Persistence module explained
A persistence module is a mathematical structure in persistent homology and topological data analysis that formally captures the persistence of topological features of an object across a range of scale parameters. A persistence module often consists of a collection of homology groups (or vector spaces if using field coefficients) corresponding to a filtration of topological spaces, and a collection of linear maps induced by the inclusions of the filtration. The concept of a persistence module was first introduced in 2005 as an application of graded modules over polynomial rings, thus importing well-developed algebraic ideas from classical commutative algebra theory to the setting of persistent homology.[1] Since then, persistence modules have been one of the primary algebraic structures studied in the field of applied topology.[2] [3] [4] [5] [6] [7]
Definition
Single Parameter Persistence Modules
Let
be a totally ordered set and let
be a
field. The set
is sometimes called the
indexing set. Then a single-parameter
persistence module
is a
functor
from the poset
category of
to the category of
vector spaces over
and
linear maps.
[8] A single-parameter persistence module indexed by a
discrete poset such as the
integers can be represented intuitively as a diagram of spaces:
To emphasize the indexing set being used, a persistence module indexed by
is sometimes called a
-persistence module, or simply a
-module.
[9] Common choices of indexing sets include
, 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
where
is a collection
of
-vector spaces and
is a collection
of linear maps where
for each
, such that
for any
(i.e., all the maps
commute).
Multiparameter Persistence Modules
Let
be a product of
totally ordered sets, i.e.,
for some totally ordered sets
. Then by endowing
with the product
partial order given by
(s1,...,sn)\leq(t1,...,tn)
only if
for all
, we can define a
multiparameter persistence module indexed by
as a functor
. This is a generalization of single-parameter persistence modules, and in particular, this agrees with the single-parameter definition when
.
In this case, a
-persistence module is referred to as an
-dimensional or
-parameter persistence module, or simply a multiparameter or multidimensional module if the number of parameters is already clear from context.
[10] Multidimensional persistence modules were first introduced in 2009 by Carlsson and Zomorodian.
[11] 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.
[12] [13] [14] 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.
[15] [16] [17] 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.[18]
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.[19]
Examples
Homology Modules
When using homology with coefficients in a field, a homology group has the structure of a vector space. Therefore, given a filtration of spaces
, by applying the homology
functor at each index we obtain a persistence module
for each
called the (
th-dimensional)
homology module of
. The vector spaces of the homology module can be defined index-wise as
for all
, and the
linear maps are
induced by the
inclusion maps of
.
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.[20] [21]
Interval Modules
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
be a nonempty subset of a poset
. Then
is an
interval in
if
if
then
there is a sequence of elements
such that
,
, and
are comparable for all
.
Now given an interval
we can define a persistence module
index-wise as follows:
:=\begin{cases}
K&ifz\inJ\\
0&otherwise
\end{cases}
:=\begin{cases}
\operatorname{id}K&ify\leqz\inJ\\
0&otherwise
\end{cases}
.
The module
is called an
interval module.
[22] Free Modules
Let
. Then we can define a persistence module
with respect to
where the spaces are given by
:=\begin{cases}
K&ifz\geqa\\
0&otherwise
\end{cases}
, and the maps defined via
:=\begin{cases}
\operatorname{id}K&ifz\geqa\\
0&otherwise
\end{cases}
.
Then
is known as a
free (persistence) module.
[23] One can also define a free module in terms of decomposition into interval modules. For each
define the interval
a\llcorner:=\{b\inP\midb\geqa\}
, sometimes called a "free interval." Then a persistence module
is a free module if there exists a
multiset
such that
. In other words, a module is a free module if it can be decomposed as a direct sum of free interval modules.
Properties
Finite Type Conditions
A persistence module
indexed over
is said to be of
finite type if the following conditions hold for all
:
- Each vector space
is finite-dimensional.
- There exists an integer
such that the map
is an isomorphism for all
.
If
satisfies the first condition, then
is commonly said to be
pointwise finite-dimensional (p.f.d.).[24] [25] [26] 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
indexed over
is of finite type if
is p.f.d., and
contains a finite number of unique vector spaces.
