Persistence module

Let $T$ be a totally ordered set and let $K$ be a field. The set $T$ is sometimes called the indexing set. Then a single-parameter persistence module $M$ is a functor $M:T\backslash to\; \backslash mathbf\{Vec\}\_K$ from the poset category of $T$ to the category of vector spaces over $K$ and linear maps.{{Cite journal |last1=Bubenik |first1=Peter |last2=Scott |first2=Jonathan A. |date=2014-04-01 |title=Categorification of Persistent Homology |url=https://doi.org/10.1007/s00454-014-9573-x |journal=Discrete & Computational Geometry |language=en |volume=51 |issue=3 |pages=600–627 |doi=10.1007/s00454-014-9573-x |s2cid=254027425 |issn=1432-0444|arxiv=1205.3669 }} A single-parameter persistence module indexed by a discrete poset such as the integers can be represented intuitively as a diagram of spaces: $$\backslash cdots\; \backslash to\; M\_\{-1\}\; \backslash to\; M\_0\; \backslash to\; M\_1\; \backslash to\; M\_2\; \backslash to\; \backslash cdots$$To emphasize the indexing set being used, a persistence module indexed by $T$ is sometimes called a $T$-persistence module, or simply a $T$-module.{{Cite journal |last=Bakke Bjerkevik |first=Håvard |date=2021 |title=On the Stability of Interval Decomposable Persistence Modules |journal=Discrete & Computational Geometry |language=en |volume=66 |issue=1 |pages=92–121 |doi=10.1007/s00454-021-00298-0 |s2cid=243797357 |issn=0179-5376|doi-access=free }} Common choices of indexing sets include $\backslash mathbb\; R,\; \backslash mathbb\; Z,\; \backslash mathbb\; N$, etc.

One can alternatively use a set-theoretic definition of a persistence module that is equivalent to the categorical viewpoint: A persistence module is a pair $(V,\backslash pi)$ where $V$ is a collection $\backslash \{V\_z\backslash \}\_\{z\backslash in\; T\}$ of $K$-vector spaces and $\backslash pi$ is a collection $\backslash \{\backslash pi\_\{y,z\}\backslash \}\_\{y\backslash leq\; z\backslash in\; T\}$ of linear maps where $\backslash pi\_\{y,z\}\; :\; V\_y\; \backslash to\; V\_z$ for each $y\backslash leq\; z\backslash in\; T$, such that $\backslash pi\_\{y,z\}\; \backslash circ\; \backslash pi\_\{x,y\}\; =\; \backslash pi\_\{x,z\}$ for any $x\; \backslash leq\; y\; \backslash leq\; z\; \backslash in\; T$ (i.e., all the maps commute).

Let $P$ be a product of $n$ totally ordered sets, i.e., $P=T\_1\; \backslash times\; \backslash dots\; \backslash times\; T\_n$ for some totally ordered sets $T\_i$. Then by endowing $P$ with the product partial order given by $(s\_1,\backslash dots,s\_n)\backslash leq\; (t\_1,\backslash dots,t\_n)$ only if $s\_i\; \backslash leq\; t\_i$ for all $i=1,\backslash dots,n$, we can define a multiparameter persistence module indexed by $P$ as a functor $M:P\backslash to\; \backslash mathbf\{Vec\}\_K$. This is a generalization of single-parameter persistence modules, and in particular, this agrees with the single-parameter definition when $n=1$.

In this case, a $P$-persistence module is referred to as an $n$-dimensional or $n$-parameter persistence module, or simply a multiparameter or multidimensional module if the number of parameters is already clear from context.{{Cite arXiv |last1=Botnan |first1=Magnus Bakke |last2=Lesnick |first2=Michael |date=2022-03-27 |title=An Introduction to Multiparameter Persistence |class=math.AT |eprint=2203.14289}}

