===============
Theory Glossary
===============
.. role:: raw-latex(raw)
:format: latex
..
.. contents::
:depth: 3
..
Symbols
=======
+------------------------------+--------------------------------------+
| :math:`\Bbbk` | An arbitrary field. |
+------------------------------+--------------------------------------+
| :math:`\mathbb R` | The field of real numbers. |
+------------------------------+--------------------------------------+
| :math:`\overline{\mathbb R}` | The two point compactification |
| | :math:`[-\infty, +\infty]` of the |
| | real numbers. |
+------------------------------+--------------------------------------+
| :math:`\mathbb N` | The counting numbers |
| | :math:`0,1,2, \ldots` as a subset of |
| | :math:`\mathbb R`. |
+------------------------------+--------------------------------------+
| :math:`\mathbb R^d` | The vector space of :math:`d`-tuples |
| | of real numbers. |
+------------------------------+--------------------------------------+
| :math:`\Delta` | The multiset |
| | :math:`\lbrace ( |
| | s, s) \mid s \in \mathbb{R} \rbrace` |
| | with multiplicity |
| | :math:`( s,s ) \mapsto +\infty`. |
+------------------------------+--------------------------------------+
Analysis
========
.. _metric_space:
Metric space
------------
A set :math:`X` with a function
.. math:: d : X \times X \to \mathbb R
is said to be a *metric space* if the values of :math:`d` are all
non-negative and for all :math:`x,y,z \in X`
.. math:: d(x,y) = 0\ \Leftrightarrow\ x = y
.. math:: d(x,y) = d(y,x)
.. math:: d(x,z) \leq d(x,y) + d(y, z).
In this case the :math:`d` is referred to as the *metric* or the
*distance function*.
.. _normed_space:
Normed space
------------
A vector space :math:`V` together with a function
.. math:: ||-|| : V \to \mathbb R
is said to be an *normed space* if the values of :math:`||-||` are all
non-negative and for all :math:`u,v \in V` and :math:`a \in \mathbb R`
.. math:: ||u|| = 0\ \Leftrightarrow\ u = 0
.. math:: ||a u || = |a|\, ||u||
.. math:: ||u + v|| \leq ||u|| + ||v||.
The function :math:`||-||` is referred to as the *norm*.
A normed space is naturally a metric space with distance function
.. math:: d(u,v) = ||u-v||.
.. _inner_product_space:
Inner product space
-------------------
A vector space :math:`V` together with a function
.. math:: \langle -, - \rangle : V \times V \to \mathbb R
is said to be an *inner product space* if for all :math:`u,v,w \in V`
and :math:`a \in \mathbb R`
.. math:: u \neq 0\ \Rightarrow\ \langle u, u \rangle > 0
.. math:: \langle u, v\rangle = \langle v, u\rangle
.. math:: \langle au+v, w \rangle = a\langle u, w \rangle + \langle v, w \rangle.
The function :math:`\langle -, - \rangle` is referred to as the *inner
product*.
An inner product space is naturally a normed space with
.. math:: ||u|| = \sqrt{\langle u, u \rangle}.
.. _vectorization_amplitude_and_kernel:
Vectorization, amplitude and kernel
-----------------------------------
Let :math:`X` be a set, for example, the set of all persistence
diagrams. A *vectorization* for :math:`X` is a function
.. math:: \phi : X \to V
where :math:`V` is a vector space.
An *amplitude* on :math:`X` is a function
.. math:: A : X \to \mathbb R
for which there exists a vectorization :math:`\phi : X \to V` with
:math:`V` a normed space such that
.. math:: A(x) = ||\phi(x)||
for all :math:`x \in X`.
A *kernel* on the set :math:`X` is a function
.. math:: k : X \times X \to \mathbb R
for which there exists a vectorization :math:`\phi : X \to V` with
:math:`V` an inner product space such that
.. math:: k(x,y) = \langle \phi(x), \phi(y) \rangle
for each :math:`x,y \in X`.
.. _euclidean_distance_and_norm:
Euclidean distance and :math:`l^p`-norms
----------------------------------------
The vector space :math:`\mathbb R^n` is an inner product space with
inner product
.. math:: \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 :math:`p \in (0,\infty]` the pair :math:`\mathbb R^n, ||-||_p`
with
.. math:: ||x||_p = (x_1^p + \cdots + x_n^p)^{1/p}
if :math:`p` is finite and
.. math:: ||x||_{\infty} = max\{x_i\ |\ i = 1,\dots,n\}
is a normed spaced and its norm is referred to as the
:math:`l^p`\ *-norm*.