[27] Formally speaking, this requires that for all but a finite number of points
there is a neighborhood
of
such that
for all
, and also that there is some
such that
for all
. A module satisfying only the former property is sometimes labeled
essentially discrete, whereas a module satisfying both properties is known as
essentially finite.[28] [29] An
-persistence module is said to be
semicontinuous if for any
and any
sufficiently close to
, the map
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.
Structure Theorem
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."
The case when
is finite is a straightforward application of the
structure theorem for finitely generated modules over a
principal ideal domain. For modules indexed over
, the first known proof of the structure theorem is due to Webb.
[30] The theorem was extended to the case of
(or any totally ordered set containing a
countable subset that is
dense in
with the
order topology) by Crawley-Boevey in 2015.
[31] 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.
[32] References
- 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 .
- Book: The structure and stability of persistence modules . 2016 . Frédéric Chazal, Vin De Silva, Marc Glisse, Steve Y. Oudot . 978-3-319-42545-0 . Switzerland . 960458101.
- Book: Oudot, Steve Y. . Persistence theory : from quiver representations to data analysis . 2015 . 978-1-4704-2545-6 . Providence, Rhode Island . 918149730.
- Book: Polterovich, Leonid . Topological persistence in geometry and analysis . 2020 . Daniel Rosen, Karina Samvelyan, Jun Zhang . 978-1-4704-5495-1 . Providence, Rhode Island . 1142009348.
- Book: Schenck, Hal . Algebraic foundations for applied topology and data analysis . 2022 . 978-3-031-06664-1 . Cham . 1351750760.
- Book: Dey, Tamal K. . Computational topology for data analysis . 2022 . Yusu Wang . 978-1-009-09995-0 . Cambridge, United Kingdom . 1281786176.
- Book: Rabadan . Raul . Topological Data Analysis for Genomics and Evolution: Topology in Biology . Blumberg . Andrew J. . 2019 . Cambridge University Press . 978-1-107-15954-9 . Cambridge . 10.1017/9781316671665. 242498045 .
- Bubenik . Peter . Scott . Jonathan A. . 2014-04-01 . Categorification of Persistent Homology . Discrete & Computational Geometry . en . 51 . 3 . 600–627 . 10.1007/s00454-014-9573-x . 254027425 . 1432-0444. 1205.3669 .
- Bakke Bjerkevik . Håvard . 2021 . On the Stability of Interval Decomposable Persistence Modules . Discrete & Computational Geometry . en . 66 . 1 . 92–121 . 10.1007/s00454-021-00298-0 . 243797357 . 0179-5376. free .
- Botnan . Magnus Bakke . Lesnick . Michael . 2022-03-27 . An Introduction to Multiparameter Persistence . math.AT . 2203.14289.
- Carlsson . Gunnar . Zomorodian . Afra . 2009-07-01 . The Theory of Multidimensional Persistence . Discrete & Computational Geometry . en . 42 . 1 . 71–93 . 10.1007/s00454-009-9176-0 . 1432-0444. free .
- Book: Cerri . Andrea . Landi . Claudia . Discrete Geometry for Computer Imagery . The Persistence Space in Multidimensional Persistent Homology . 2013 . Gonzalez-Diaz . Rocio . Jimenez . Maria-Jose . Medrano . Belen . Lecture Notes in Computer Science . 7749 . en . Berlin, Heidelberg . Springer . 180–191 . 10.1007/978-3-642-37067-0_16 . 978-3-642-37067-0. free .
- Cagliari . F. . Di Fabio . B. . Ferri . M. . 2008-07-28 . One-Dimensional Reduction of Multidimensional Persistent Homology . math/0702713 .
- Allili . Madjid . Kaczynski . Tomasz . Landi . Claudia . 2017-01-01 . Reducing complexes in multidimensional persistent homology theory . Journal of Symbolic Computation . Algorithms and Software for Computational Topology . en . 78 . 61–75 . 10.1016/j.jsc.2015.11.020 . 14185228 . 0747-7171. 11380/1123249 . free .
