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. ^[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 ${\displaystyle \mathbb {F} }$ be a fixed field. Consider a real-valued function on a chain complex ${\displaystyle f:K\rightarrow \mathbb {R} }$ compatible with the differential, so that ${\displaystyle f(\sigma _{i})\leq f(\tau )}$ whenever ${\displaystyle \partial \tau =\sum _{i}\sigma _{i}}$ in ${\displaystyle K}$. Then for every ${\displaystyle a\in \mathbb {R} }$ the sublevel set ${\displaystyle K_{a}=f^{-1}((-\infty ,a])}$ is a subcomplex of K, and the values of ${\displaystyle f}$ on the generators in ${\displaystyle K}$ define a filtration (which is in practice always finite):

${\displaystyle \emptyset =K_{0}\subseteq K_{1}\subseteq \cdots \subseteq K_{n}=K}$.

Then, the filtered complexes classification theorem states that for any filtered chain complex over ${\displaystyle \mathbb {F} }$, 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 ${\displaystyle d(e_{a_{j}})=e_{a_{i}}}$ and one-dimensional complexes with trivial differential ${\displaystyle d(e_{a'_{i}})=0}$. ^[2] The multiset ${\displaystyle {\mathcal {B}}_{f}}$ of the intervals ${\displaystyle [a_{i},a_{j})}$ or ${\displaystyle [a_{i}',\infty )}$ 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 ${\displaystyle M}$ indexed over ${\displaystyle \mathbb {R} }$ consists of a family of ${\displaystyle \mathbb {F} }$-vector spaces ${\displaystyle \{M_{t}\}_{t\in \mathbb {R} }}$ and linear maps ${\displaystyle \varphi _{s,t}:M_{s}\to M_{t}}$ for each ${\displaystyle s\leq t}$ such that ${\displaystyle \varphi _{s,t}\circ \varphi _{r,s}=\varphi _{r,t}}$ for all ${\displaystyle r\leq s\leq t}$. ^[4] This construction is not specific to ${\displaystyle \mathbb {R} }$; 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 ${\displaystyle M}$ 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 ${\displaystyle I}$ be an interval in ${\displaystyle \mathbb {R} }$. Define a persistence module ${\displaystyle Q(I)}$ via ${\displaystyle Q(I_{s})={\begin{cases}0,&{\text{if }}s\notin I;\\\mathbb {F} ,&{\text{otherwise}}\end{cases}}}$, where the linear maps are the identity map inside the interval. The module ${\displaystyle Q(I)}$ is sometimes referred to as an interval module. ^[6]

Then for any ${\displaystyle \mathbb {R} }$-indexed persistence module ${\displaystyle M}$ of finite type, there exists a multiset ${\displaystyle {\mathcal {B}}_{M}}$ of intervals such that ${\displaystyle M\cong \bigoplus _{I\in {\mathcal {B}}_{M}}Q(I)}$, where the direct sum of persistence modules is carried out index-wise. The multiset ${\displaystyle {\mathcal {B}}_{M}}$ is called the barcode of ${\displaystyle M}$, 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". Bulletin of the American Mathematical Society. 45 (1): 61–76. doi: 10.1090/S0273-0979-07-01191-3 . ISSN   0273-0979.
2. Barannikov, Sergey (1994). "Framed Morse complex and its invariants" (PDF). Advances in Soviet Mathematics. ADVSOV. 21: 93–115. doi:10.1090/advsov/021/03. ISBN   9780821802373. S2CID   125829976.
3. Carlsson, Gunnar; Zomorodian, Afra; Collins, Anne; Guibas, Leonidas (2004-07-08). "Persistence barcodes for shapes". 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. S2CID   456712.
4. Zomorodian, Afra; Carlsson, Gunnar (2005). "Computing Persistent Homology". 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". Journal of Algebra and Its Applications. 14 (5): 1550066. arXiv: 1210.0819 . doi:10.1142/S0219498815500668. ISSN   0219-4988. S2CID   119635797.
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.