Theory Glossary ¶

Contents

Theory Glossary

Symbols ¶

\(\Bbbk\)	An arbitrary field.
\(\mathbb R\)	The field of real numbers.
\(\overline{\mathbb R}\)	The two point compactification \([-\infty, +\infty]\) of the real numbers.
\(\mathbb N\)	The counting numbers \(0,1,2, \ldots\) as a subset of \(\mathbb R\).
\(\mathbb R^d\)	The vector space of \(d\)-tuples of real numbers.
\(\Delta\)	The multiset \(\lbrace (s , s) \mid s \in \mathbb{R} \rbrace\) with multiplicity \(( s,s ) \mapsto +\infty\).

An elementary interval \(I_a\) is a subset of \(\mathbb{R}\) of the form \([a, a+1]\) or \([a,a] = \{a\}\) for some \(a \in \mathbb{R}\). These two types are called respectively non-degenerate and degenerate. To a non-degenerate elementary interval we assign two degenerate elementary intervals

\[d^+ I_a = [a+1, a+1] \qquad \text{and} \qquad d^- I_a = [a, a].\]

An elementary cube is a subset of the form

\[I_{a_1} \times \cdots \times I_{a_N} \subset \mathbb{R}^N\]

where each \(I_{a_i}\) is an elementary interval. We refer to the total number of its non-degenerate factors \(I_{a_{k_1}}, \dots, I_{a_{k_n}}\) as its dimension and, assuming

\[a_{k_1} < \cdots < a_{k_{n,}}\]

we define for \(i = 1, \dots, n\) the following two elementary cubes

\[d_i^\pm I^N = I_{a_1} \times \cdots \times d^\pm I_{a_{k_i}} \times \cdots \times I_{a_{N.}}\]

A cubical complex is a finite set of elementary cubes of \(\mathbb{R}^N\), and a subcomplex of \(X\) is a cubical complex whose elementary cubes are also in \(X\). We denote the set of \(n\)-dimensional cubes as \(X_n\).

Reference:¶

(Kaczynski, Mischaikow, and Mrozek 2004)

Simplicial complex ¶

A set \(\{v_0, \dots, v_n\} \subset \mathbb{R}^N\) is said to be geometrically independent if the vectors \(\{v_0-v_1, \dots, v_0-v_n\}\) are linearly independent. In this case, we refer to their convex closure as a simplex, explicitly

\[\lbrack v_0, \ldots , v_n \rbrack = \left\{ \sum c_i (v_0 - v_i)\ \big|\ c_1+\dots+c_n = 1,\ c_i \geq 0 \right\}\]

and to \(n\) as its dimension. The \(i\)-th face of \([v_0, \dots, v_n]\) is defined by

\[d_i[v_0, \ldots, v_n] = [v_0, \dots, \widehat{v}_i, \dots, v_n]\]

where \(\widehat{v}_i\) denotes the absence of \(v_i\) from the set.

A simplicial complex \(X\) is a finite union of simplices in \(\mathbb{R}^N\) satisfying that every face of a simplex in \(X\) is in \(X\) and that the non-empty intersection of two simplices in \(X\) is a face of each. Every simplicial complex defines an abstract simplicial complex.

Abstract simplicial complex ¶

An abstract simplicial complex is a pair of sets \((V, X)\) with the elements of \(X\) being subsets of \(V\) such that:

for every \(v\) in \(V\), the singleton \(\{v\}\) is in \(X\) and
if \(x\) is in \(X\) and \(y\) is a subset of \(x\), then \(y\) is in \(X\).

We abuse notation and denote the pair \((V, X)\) simply by \(X\).

The elements of \(X\) are called simplices and the dimension of a simplex \(x\) is defined by \(|x| = \# x - 1\) where \(\# x\) denotes the cardinality of \(x\). Simplices of dimension \(d\) are called \(d\)-simplices. We abuse terminology and refer to the elements of \(V\) and to their associated \(0\)-simplices both as vertices.