File:Two-Parameter Persistence Module.png

Multidimensional persistence modules were first introduced in 2009 by Carlsson and Zomorodian.{{Cite journal |last1=Carlsson |first1=Gunnar |last2=Zomorodian |first2=Afra |date=2009-07-01 |title=The Theory of Multidimensional Persistence |journal=Discrete & Computational Geometry |language=en |volume=42 |issue=1 |pages=71–93 |doi=10.1007/s00454-009-9176-0 |issn=1432-0444|doi-access=free }} Since then, there has been a significant amount of research into the theory and practice of working with multidimensional modules, since they provide more structure for studying the shape of data.{{Cite book |last1=Cerri |first1=Andrea |last2=Landi |first2=Claudia |title=Discrete Geometry for Computer Imagery |chapter=The Persistence Space in Multidimensional Persistent Homology |date=2013 |editor-last=Gonzalez-Diaz |editor-first=Rocio |editor2-last=Jimenez |editor2-first=Maria-Jose |editor3-last=Medrano |editor3-first=Belen |series=Lecture Notes in Computer Science |volume=7749 |language=en |location=Berlin, Heidelberg |publisher=Springer |pages=180–191 |doi=10.1007/978-3-642-37067-0_16 |isbn=978-3-642-37067-0|doi-access=free }}{{Cite arXiv |last1=Cagliari |first1=F. |last2=Di Fabio |first2=B. |last3=Ferri |first3=M. |date=2008-07-28 |title=One-Dimensional Reduction of Multidimensional Persistent Homology |eprint=math/0702713 }}{{Cite journal |last1=Allili |first1=Madjid |last2=Kaczynski |first2=Tomasz |last3=Landi |first3=Claudia |date=2017-01-01 |title=Reducing complexes in multidimensional persistent homology theory |url=https://www.sciencedirect.com/science/article/pii/S0747717116300086 |journal=Journal of Symbolic Computation |series=Algorithms and Software for Computational Topology |language=en |volume=78 |pages=61–75 |doi=10.1016/j.jsc.2015.11.020 |s2cid=14185228 |issn=0747-7171|hdl=11380/1123249 |hdl-access=free }} Namely, multiparameter modules can have greater density sensitivity and robustness to outliers than single-parameter modules, making them a potentially useful tool for data analysis.{{Cite journal |last1=Blumberg |first1=Andrew J. |last2=Lesnick |first2=Michael |date=2022-10-17 |title=Stability of 2-Parameter Persistent Homology |url=https://doi.org/10.1007/s10208-022-09576-6 |journal=Foundations of Computational Mathematics |language=en |doi=10.1007/s10208-022-09576-6 |arxiv=2010.09628 |s2cid=224705357 |issn=1615-3383}}{{Cite journal |last1=Cerri |first1=Andrea |last2=Fabio |first2=Barbara Di |last3=Ferri |first3=Massimo |last4=Frosini |first4=Patrizio |last5=Landi |first5=Claudia |date=2013 |title=Betti numbers in multidimensional persistent homology are stable functions |url=https://onlinelibrary.wiley.com/doi/10.1002/mma.2704 |journal=Mathematical Methods in the Applied Sciences |language=en |volume=36 |issue=12 |pages=1543–1557 |doi=10.1002/mma.2704|bibcode=2013MMAS...36.1543C |s2cid=9938133 }}{{Cite arXiv |last1=Cerri |first1=Andrea |last2=Di Fabio |first2=Barbara |last3=Ferri |first3=Massimo |last4=Frosini |first4=Patrizio |last5=Landi |first5=Claudia |date=2009-08-01 |title=Multidimensional persistent homology is stable |class=math.AT |eprint=0908.0064 }}

One downside of multiparameter persistence is its inherent complexity. This makes performing computations related to multiparameter persistence modules difficult. In the worst case, the computational complexity of multidimensional persistent homology is exponential.{{Cite journal |last1=Skryzalin |first1=Jacek |last2=Vongmasa |first2=Pawin |date=2017 |title=The Computational Complexity of Multidimensional Persistence |url=https://www.osti.gov/biblio/1429696 |journal=Proposed Journal Article, Unpublished |language=English |volume=2017 |osti=1429696 |issn=}}

