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 \(( s,s ) \mapsto +\infty\).

Analysis

Metric space

A set \(X\) with a function

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

is said to be 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.

Normed space

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

\[||-|| : V \to \mathbb R\]

is said to be an normed space if the values of \(||-||\) are all non-negative and for all \(u,v \in V\) and \(a \in \mathbb R\)

\[||u|| = 0\ \Leftrightarrow\ u = 0\]
\[||a u || = |a|\, ||u||\]
\[||u + v|| \leq ||u|| + ||v||.\]

The function \(||-||\) is referred to as the norm.

A normed space is naturally a metric space with distance function

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

Inner product space

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

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

is said to be an inner product space if for all \(u,v,w \in V\) and \(a \in \mathbb R\)

\[u \neq 0\ \Rightarrow\ \langle u, u \rangle > 0\]
\[\langle u, v\rangle = \langle v, u\rangle\]
\[\langle au+v, w \rangle = a\langle u, w \rangle + \langle v, w \rangle.\]

The function \(\langle -, - \rangle\) is referred to as the inner product.

An inner product space is naturally a normed space with

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

Vectorization, amplitude and kernel

Let \(X\) be a set, for example, the set of all persistence diagrams. A vectorization for \(X\) is a function

\[\phi : X \to V\]

where \(V\) is a vector space.

An amplitude on \(X\) is a function

\[A : X \to \mathbb R\]

for which there exists a vectorization \(\phi : X \to V\) with \(V\) a normed space such that

\[A(x) = ||\phi(x)||\]

for all \(x \in X\).

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

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

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

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

for each \(x,y \in X\).

Euclidean distance and \(l^p\)-norms

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

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

This inner product is referred to as dot product and the associated norm and distance function are respectively named Euclidean norm and Euclidean distance.

For any \(p \in (0,\infty]\) the pair \(\mathbb R^n, ||-||_p\) with

\[||x||_p = (x_1^p + \cdots + x_n^p)^{1/p}\]

if \(p\) is finite and

\[||x||_{\infty} = max\{x_i\ |\ i = 1,\dots,n\}\]

is a normed spaced and its norm is referred to as the \(l^p\)-norm.

Distance matrices and point clouds

Let \((X, d)\) be a finite metric space. A distance matrix associated to it is obtained by choosing a total order on \(X = {x_1 < \cdots < x_m}\) and setting the \((i,j)\)-entry to be equal to \(d(x_i, x_j)\).

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

\(L^p\)-norms

Let \(U \subseteq \mathbb R^n\) and \(C(U, \mathbb R)\) be the set of continuous real-valued functions on \(U\). A function \(f \in C(U, \mathbb R)\) is said to be \(p\)-integrable if

\[\int_U |f(x)|^p dx\]

is finite. The subset of \(p\)-integrable functions together with the assignment \(||-||_p\)

\[f \mapsto \left( \int_U |f(x)|^p dx \right)^{1/p}\]

is a normed space and \(||-||_p\) is referred to as the \(L^p\)-norm.

The only \(L^p\)-norm that is induced from an inner product is \(L^2\), and the inner product is given by

\[\langle f, g \rangle = \left(\int_U |f(x)-g(x)|^2 dx\right)^{1/2}.\]

Homology

Cubical complex

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

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

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 \(X\).

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

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

and to \(n\) as its dimension. The \(i\)-th face of \(\lbrack v_0, \dots, v_n \rbrack\) 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:

  1. for every \(v\) in \(V\), the singleton \(\{v\}\) is in \(X\) and

  2. 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.

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 \(\lbrack v_0, \dots, v_n \rbrack\).

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:

  1. for every \(v\) in \(V\), the tuple \(v\) is in \(X\)

  2. 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 and denote the subset of \(n\)-simplices by \(X_n\). 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 and denote the subset of \(n\)-cubes by \(X_n\). Define the 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.

Persistence

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

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

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

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

The Čech persistence (co)homology of \((X,d)\) is the persistent simplicial (co)homology 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

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

The set of persistence diagrams together with any of the distances above is a metric space.