The \(k\)-skeleton \(X_k\) of a simplicial complex \(X\) is the subcomplex containing all simplices of dimension at most \(k\). A simplicial complex is said to be \(d\)-dimensional if \(d\) is the smallest integer satisfying \(X = X_d\).

A simplicial map between simplicial complexes is a function between their vertices such that the image of any simplex via the induced map is a simplex.

A simplicial complex \(X\) is a subcomplex of a simplicial complex \(Y\) if every simplex of \(X\) is a simplex of \(Y\).

Given a finite abstract simplicial complex \(X = (V, X)\) we can choose a bijection from \(V\) to a geometrically independent subset of \(\mathbb R^N\) and associate a simplicial complex to \(X\) called its geometric realization.

Ordered simplicial complex ¶

An ordered simplicial complex is an abstract simplicial complex where the set of vertices is equipped with a partial order such that the restriction of this partial order to any simplex is a total order. We denote an \(n\)-simplex using its ordered vertices by \([v_0, \dots, v_n]\).

A simplicial map between ordered simplicial complexes is a simplicial map \(f\) between their underlying simplicial complexes preserving the order, i.e., \(v \leq w\) implies \(f(v) \leq f(w)\).

Directed simplicial complex ¶

A directed simplicial complex is a pair of sets \((V, X)\) with the elements of \(X\) being tuples of elements of \(V\), i.e., elements in \(\bigcup_{n\geq1} V^{\times n}\) such that:

for every \(v\) in \(V\), the tuple \(v\) is in \(X\)
if \(x\) is in \(X\) and \(y\) is a subtuple of \(x\), then \(y\) is in \(X\).

With appropriate modifications the same terminology and notation introduced for ordered simplicial complex applies to directed simplicial complex.

Chain complex ¶

A chain complex of is a pair \((C_*, \partial)\) where

\[C_* = \bigoplus_{n \in \mathbb Z} C_n \quad \mathrm{and} \quad \partial = \bigoplus_{n \in \mathbb Z} \partial_n\]

with \(C_n\) a \(\Bbbk\)-vector space and \(\partial_n : C_{n+1} \to C_n\) is a \(\Bbbk\)-linear map such that \(\partial_{n+1} \partial_n = 0\). We refer to \(\partial\) as the boundary map of the chain complex.

The elements of \(C\) are called chains and if \(c \in C_n\) we say its degree is \(n\) or simply that it is an \(n\)-chain. Elements in the kernel of \(\partial\) are called cycles, and elements in the image of \(\partial\) are called boundaries. Notice that every boundary is a cycle. This fact is central to the definition of homology.

A chain map is a \(\Bbbk\)-linear map \(f : C \to C'\) between chain complexes such that \(f(C_n) \subseteq C'_n\) and \(\partial f = f \partial\).

Given a chain complex \((C_*, \partial)\), its linear dual \(C^*\) is also a chain complex with \(C^{-n} = \mathrm{Hom_\Bbbk}(C_n, \Bbbk)\) and boundary map \(\delta\) defined by \(\delta(\alpha)(c) = \alpha(\partial c)\) for any \(\alpha \in C^*\) and \(c \in C_*\).

Homology and cohomology ¶

Let \((C_*, \partial)\) be a chain complex. Its \(n\)-th homology group is the quotient of the subspace of \(n\)-cycles by the subspace of \(n\)-boundaries, that is, \(H_n(C_*) = \mathrm{ker}(\partial_n)/ \mathrm{im}(\partial_{n+1})\). The homology of \((C, \partial)\) is defined by \(H_*(C) = \bigoplus_{n \in \mathbb Z} H_n(C)\).

When the chain complex under consideration is the linear dual of a chain complex we sometimes refer to its homology as the cohomology of the predual complex and write \(H^n\) for \(H_{-n}\).

A chain map \(f : C \to C'\) induces a map between the associated homologies.

