Persistence module

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This is the current revision of this page, as edited by 204.84.244.11 (talk) at 02:55, 12 December 2023 (I changed {V_z\}_{z\in P} to {V_z\}_{z\in T} since T is the index set, not P. P is defined later in the multiparameter persistence modules section, but it is not used at all in the single parameter section). The present address (URL) is a permanent link to this version.

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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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi } is a collection Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{\pi_{y,z}\}_{y\leq z\in T} } of linear maps where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi_{y,z} : V_y \to V_z } for each Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y\leq z\in T } , such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi_{y,z} \circ \pi_{x,y} = \pi_{x,z} } for any Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \leq y \leq z \in T } (i.e., all the maps commute).[4]

Multiparameter Persistence Modules

Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P} be a product of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} totally ordered sets, i.e., Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P=T_1 \times \dots \times T_n} for some totally ordered sets Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_i } . Then by endowing Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P} with the product partial order given by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (s_1,\dots,s_n)\leq (t_1,\dots,t_n)} only if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s_i \leq t_i} for all Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i=1,\dots,n} , we can define a multiparameter persistence module indexed by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P} as a functor Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M:P\to \mathbf{Vec}_K} . This is a generalization of single-parameter persistence modules, and in particular, this agrees with the single-parameter definition when Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n=1} .

In this case, a Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P} -persistence module is referred to as an Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} -dimensional or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} -parameter persistence module, or simply a multiparameter or multidimensional module if the number of parameters is already clear from context.[10]

An example of a two-parameter persistence module indexed over the 5x5 grid, considered as a finite poset.

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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F:P \to \mathbf{Top} } , by applying the homology functor at each index we obtain a persistence module Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_i(F) : P \to \mathbf{Vec}_K } for each Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i=1,2,\dots } called the (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i } th-dimensional) homology module of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F } . The vector spaces of the homology module can be defined index-wise as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_i(F)_z = H_i (F_z) } for all , and the linear maps are induced by the inclusion maps of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F } .[1]

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.[5][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.[1]

Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J } be a nonempty subset of a poset . Then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J } is an interval in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P } if

  • For every Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x,z \in J } if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \leq y \leq z \in P } then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y \in J }
  • For every Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x,z \in J } there is a sequence of elements Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p_1,p_2,\dots, p_n \in J } such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p_1=x } , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p_n=z } , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p_i, p_j } are comparable for all Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i,j \in \{1,\dots , n\} } .

Now given an interval Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J\subseteq P } we can define a persistence module Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb I^J } index-wise as follows:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb I^J_z := \begin{cases} K & \text{if } z \in J\\ 0 & \text{otherwise } \end{cases} } ; Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb I^J_{y,z} := \begin{cases} \operatorname{id}_K & \text{if } y\leq z \in J\\ 0 & \text{otherwise } \end{cases} } .

The module Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb I^J } is called an interval module.[9][22]

Free Modules

Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a\in P } . Then we can define a persistence module Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q^a } with respect to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a } where the spaces are given by

, and the maps defined via Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q^a_{y,z} := \begin{cases} \operatorname{id}_K & \text{if } z \geq a\\ 0 & \text{otherwise } \end{cases} } .

Then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q^a } is known as a free (persistence) module.[23]

One can also define a free module in terms of decomposition into interval modules. For each Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a\in P } define the interval Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a^\llcorner := \{ b \in P \mid b \geq a \} } , sometimes called a "free interval."[9] Then a persistence module Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F } is a free module if there exists a multiset Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak J(F) \subseteq P } such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F = \bigoplus_{a\in \mathfrak J(F)}\mathbb I^{a^\llcorner} } .[22] 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M } indexed over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb N } is said to be of finite type if the following conditions hold for all Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n \in \mathbb N } :

  1. Each vector space Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M_n } is finite-dimensional.
  2. There exists an integer Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N } such that the map Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M_{N,n} } is an isomorphism for all Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n \geq N } .

If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M } satisfies the first condition, then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M } 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M } indexed over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb R } is of finite type if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M } is p.f.d., and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M } contains a finite number of unique vector spaces.[27] Formally speaking, this requires that for all but a finite number of points Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x\in \mathbb R } there is a neighborhood Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N } of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x } such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M_y \cong M_z } for all Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y,z \in N } , and also that there is some Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle w \in \mathbb R } such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M_v = 0 } for all Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v \leq w } .[4] A module satisfying only the former property is sometimes labeled essentially discrete, whereas a module satisfying both properties is known as essentially finite.[28][23][29]

An Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb R } -persistence module is said to be semicontinuous if for any Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x\in \mathbb R } and any Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y\leq x } sufficiently close to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x } , the map Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M_{y,x}: M_y \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.[4]

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."[24]

An example of a 2-D persistence module in the plane with its interval decompositions.