The most common way to measure the similarity of two multiparameter persistence modules is using the interleaving distance, which is an extension of the bottleneck distance.{{Cite journal |last=Lesnick |first=Michael |date=2015 |title=The Theory of the Interleaving Distance on Multidimensional Persistence Modules |url=http://link.springer.com/10.1007/s10208-015-9255-y |journal=Foundations of Computational Mathematics |language=en |volume=15 |issue=3 |pages=613–650 |doi=10.1007/s10208-015-9255-y |arxiv=1106.5305 |s2cid=254158297 |issn=1615-3375}}

When using homology with coefficients in a field, a homology group has the structure of a vector space. Therefore, given a filtration of spaces $F:P\; \backslash to\; \backslash mathbf\{Top\}$, by applying the homology functor at each index we obtain a persistence module $H\_i(F)\; :\; P\; \backslash to\; \backslash mathbf\{Vec\}\_K$ for each $i=1,2,\backslash dots$ called the ($i$th-dimensional) homology module of $F$. The vector spaces of the homology module can be defined index-wise as $H\_i(F)\_z\; =\; H\_i\; (F\_z)$ for all $z\backslash in\; P$, and the linear maps are induced by the inclusion maps of $F$.

Homology modules are the most ubiquitous examples of persistence modules, as they encode information about the number and scale of topological features of an object (usually derived from building a filtration on a point cloud) in a purely algebraic structure, thus making understanding the shape of the data amenable to algebraic techniques, imported from well-developed areas of mathematics such as commutative algebra and representation theory.{{Cite journal |last=Carlsson |first=Gunnar |date=2009 |title=Topology and data |url=https://www.ams.org/bull/2009-46-02/S0273-0979-09-01249-X/ |journal=Bulletin of the American Mathematical Society |language=en |volume=46 |issue=2 |pages=255–308 |doi=10.1090/S0273-0979-09-01249-X |issn=0273-0979|doi-access=free }}{{Cite journal |last1=Chazal |first1=Frédéric |last2=Michel |first2=Bertrand |date=2021 |title=An Introduction to Topological Data Analysis: Fundamental and Practical Aspects for Data Scientists |journal=Frontiers in Artificial Intelligence |volume=4 |page=667963 |doi=10.3389/frai.2021.667963 |pmid=34661095 |pmc=8511823 |issn=2624-8212|doi-access=free }}

A primary concern in the study of persistence modules is whether modules can be decomposed into "simpler pieces", roughly speaking. In particular, it is algebraically and computationally convenient if a persistence module can be expressed as a direct sum of smaller modules known as interval modules.

Let $J$ be a nonempty subset of a poset $P$. Then $J$ is an interval in $P$ if

• For every $x,z\; \backslash in\; J$ if $x\; \backslash leq\; y\; \backslash leq\; z\; \backslash in\; P$ then $y\; \backslash in\; J$
• For every $x,z\; \backslash in\; J$ there is a sequence of elements $p\_1,p\_2,\backslash dots,\; p\_n\; \backslash in\; J$ such that $p\_1=x$, $p\_n=z$, and $p\_i,\; p\_j$ are comparable for all $i,j\; \backslash in\; \backslash \{1,\backslash dots\; ,\; n\backslash \}$.

Now given an interval $J\backslash subseteq\; P$ we can define a persistence module $\backslash mathbb\; I^J$index-wise as follows:

$\backslash mathbb\; I^J\_z\; :=$

\begin{cases}

K & \text{if } z \in J\\

0 & \text{otherwise }

\end{cases} ; $\backslash mathbb\; I^J\_\{y,z\}\; :=$

\begin{cases}

\operatorname{id}_K & \text{if } y\leq z \in J\\

0 & \text{otherwise }

\end{cases} .