Simplicial chains and simplicial homology ¶

Let \(X\) be an ordered or directed simplicial complex. Define its simplicial chain complex with \(\Bbbk\)-coefficients \(C_*(X; \Bbbk)\) by

\[C_n(X; \Bbbk) = \Bbbk\{X_n\}, \qquad \partial_n(x) = \sum_{i=0}^{n} (-1)^i d_ix\]

and its homology and cohomology with \(\Bbbk\)-coefficients as the homology and cohomology of this chain complex. We use the notation \(H_*(X; \Bbbk)\) and \(H^*(X; \Bbbk)\) for these.

A simplicial map induces a chain map between the associated simplicial chain complexes and, therefore, between the associated simplicial (co)homologies.

Cubical chains and cubical homology ¶

Let \(X\) be a cubical complex. Define its cubical chain complex with \(\Bbbk\)-coefficients \(C_*(X; \Bbbk)\) by

\[C_n(X; \Bbbk) = \Bbbk\{X_n\}, \qquad \partial_n x = \sum_{i = 1}^{n} (-1)^{i-1}(d^+_i x - d^-_i x)\]

where \(x = I_1 \times \cdots \times I_N\) and \(s(i)\) is the dimension of \(I_1 \times \cdots \times I_i\). Its homology and cohomology with \(\Bbbk\)-coefficients is the homology and cohomology of this chain complex. We use the notation \(H_*(X; \Bbbk)\) and \(H^*(X; \Bbbk)\) for these.

Filtered complex ¶

A filtered complex is a collection of simplicial or cubical complexes \(\{X_s\}_{s \in \mathbb R}\) such that \(X_s\) is a subcomplex of \(X_t\) for each \(s \leq t\).

Cellwise filtration ¶

A cellwise filtration is a simplicial or cubical complex \(X\) together with a total order \(\leq\) on its simplices or elementary cubes such that for each \(y \in X\) the set \(\{x \in X\ :\ x \leq y\}\) is a subcomplex of \(X\). A cellwise filtration can be naturally thought of as a filtered complex.

Clique and flag complexes ¶

Let \(G\) be a \(1\)-dimensional abstract (resp. directed) simplicial complex. The abstract (resp. directed) simplicial complex \(\langle G \rangle\) has the same set of vertices as \(G\) and \(\{v_0, \dots, v_n\}\) (resp. \((v_0, \dots, v_n)\)) is a simplex in \(\langle G \rangle\) if an only if \(\{v_i, v_j\}\) (resp. \((v_i, v_j)\)) is in \(G\) for each pair of vertices \(v_i, v_j\).

An abstract (resp. directed) simplicial complex \(X\) is a clique (resp. flag) complex if \(X = \langle G \rangle\) for some \(G\).

Given a function

\[w : G \to \mathbb R \cup \{\infty\}\]

consider the extension

\[w : \langle G \rangle \to \mathbb R \cup \{\infty\}\]

defined respectively by

\[\begin{split}\begin{aligned} w\{v_0, \dots, v_n\} & = \max\{ w\{v_i, v_j\}\ |\ i \neq j\} \\ w(v_0, \dots, v_n) & = \max\{ w(v_i, v_j)\ |\ i < j\} \end{aligned}\end{split}\]

and define the filtered complex \(\{\langle G \rangle_{s}\}_{s \in \mathbb R}\) by

\[\langle G \rangle_s = \{\sigma \in \langle G \rangle\ |\ w(\sigma) \leq s\}.\]

A filtered complex \(\{X_s\}_{s \in \mathbb R}\) is a filtered clique (resp. flag) complex if \(X_s = \langle G \rangle_s\) for some \((G,w)\).