Reference:

(Kerber, Morozov, and Nigmetov 2017)

Persistence landscape

Let \(\{(b_i, d_i)\}_{i \in I}\) be a persistence diagram. Its persistence landscape is the set \(\{\lambda_k\}_{k \in \mathbb N}\) of functions

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

defined by letting \(\lambda_k(t)\) 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 \(c_+ := \max(c,0)\). The function \(\lambda_k\) is referred to as the \(k\)-layer of the persistence landscape.

We describe the graph of each \(\lambda_k\) intuitively. For each \(i \in I\), draw an isosceles triangle with base the interval \((b_i, d_i)\) on the horizontal \(t\)-axis, and sides with slope 1 and \(-1\). This subdivides the plane into a number of polygonal regions. Label each of these regions by the number of triangles containing it. If \(P_k\) is the union of the polygonal regions with values at least \(k\), then the graph of \(\lambda_k\) is the upper contour of \(P_k\), with \(\lambda_k(a) = 0\) if the vertical line \(t=a\) does not intersect \(P_k\).

The persistence landscape construction defines a vectorization of the set of persistence diagrams with target the vector space of real-valued function on \(\mathbb N \times \mathbb R\). For any \(p = 1,\dots,\infty\) we can restrict attention to persistence diagrams \(D\) whose associated persistence landscape \(\lambda\) is \(p\)-integrable, that is to say,

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

where

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

is finite. In this case we refer to [equation:persistence_landscape_norm] as the \(p\)-landscape norm of \(D\) and, for \(p = 2\), define the value of the landscape kernel on two persistence diagrams \(D\) and \(E\) as

\[\langle \lambda, \mu \rangle = \left(\sum_{i \in \mathbb N} \int_{\mathbb R} |\lambda_i(x) - \mu_i(x)|^2\, dx\right)^{1/2}\]

where \(\lambda\) and \(\mu\) are their associated persistence landscapes.

References:

(Bubenik 2015)

Weighted silhouette

Let \(D = \{(b_i, d_i)\}_{i \in I}\) be a persistence diagram and \(w = \{w_i\}_{i \in I}\) a set of positive real numbers. The silhouette of \(D\) weighted by \(w\) is the function \(\phi : \mathbb R \to \mathbb R\) defined by

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

where

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

and \(c_+ := \max(c,0)\). When \(w_i = \vert d_i - b_i \vert^p\) for \(0 < p \leq \infty\) we refer to \(\phi\) as the \(p\)-power-weighted silhouette of \(D\). The silhouette construction defines a vectorization of the set of persistence diagrams with target the vector space of continuous real-valued functions on \(\mathbb R\).

References:

(Chazal et al. 2014)

Heat vectorizations

Considering the points in a persistence diagram as the support of Dirac deltas one can construct, for any \(t > 0\), two vectorizations of the set of persistence diagrams to the set of continuous real-valued function on the first quadrant \(\mathbb{R}^2_{>0}\). The symmetry heat vectorization is constructed for every persistence diagram \(D\) by solving the heat equation

\[\begin{split}\begin{aligned} \label{equation: heat equation} \Delta_x(u) &= \partial_t u && \text{on } \Omega \times \mathbb R_{>0} \nonumber \\ u &= 0 && \text{on } \{x_1 = x_2\} \times \mathbb R_{\geq 0} \\ u &= \sum_{p \in D} \delta_p && \text{on } \Omega \times {0} \nonumber \end{aligned}\end{split}\]

where \(\Omega = \{(x_1, x_2) \in \mathbb R^2\ |\ x_1 \leq x_2\}\), then solving the same equation after precomposing the data of [equation: heat equation] with the change of coordinates \((x_1, x_2) \mapsto (x_2, x_1)\), and defining the image of \(D\) to be the difference between these two solutions at the chosen time \(t\).

Similarly, the rotation heat vectorization is defined by sending \(D\) to the solution, evaluated at time \(t\), of the equation obtained by precomposing the data of [equation: heat equation] with the change of coordinates \((x_1, x_2) \mapsto (x_1, x_2-x_1)\).

We recall that the solution to the heat equation with initial condition given by a Dirac delta supported at \(p \in \mathbb R^2\) is

\[\frac{1}{4 \pi t} \exp\left(-\frac{||p-x||^2}{4t}\right)\]

and, to highlight the connection with normally distributed random variables, it is customary to use the the change of variable \(\sigma = \sqrt{2t}\).

References:

(Reininghaus et al. 2015; Adams et al. 2017)

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.

Time series

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.

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