Theory Glossary

Contents

• Theory Glossary

  • Symbols

  • Analysis

    • Metric space

    • Normed space

    • Inner product space

    • Vectorization, amplitude and kernel

    • Euclidean distance and $l^p$-norms

    • Distance matrices and point clouds

    • $L^p$-norms

  • Homology

    • Cubical complex

    • Simplicial complex

    • Abstract simplicial complex

    • Ordered simplicial complex

    • Directed simplicial complex

    • Chain complex

    • Homology and cohomology

    • Simplicial chains and simplicial homology

    • Cubical chains and cubical homology

  • Persistence

    • Filtered complex

    • Cellwise filtration

    • Clique and flag complexes

    • Persistence module

    • Persistent simplicial (co)homology

    • Vietoris-Rips complex and Vietoris-Rips persistence

    • Čech complex and Čech persistence

    • Multiset

    • Persistence diagram

    • Wasserstein and bottleneck distance

    • Persistence landscape

    • Weighted silhouette

    • Heat vectorizations

    • Persistence entropy

    • Betti curve

  • Time series

    • Time series

    • Takens embedding

    • Manifold

    • Compact subset

  • Bibliography

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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