Persistence module ¶

A persistence module is a collection containing a \(\Bbbk\)-vector spaces \(V(s)\) for each real number \(s\) together with \(\Bbbk\)-linear maps \(f_{st} : V(s) \to V(t)\), referred to as structure maps, for each pair \(s \leq t\), satisfying naturality, i.e., if \(r \leq s \leq t\), then \(f_{rt} = f_{st} \circ f_{rs}\) and tameness, i.e., all but finitely many structure maps are isomorphisms.

A morphism of persistence modules \(F : V \to W\) is a collection of linear maps \(F(s) : V(s) \to W(s)\) such that \(F(t) \circ f_{st} = f_{st} \circ F(s)\) for each par of reals \(s \leq t\). We say that \(F\) is an isomorphisms if each \(F(s)\) is.

Persistent simplicial (co)homology ¶

Let \(\{X(s)\}_{s \in \mathbb R}\) be a set of ordered or directed simplicial complexes together with simplicial maps \(f_{st} : X(s) \to X(t)\) for each pair \(s \leq t\), such that

\[r \leq s \leq t\ \quad\text{implies} \quad f_{rt} = f_{st} \circ f_{rs}\]

for example, a filtered complex. Its persistent simplicial homology with \(\Bbbk\)-coefficients is the persistence module

\[H_*(X(s); \Bbbk)\]

with structure maps \(H_*(f_{st}) : H_*(X(s); \Bbbk) \to H_*(X(t); \Bbbk)\) induced form the maps \(f_{st.}\) In general, the collection constructed this way needs not satisfy the tameness condition of a persistence module, but we restrict attention to the cases where it does. Its persistence simplicial cohomology with \(\Bbbk\)-coefficients is defined analogously.

Vietoris-Rips complex and Vietoris-Rips persistence ¶

Let \((X, d)\) be a finite metric space. Define the Vietoris-Rips complex of \(X\) as the filtered complex \(VR_s(X)\) that contains a subset of \(X\) as a simplex if all pairwise distances in the subset are less than or equal to \(s\), explicitly

\[VR_s(X) = \Big\{ [v_0,\dots,v_n]\ \Big|\ \forall i,j\ \,d(v_i, v_j) \leq s \Big\}.\]

The Vietoris-Rips persistence of \((X, d)\) is the persistent simplicial (co)homology of \(VR_s(X)\).

A more general version is obtained by replacing the distance function with an arbitrary function

\[w : X \times X \to \mathbb R \cup \{\infty\}\]

and defining \(VR_s(X)\) as the filtered clique complex associated to \((X \times X ,w)\).

Čech complex and Čech persistence ¶

Let \((X, d)\) be a point cloud. Define the Čech complex of \(X\) as the filtered complex \(\check{C}_s(X)\) that is empty if \(s<0\) and, if \(s \geq 0\), contains a subset of \(X\) as a simplex if the balls of radius \(s\) with centers in the subset have a non-empty intersection, explicitly

\[\check{C}_s(X) = \Big\{ [v_0,\dots,v_n]\ \Big|\ \bigcap_{i=0}^n B_s(x_i) \neq \emptyset \Big\}.\]

The Čech persistence (co)homology of \((X,d)\) is the persistent simplicial (co)homo-logy of \(\check{C}_s(X)\).

Multiset ¶

A multiset is a pair \((S, \phi)\) where \(S\) is a set and \(\phi : S \to \mathbb N \cup \{+\infty\}\) is a function attaining positive values. For \(s \in S\) we refer to \(\phi(s)\) as its multiplicity. The union of two multisets \((S_1, \phi_1), (S_2, \phi_2)\) is the multiset \((S_1 \cup S_2, \phi_1 \cup \phi_2)\) with

\[\begin{split}(\phi_1 \cup \phi_2)(s) = \begin{cases} \phi_1(s) & s \in S_1, s \not\in S_2 \\ \phi_2(s) & s \in S_2, s \not\in S_1 \\ \phi_1(s) + \phi_2(s) & s \in S_1, s \in S_2. \\ \end{cases}\end{split}\]

Persistence diagram ¶

A persistence diagram is a multiset of points in

\[\mathbb R \times \big( \mathbb{R} \cup \{+\infty\} \big).\]

Given a persistence module, its associated persistence diagram is determined by the following condition: for each pair \(s,t\) the number counted with multiplicity of points \((b,d)\) in the multiset, satisfying \(b \leq s \leq t < d\) is equal to the rank of \(f_{st.}\)

A well known result establishes that there exists an isomorphism between two persistence module if and only if their persistence diagrams are equal.

