcode

Short description: Technique in topological data analysis

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(σi)≤f(τ) whenever ∂τ=∑iσi in K. Then for every a∈ℝ the sublevel set Ka=f−1((−∞,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):

∅=K0⊆K1⊆⋯⊆Kn=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(eaj)=eai and one-dimensional complexes with trivial differential d(ea'i)=0.[2] The multiset ℬf of the intervals [ai,aj) or [ai′,∞) 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 ℝ consists of a family of 𝔽-vector spaces {Mt}t∈ℝ and linear maps φs,t:Ms->Mt for each s≤t such that φs,t∘φr,s=φr,t for all r≤s≤t.[4] This construction is not specific to ℝ; indeed, it works identically with any totally-ordered set.

A series of four nested simplicial complexes and the 0-dimensional persistence barcode of the resulting filtration.

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.[5]

Let I be an interval in ℝ. Define a persistence module Q(I) via Q(Is)={0,if s∉I;𝔽,otherwise, where the linear maps are the identity map inside the interval. The module Q(I) is sometimes referred to as an interval module.[6]

Then for any ℝ-indexed persistence module M of finite type, there exists a multiset ℬM of intervals such that M≅⨁I∈ℬMQ(I), where the direct sum of persistence modules is carried out index-wise. The multiset ℬM is called the barcode of 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]

## References

1. ↑ Ghrist, Robert (2007-10-26). "Barcodes: The persistent topology of data" (in en). Bulletin of the American Mathematical Society 45 (1): 61–76. doi:10.1090/S0273-0979-07-01191-3. ISSN 0273-0979. http://www.ams.org/journal-getitem?pii=S0273-0979-07-01191-3.
2. ↑ 2.0 2.1 2.2 Barannikov, Sergey (1994). "Framed Morse complex and its invariants". Advances in Soviet Mathematics. ADVSOV 21: 93–115. doi:10.1090/advsov/021/03. ISBN 9780821802373. https://hal.archives-ouvertes.fr/hal-01745109/file/GeneralizedMorse.pdf.
3. ↑ 3.0 3.1 Carlsson, Gunnar; Zomorodian, Afra; Collins, Anne; Guibas, Leonidas (2004-07-08). "Persistence barcodes for shapes" (in en). Proceedings of the 2004 Eurographics/ACM SIGGRAPH symposium on Geometry processing. Nice France: ACM. pp. 124–135. doi:10.1145/1057432.1057449. ISBN 978-3-905673-13-5. https://dl.acm.org/doi/10.1145/1057432.1057449.
4. ↑ Zomorodian, Afra; Carlsson, Gunnar (2005). "Computing Persistent Homology" (in en). Discrete & Computational Geometry 33 (2): 249–274. doi:10.1007/s00454-004-1146-y. ISSN 0179-5376.
5. ↑ Crawley-Boevey, William (2015). "Decomposition of pointwise finite-dimensional persistence modules" (in en). Journal of Algebra and Its Applications 14 (5): 1550066. doi:10.1142/S0219498815500668. ISSN 0219-4988. https://www.worldscientific.com/doi/abs/10.1142/S0219498815500668.
6. ↑ Chazal, Fréderic; de Silva, Vin; Glisse, Marc; Oudot, Steve (2016). The structure and stability of persistence modules. Switzerland. ISBN 978-3-319-42545-0. OCLC 960458101.
7. ↑ Botnan, Magnus, and William Crawley-Boevey. "Decomposition of persistence modules." Proceedings of the American Mathematical Society 148, no. 11 (2020): 4581-4596.
8. ↑ Webb, Cary. "Decomposition of graded modules." Proceedings of the American Mathematical Society 94, no. 4 (1985): 565-571.

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