- Blumberg . Andrew J. . Lesnick . Michael . 2022-10-17 . Stability of 2-Parameter Persistent Homology . Foundations of Computational Mathematics . en . 10.1007/s10208-022-09576-6 . 2010.09628 . 224705357 . 1615-3383.
- Cerri . Andrea . Fabio . Barbara Di . Ferri . Massimo . Frosini . Patrizio . Landi . Claudia . 2013 . Betti numbers in multidimensional persistent homology are stable functions . Mathematical Methods in the Applied Sciences . en . 36 . 12 . 1543–1557 . 10.1002/mma.2704. 2013MMAS...36.1543C . 9938133 .
- Cerri . Andrea . Di Fabio . Barbara . Ferri . Massimo . Frosini . Patrizio . Landi . Claudia . 2009-08-01 . Multidimensional persistent homology is stable . math.AT . 0908.0064 .
- Skryzalin . Jacek . Vongmasa . Pawin . 2017 . The Computational Complexity of Multidimensional Persistence . Proposed Journal Article, Unpublished . English . 2017 . 1429696 .
- Lesnick . Michael . 2015 . The Theory of the Interleaving Distance on Multidimensional Persistence Modules . Foundations of Computational Mathematics . en . 15 . 3 . 613–650 . 10.1007/s10208-015-9255-y . 1106.5305 . 254158297 . 1615-3375.
- Carlsson . Gunnar . 2009 . Topology and data . Bulletin of the American Mathematical Society . en . 46 . 2 . 255–308 . 10.1090/S0273-0979-09-01249-X . 0273-0979. free .
- Chazal . Frédéric . Michel . Bertrand . 2021 . An Introduction to Topological Data Analysis: Fundamental and Practical Aspects for Data Scientists . Frontiers in Artificial Intelligence . 4 . 667963 . 10.3389/frai.2021.667963 . 34661095 . 8511823 . 2624-8212. free .
- Botnan . Magnus . Lesnick . Michael . 2018-10-18 . Algebraic stability of zigzag persistence modules . Algebraic & Geometric Topology . en . 18 . 6 . 3133–3204 . 10.2140/agt.2018.18.3133 . 1604.00655 . 14072359 . 1472-2739.
- Lesnick . Michael . 2022 . Lecture Notes for AMAT 840: Multiparameter Persistence . University at Albany, SUNY.
- Botnan . Magnus Bakke . Crawley-Boevey . William . 2019-10-04 . Decomposition of persistence modules . math.RT . 1811.08946.
- Schmahl . Maximilian . 2022 . Structure of semi-continuous $q$-tame persistence modules . Homology, Homotopy and Applications . EN . 24 . 1 . 117–128 . 10.4310/HHA.2022.v24.n1.a6 . 2008.09493 . 221246111 . 1532-0081.
- Hanson . Eric J. . Rock . Job D. . 2020-07-17 . Decomposition of Pointwise Finite-Dimensional S^1 Persistence Modules . math.RT . 2006.13793.
- 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 .
- Lesnick . Michael . 2012-06-06 . Multidimensional Interleavings and Applications to Topological Inference . math.AT . 1206.1365 .
- Web site: 3. Mathematical Preliminaries — RIVET 1.0 documentation . 2023-02-27 . rivet.readthedocs.io.
- Webb . Cary . 1985 . Decomposition of graded modules . Proceedings of the American Mathematical Society . en . 94 . 4 . 565–571 . 10.1090/S0002-9939-1985-0792261-6 . 115146035 . 0002-9939. free .
- Crawley-Boevey . William . 2015-06-01 . Decomposition of pointwise finite-dimensional persistence modules . Journal of Algebra and Its Applications . 14 . 5 . 1550066 . 10.1142/S0219498815500668 . 1210.0819 . 119635797 . 0219-4988.
- Botnan . Magnus . Crawley-Boevey . William . 2020 . Decomposition of persistence modules . Proceedings of the American Mathematical Society . en . 148 . 11 . 4581–4596 . 10.1090/proc/14790 . 119711245 . 0002-9939. 1811.08946 .