Wasserstein and bottleneck distance ¶

The \(p\)-Wasserstein distance between two persistence diagrams \(D_1\) and \(D_2\) is the infimum over all bijections \(\gamma: D_1 \cup \Delta \to D_2 \cup \Delta\) of

\[\Big(\sum_{x \in D_1 \cup \Delta} ||x - \gamma(x)||_\infty^p \Big)^{1/p}\]

where \(||-||_\infty\) is defined for \((x,y) \in \mathbb R^2\) by \(\max\{|x|, |y|\}\).

The limit \(p \to \infty\) defines the bottleneck distance. More explicitly, it is the infimum over the same set of bijections of the value

\[\sup_{x \in D_1 \cup \Delta} ||x - \gamma(x)||_{\infty.}\]

Reference:¶

(Kerber, Morozov, and Nigmetov 2017)

Persistence landscape ¶

A persistence landscape is a set \(\{\lambda_k\}_{k \in \mathbb N}\) of functions

\[\lambda : \mathbb R \to \overline{\mathbb R}\]

where \(\lambda_k\) is referred to as the \(k\)-layer of the persistence landscape.

Let \(\{(b_i, d_i)\}_{i \in I}\) be a persistence diagram. Its associated persistence landscape \(\lambda\) is defined by letting \(\lambda_k\) be the \(k\)-th largest value of the set \(\{\Lambda_i(t)\}_ {i \in I}\) where

\[\Lambda_i(t) = \left[ \min \{t-b_i, d_i-t\}\right]_+\]

and \(c_+ := \max(c,0)\).

Intuitively, we can describe the set of graphs of a persistence landscape by first joining each of the points in the multiset to the diagonal via a horizontal as well as a vertical line, then clockwise rotating the figure 45 degrees and rescaling it by \(1/\sqrt{2}\).

Reference:¶

(Bubenik 2015)

Persistence landscape norm ¶

Given a function \(f : \mathbb R \to \overline{\mathbb R}\) define

\[||f||_p = \left( \int_{\mathbb R} f^p(x)\, dx \right)^{1/p}\]

whenever the right hand side exists and is finite.

The \(p\)-norm of a persistence landscape \(\lambda = \{\lambda_k\}_{k \in \mathbb N}\) is defined to be

\[||\lambda||_p = \left( \sum_{i \in \mathbb N} ||\lambda_i||^p_p \right)^{1/p}\]

whenever the right hand side exists and is finite.

References:¶

(Stein and Shakarchi 2011; Bubenik 2015)

Weighted silhouette ¶

Let \(D = {(b_i, d_i)}_{i \in I}\) be a persistence diagram. A weighted silhouette associated to \(D\) is a continuous function \(\phi : \mathbb R \to \mathbb R\) of the form

\[\phi(t) = \frac{\sum_{i \in I}w_i \Lambda_i(t)}{\sum_{i \in I}w_i},\]

where \(\{w_i\}_{i \in I}\) is a set of positive real numbers and

\[\label{equation:lambda_for_persistence_landscapes} \Lambda_i(t) = \left[ \min \{t-b_i, d_i-t\}\right]_+\]

with \(c_+ := \max(c,0)\). The particular choice \(w_i = \vert d_i - b_i \vert^p\) for \(0 < p \leq \infty\) is referred to as power-weighted silhouettes.