The module $\backslash mathbb\; I^J$ is called an interval module.{{Cite journal |last1=Botnan |first1=Magnus |last2=Lesnick |first2=Michael |date=2018-10-18 |title=Algebraic stability of zigzag persistence modules |url=https://msp.org/agt/2018/18-6/p01.xhtml |journal=Algebraic & Geometric Topology |language=en |volume=18 |issue=6 |pages=3133–3204 |doi=10.2140/agt.2018.18.3133 |arxiv=1604.00655 |s2cid=14072359 |issn=1472-2739}}

Let $a\backslash in\; P$. Then we can define a persistence module $Q^a$ with respect to $a$ where the spaces are given by

$Q^a\_z\; :=$

\begin{cases}

K & \text{if } z \geq a\\

0 & \text{otherwise }

\end{cases} , and the maps defined via $Q^a\_\{y,z\}\; :=$

\begin{cases}

\operatorname{id}_K & \text{if } z \geq a\\

0 & \text{otherwise }

\end{cases} .

Then $Q^a$ is known as a free (persistence) module.{{Cite journal |last=Lesnick |first=Michael |date=2022 |title=Lecture Notes for AMAT 840: Multiparameter Persistence |url=https://www.albany.edu/~ML644186/840_2022/Math840_Notes_22.pdf |journal=University at Albany, SUNY}}

One can also define a free module in terms of decomposition into interval modules. For each $a\backslash in\; P$ define the interval $a^\backslash llcorner\; :=\; \backslash \{\; b\; \backslash in\; P\; \backslash mid\; b\; \backslash geq\; a\; \backslash \}$, sometimes called a "free interval." Then a persistence module $F$ is a free module if there exists a multiset $\backslash mathfrak\; J(F)\; \backslash subseteq\; P$ such that $F\; =\; \backslash bigoplus\_\{a\backslash in\; \backslash mathfrak\; J(F)\}\backslash mathbb\; I^\{a^\backslash llcorner\}$. In other words, a module is a free module if it can be decomposed as a direct sum of free interval modules.

A persistence module $M$ indexed over $\backslash mathbb\; N$ is said to be of finite type if the following conditions hold for all $n\; \backslash in\; \backslash mathbb\; N$:

1. Each vector space $M\_n$ is finite-dimensional.
2. There exists an integer $N$ such that the map $M\_\{N,n\}$ is an isomorphism for all $n\; \backslash geq\; N$.

If $M$ satisfies the first condition, then $M$ is commonly said to be pointwise finite-dimensional (p.f.d.).{{Cite arXiv |last1=Botnan |first1=Magnus Bakke |last2=Crawley-Boevey |first2=William |date=2019-10-04 |title=Decomposition of persistence modules |class=math.RT |eprint=1811.08946}}{{Cite journal |last=Schmahl |first=Maximilian |date=2022 |title=Structure of semi-continuous $q$-tame persistence modules |url=https://www.intlpress.com/site/pub/pages/journals/items/hha/content/vols/0024/0001/a006/abstract.php |journal=Homology, Homotopy and Applications |language=EN |volume=24 |issue=1 |pages=117–128 |doi=10.4310/HHA.2022.v24.n1.a6 |arxiv=2008.09493 |s2cid=221246111 |issn=1532-0081}}{{Cite arXiv |last1=Hanson |first1=Eric J. |last2=Rock |first2=Job D. |date=2020-07-17 |title=Decomposition of Pointwise Finite-Dimensional S^1 Persistence Modules |class=math.RT |eprint=2006.13793}} The notion of pointwise finite-dimensionality immediately extends to arbitrary indexing sets.

