# Theory Glossary¶

Contents

## 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

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

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

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

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

A normed space is naturally a metric space with distance function

### Inner product space¶

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

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

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

An inner product space is naturally a normed space with

### Vectorization, amplitude and kernel¶

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

where \(V\) is a vector space.

An *amplitude* on \(X\) is a function

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

for all \(x \in X\).

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

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

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

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

if \(p\) is finite and

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

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

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

## 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

An *elementary cube* is a subset of the form

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

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

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

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

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*.

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:

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

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

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

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

consider the extension

defined respectively by

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

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

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

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

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

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

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

### Persistence diagram¶

A *persistence diagram* is a multiset of points in

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

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

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

defined by letting \(\lambda_k(t)\) be the \(k\)-th largest value of the set \(\{\Lambda_i(t)\}_ {i \in I}\) where

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,

where

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

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

where

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

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

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

where

#### 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

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

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

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

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

### Takens embedding¶

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

and

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

defined by

where

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

#### 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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