References:¶

(Chazal et al. 2014)

Amplitude ¶

Given a function assigning a real number to a pair of persistence diagrams, we define the amplitude of a persistence diagram \(D\) to be the value assigned to the pair \((D \cup \Delta, \Delta)\). Important examples of such functions are: Wasserstein and bottleneck distances and landscape distance.

Persistence entropy ¶

Intuitively, this is a measure of the entropy of the points in a persistence diagram. Precisely, let \(D = \{(b_i, d_i)\}_{i \in I}\) be a persistence diagram with each \(d_i < +\infty\). The persistence entropy of \(D\) is defined by

\[E(D) = - \sum_{i \in I} p_i \log(p_i)\]

where

\[p_i = \frac{(d_i - b_i)}{L_D} \qquad \text{and} \qquad L_D = \sum_{i \in I} (d_i - b_i) .\]

References:¶

(Rucco et al. 2016)

Betti curve ¶

Let \(D\) be a persistence diagram. Its Betti curve is the function \(\beta_D : \mathbb R \to \mathbb N\) whose value on \(s \in \mathbb R\) is the number, counted with multiplicity, of points \((b_i,d_i)\) in \(D\) such that \(b_i \leq s <d_i\).

The name is inspired from the case when the persistence diagram comes from persistent homology.

Distances, inner products and kernels ¶

A set \(X\) with a function

\[d : X \times X \to \mathbb R\]

is called a metric space if the values of \(d\) are all non-negative and for all \(x,y,z \in X\)

\[d(x,y) = 0\ \Leftrightarrow\ x = y\]

\[d(x,y) = d(y,x)\]

\[d(x,z) \leq d(x,y) + d(y, z).\]

In this case the \(d\) is referred to as the metric or the distance function.

A vector space \(V\) together with a function

\[\langle -, - \rangle : V \times V \to \mathbb R\]

is called and inner product space if for all \(u,v,w \in V\)

\[u \neq 0\ \Rightarrow\ \langle u, u \rangle > 0\]

\[\langle u, v\rangle = \langle v, u\rangle\]

\[\langle au+v, w \rangle = a\langle u, w \rangle + \langle v, w \rangle.\]

In this case the function \(\langle -, - \rangle\) is referred to as the inner product and the function given by

\[||u|| = \sqrt{\langle u, u \rangle}\]

as its associated norm. An inner product space is naturally a metric space with distance function

\[d(u,v) = ||u-v||.\]

A kernel on a set \(X\) is a function

\[k : X \times X\]

for which there exists a function \(\phi : X \to V\) to an inner product space such that

\[k(x, y) = \langle \phi(x), \phi(y) \rangle.\]

Euclidean distance and norm ¶

The vector space \(\mathbb R^n\) is an inner product space with inner product

\[\langle x, y \rangle = (x_1-y_1)^2 + \cdots + (x_n-y_n)^2.\]

The associated norm and distance function are referred to as Euclidean norm and Euclidean distance.

Finite metric spaces and point clouds ¶

A finite metric space is a finite set together with a metric. A distance matrix associated to a finite metric space is obtained by choosing a total order on the finite set and setting the \((i,j)\)-entry to be equal to the distance between the \(i\)-th and \(j\)-th elements.

A point cloud is a finite subset of \(\mathbb{R}^n\) (for some \(n\)) together with the metric induced from the euclidean distance.

Time series ¶

A time series is a sequence \(\{y_i\}_{i = 0}^n\) of real numbers.

A common construction of a times series \(\{x_i\}_{i = 0}^n\) is given by choosing \(x_0\) arbitrarily as well as a step parameter \(h\) and setting

\[x_i = x_0 + h\cdot i.\]

Another usual construction is as follows: given a time series \(\{x_i\}_{i = 0}^n \subseteq U\) and a function

\[f : U \subseteq \mathbb R \to \mathbb R\]

we obtain a new time series \(\{f(x_i)\}_{i = 0.}^n\)

Generalizing the previous construction we can define a time series from a function

\[\varphi : U \times M \to M, \qquad U \subseteq \mathbb R, \qquad M \subseteq \mathbb R^d\]

using a function \(f : M \to \mathbb R\) as follows: let \(\{t_i\}_{i=0}^n\) be a time series taking values in \(U\), then

\[\{f(\varphi(t_i, m))\}_{i=0}^n\]

for an arbitrarily chosen \(m \in M\).