The definition of finite type can also be adapted to continuous indexing sets. Namely, a module $M$ indexed over $\backslash mathbb\; R$ is of finite type if $M$ is p.f.d., and $M$ contains a finite number of unique vector spaces.{{Cite book |last1=Carlsson |first1=Gunnar |last2=Zomorodian |first2=Afra |last3=Collins |first3=Anne |last4=Guibas |first4=Leonidas |title=Proceedings of the 2004 Eurographics/ACM SIGGRAPH symposium on Geometry processing |chapter=Persistence barcodes for shapes |date=2004-07-08 |chapter-url=https://dl.acm.org/doi/10.1145/1057432.1057449 |language=en |location=Nice France |publisher=ACM |pages=124–135 |doi=10.1145/1057432.1057449 |isbn=978-3-905673-13-5|s2cid=456712 }} Formally speaking, this requires that for all but a finite number of points $x\backslash in\; \backslash mathbb\; R$ there is a neighborhood $N$ of $x$ such that $M\_y\; \backslash cong\; M\_z$ for all $y,z\; \backslash in\; N$, and also that there is some $w\; \backslash in\; \backslash mathbb\; R$ such that $M\_v\; =\; 0$ for all $v\; \backslash leq\; w$. A module satisfying only the former property is sometimes labeled essentially discrete, whereas a module satisfying both properties is known as essentially finite.{{Cite arXiv|last=Lesnick |first=Michael |date=2012-06-06 |title=Multidimensional Interleavings and Applications to Topological Inference |class=math.AT |eprint=1206.1365 }}{{Cite web |title=3. Mathematical Preliminaries — RIVET 1.0 documentation |url=https://rivet.readthedocs.io/en/latest/preliminaries.html |access-date=2023-02-27 |website=rivet.readthedocs.io}}

An $\backslash mathbb\; R$-persistence module is said to be semicontinuous if for any $x\backslash in\; \backslash mathbb\; R$ and any $y\backslash leq\; x$ sufficiently close to $x$, the map $M\_\{y,x\}:\; M\_y\; \backslash to\; M\_x$ is an isomorphism. Note that this condition is redundant if the other finite type conditions above are satisfied, so it is not typically included in the definition, but is relevant in certain circumstances.

One of the primary goals in the study of persistence modules is to classify modules according to their decomposability into interval modules. A persistence module that admits a decomposition as a direct sum of interval modules is often simply called "interval decomposable." One of the primary results in this direction is that any p.f.d. persistence module indexed over a totally ordered set is interval decomposable. This is sometimes referred to as the "structure theorem for persistence modules."

File:Interval Decomposable 2-D Persistence Module.png

The case when $P$ is finite is a straightforward application of the structure theorem for finitely generated modules over a principal ideal domain. For modules indexed over $\backslash mathbb\; Z$, the first known proof of the structure theorem is due to Webb.{{Cite journal |last=Webb |first=Cary |date=1985 |title=Decomposition of graded modules |url=https://www.ams.org/proc/1985-094-04/S0002-9939-1985-0792261-6/ |journal=Proceedings of the American Mathematical Society |language=en |volume=94 |issue=4 |pages=565–571 |doi=10.1090/S0002-9939-1985-0792261-6 |s2cid=115146035 |issn=0002-9939|doi-access=free }} The theorem was extended to the case of $\backslash mathbb\; R$ (or any totally ordered set containing a countable subset that is dense in $\backslash mathbb\; R$ with the order topology) by Crawley-Boevey in 2015.{{Cite journal |last=Crawley-Boevey |first=William |date=2015-06-01 |title=Decomposition of pointwise finite-dimensional persistence modules |url=https://www.worldscientific.com/doi/abs/10.1142/S0219498815500668 |journal=Journal of Algebra and Its Applications |volume=14 |issue=5 |pages=1550066 |doi=10.1142/S0219498815500668 |arxiv=1210.0819 |s2cid=119635797 |issn=0219-4988}} The generalized version of the structure theorem, i.e., for p.f.d. modules indexed over arbitrary totally ordered sets, was established by Botnan and Crawley-Boevey in 2019.{{Cite journal |last1=Botnan |first1=Magnus |last2=Crawley-Boevey |first2=William |date=2020 |title=Decomposition of persistence modules |url=https://www.ams.org/proc/2020-148-11/S0002-9939-2020-14790-9/ |journal=Proceedings of the American Mathematical Society |language=en |volume=148 |issue=11 |pages=4581–4596 |doi=10.1090/proc/14790 |s2cid=119711245 |issn=0002-9939|arxiv=1811.08946 }}

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