The case when Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P } is finite is a straightforward application of the structure theorem for finitely generated modules over a principal ideal domain. For modules indexed over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb Z } , the first known proof of the structure theorem is due to Webb.[30] The theorem was extended to the case of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb R } (or any totally ordered set containing a countable subset that is dense in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb R } 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

  1. ^ 1.0 1.1 1.2 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.
  2. ^ The structure and stability of persistence modules. Frédéric Chazal, Vin De Silva, Marc Glisse, Steve Y. Oudot. Switzerland. 2016. ISBN 978-3-319-42545-0. OCLC 960458101.{{cite book}}: CS1 maint: location missing publisher (link) CS1 maint: others (link)
  3. ^ Oudot, Steve Y. (2015). Persistence theory : from quiver representations to data analysis. Providence, Rhode Island. ISBN 978-1-4704-2545-6. OCLC 918149730.{{cite book}}: CS1 maint: location missing publisher (link)
  4. ^ 4.0 4.1 4.2 4.3 Polterovich, Leonid (2020). Topological persistence in geometry and analysis. Daniel Rosen, Karina Samvelyan, Jun Zhang. Providence, Rhode Island. ISBN 978-1-4704-5495-1. OCLC 1142009348.{{cite book}}: CS1 maint: location missing publisher (link)
  5. ^ 5.0 5.1 Schenck, Hal (2022). Algebraic foundations for applied topology and data analysis. Cham. ISBN 978-3-031-06664-1. OCLC 1351750760.{{cite book}}: CS1 maint: location missing publisher (link)
  6. ^ Dey, Tamal K. (2022). Computational topology for data analysis. Yusu Wang. Cambridge, United Kingdom. ISBN 978-1-009-09995-0. OCLC 1281786176.{{cite book}}: CS1 maint: location missing publisher (link)
  7. ^ Rabadan, Raul; Blumberg, Andrew J. (2019). Topological Data Analysis for Genomics and Evolution: Topology in Biology. Cambridge: Cambridge University Press. doi:10.1017/9781316671665. ISBN 978-1-107-15954-9. S2CID 242498045.
  8. ^ Bubenik, Peter; Scott, Jonathan A. (2014-04-01). "Categorification of Persistent Homology". Discrete & Computational Geometry. 51 (3): 600–627. arXiv:1205.3669. doi:10.1007/s00454-014-9573-x. ISSN 1432-0444. S2CID 254027425.
  9. ^ 9.0 9.1 9.2 Bakke Bjerkevik, Håvard (2021). "On the Stability of Interval Decomposable Persistence Modules". Discrete & Computational Geometry. 66 (1): 92–121. doi:10.1007/s00454-021-00298-0. ISSN 0179-5376. S2CID 243797357.
  10. ^ Botnan, Magnus Bakke; Lesnick, Michael (2022-03-27). "An Introduction to Multiparameter Persistence". arXiv:2203.14289 [math.AT].
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  12. ^ Cerri, Andrea; Landi, Claudia (2013). "The Persistence Space in Multidimensional Persistent Homology". In Gonzalez-Diaz, Rocio; Jimenez, Maria-Jose; Medrano, Belen (eds.). Discrete Geometry for Computer Imagery. Lecture Notes in Computer Science. Vol. 7749. Berlin, Heidelberg: Springer. pp. 180–191. doi:10.1007/978-3-642-37067-0_16. ISBN 978-3-642-37067-0.
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  16. ^ 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. 36 (12): 1543–1557. Bibcode:2013MMAS...36.1543C. doi:10.1002/mma.2704. S2CID 9938133.
  17. ^ Cerri, Andrea; Di Fabio, Barbara; Ferri, Massimo; Frosini, Patrizio; Landi, Claudia (2009-08-01). "Multidimensional persistent homology is stable". arXiv:0908.0064 [math.AT].
  18. ^ Skryzalin, Jacek; Vongmasa, Pawin (2017). "The Computational Complexity of Multidimensional Persistence". Proposed Journal Article, Unpublished. 2017. OSTI 1429696.
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