Takens embedding ¶

Let \(M \subset \mathbb R^d\) be a compact manifold of dimension \(n\). Let

\[\varphi : \mathbb R \times M \to M\]

and

\[f : M \to \mathbb R\]

be generic smooth functions. Then, for any \(\tau > 0\) the map

\[M \to \mathbb R^{2n+1}\]

defined by

\[x \mapsto\big( f(x), f(x_1), f(x_2), \dots, f(x_{2n}) \big)\]

where

\[x_i = \varphi(i \cdot \tau, x)\]

is an injective map with full rank.

Reference:¶

(Takens 1981)

Manifold ¶

Intuitively, a manifold of dimension \(n\) is a space locally equivalent to \(\mathbb R^n\). Formally, a subset \(M\) of \(\mathbb R^d\) is an \(n\)-dimensional manifold if for each \(x \in M\) there exists an open ball \(B(x) = \{ y \in M\,;\ d(x,y) < \epsilon\}\) and a smooth function with smooth inverse

\[\phi_x : B(x) \to \{v \in \mathbb R^n\,;\ ||v||<1\}.\]

References:¶

(Milnor and Weaver 1997; Guillemin and Pollack 2010)

Compact subset ¶

A subset \(K\) of a metric space \((X,d)\) is said to be bounded if there exist a real number \(D\) such that for each pair of elements in \(K\) the distance between them is less than \(D\). It is said to be complete if for any \(x \in X\) it is the case that \(x \in K\) if for any \(\epsilon > 0\) the intersection between \(K\) and \(\{y \,;\ d(x,y) < \epsilon \}\) is not empty. It is said to be compact if it is both bounded and complete.

Bibliography ¶

Bubenik, Peter. 2015. “Statistical Topological Data Analysis Using Persistence Landscapes.” The Journal of Machine Learning Research 16 (1): 77–102.

Chazal, Frédéric, Brittany Terese Fasy, Fabrizio Lecci, Alessandro Rinaldo, and Larry Wasserman. 2014. “Stochastic Convergence of Persistence Landscapes and Silhouettes.” In Proceedings of the Thirtieth Annual Symposium on Computational Geometry, 474–83. SOCG’14. Kyoto, Japan: Association for Computing Machinery. https://doi.org/10.1145/2582112.2582128.

Guillemin, Victor, and Alan Pollack. 2010. Differential Topology. Vol. 370. American Mathematical Soc.

Kaczynski, Tomasz, Konstantin Mischaikow, and Marian Mrozek. 2004. Computational Homology. Vol. 157. Applied Mathematical Sciences. Springer-Verlag, New York. https://doi.org/10.1007/b97315.

Kerber, Michael, Dmitriy Morozov, and Arnur Nigmetov. 2017. “Geometry Helps to Compare Persistence Diagrams.” Journal of Experimental Algorithmics (JEA) 22: 1–4.

Milnor, John Willard, and David W Weaver. 1997. Topology from the Differentiable Viewpoint. Princeton university press.

Rucco, Matteo, Filippo Castiglione, Emanuela Merelli, and Marco Pettini. 2016. “Characterisation of the Idiotypic Immune Network Through Persistent Entropy.” In Proceedings of Eccs 2014, 117–28. Springer.

Stein, Elias M, and Rami Shakarchi. 2011. Functional Analysis: Introduction to Further Topics in Analysis. Vol. 4. Princeton University Press.

Takens, Floris. 1981. “Detecting Strange Attractors in Turbulence.” In Dynamical Systems and Turbulence, Warwick 1980, 366–81. Springer.