.. _distance_matrices_and_point_clouds:
Distance matrices and point clouds
----------------------------------
Let :math:`(X, d)` be a finite metric space. A *distance matrix*
associated to it is obtained by choosing a total order on
:math:`X = {x_1 < \cdots < x_m}` and setting the :math:`(i,j)`-entry to
be equal to :math:`d(x_i, x_j)`.
A *point cloud* is a finite subset of :math:`\mathbb{R}^n` (for some
:math:`n`) together with the metric induced from the Euclidean distance.
.. _functional_lp:
:math:`L^p`-norms
-----------------
Let :math:`U \subseteq \mathbb R^n` and :math:`C(U, \mathbb R)` be the
set of continuous real-valued functions on :math:`U`. A function
:math:`f \in C(U, \mathbb R)` is said to be :math:`p`\ *-integrable* if
.. math:: \int_U |f(x)|^p dx
is finite. The subset of :math:`p`-integrable functions together with
the assignment :math:`||-||_p`
.. math:: f \mapsto \left( \int_U |f(x)|^p dx \right)^{1/p}
is a normed space and :math:`||-||_p` is referred to as the
:math:`L^p`\ *-norm*.
The only :math:`L^p`-norm that is induced from an inner product is
:math:`L^2`, and the inner product is given by
.. math:: \langle f, g \rangle = \left(\int_U |f(x)-g(x)|^2 dx\right)^{1/2}.
Homology
========
.. _cubical_complex:
Cubical complex
---------------
An *elementary interval* :math:`I_a` is a subset of :math:`\mathbb{R}`
of the form :math:`[a, a+1]` or :math:`[a,a] = \{a\}` for some
:math:`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
.. math:: 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
.. math:: I_{a_1} \times \cdots \times I_{a_N} \subset \mathbb{R}^N
where each :math:`I_{a_i}` is an elementary interval. We refer to the
total number of its non-degenerate factors
:math:`I_{a_{k_1}}, \dots, I_{a_{k_n}}` as its *dimension* and, assuming
.. math:: a_{k_1} < \cdots < a_{k_{n,}}
we define for :math:`i = 1, \dots, n` the following two elementary cubes
.. math:: 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
:math:`\mathbb{R}^N`, and a *subcomplex* of :math:`X` is a cubical
complex whose elementary cubes are also in :math:`X`.
Reference:
~~~~~~~~~~
(Kaczynski, Mischaikow, and Mrozek 2004)
.. _simplicial_complex:
Simplicial complex
------------------
A set :math:`\{v_0, \dots, v_n\} \subset \mathbb{R}^N` is said to be
*geometrically independent* if the vectors
:math:`\{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
.. math:: \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 :math:`n` as its *dimension*. The :math:`i`\ *-th face* of
:math:`\lbrack v_0, \dots, v_n \rbrack` is defined by
.. math:: d_i[v_0, \ldots, v_n] = [v_0, \dots, \widehat{v}_i, \dots, v_n]
where :math:`\widehat{v}_i` denotes the absence of :math:`v_i` from the
set.
A *simplicial complex* :math:`X` is a finite union of simplices in
:math:`\mathbb{R}^N` satisfying that every face of a simplex in
:math:`X` is in :math:`X` and that the non-empty intersection of two
simplices in :math:`X` is a face of each. Every simplicial complex
defines an abstract simplicial complex.
.. _abstract_simplicial_complex:
Abstract simplicial complex
---------------------------
An *abstract simplicial complex* is a pair of sets :math:`(V, X)` with
the elements of :math:`X` being subsets of :math:`V` such that:
#. for every :math:`v` in :math:`V`, the singleton :math:`\{v\}` is in
:math:`X` and
#. if :math:`x` is in :math:`X` and :math:`y` is a subset of :math:`x`,
then :math:`y` is in :math:`X`.
We abuse notation and denote the pair :math:`(V, X)` simply by
:math:`X`.
The elements of :math:`X` are called *simplices* and the *dimension* of
a simplex :math:`x` is defined by :math:`|x| = \# x - 1` where
:math:`\# x` denotes the cardinality of :math:`x`. Simplices of
dimension :math:`d` are called :math:`d`-simplices. We abuse terminology
and refer to the elements of :math:`V` and to their associated
:math:`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 :math:`X` is a *subcomplex* of a simplicial complex
:math:`Y` if every simplex of :math:`X` is a simplex of :math:`Y`.
Given a finite abstract simplicial complex :math:`X = (V, X)` we can
choose a bijection from :math:`V` to a geometrically independent subset
of :math:`\mathbb R^N` and associate a simplicial complex to :math:`X`
called its *geometric realization*.
.. _ordered_simplical_complex:
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 :math:`n`-simplex using its ordered vertices by
:math:`\lbrack v_0, \dots, v_n \rbrack`.
A *simplicial map* between ordered simplicial complexes is a simplicial
map :math:`f` between their underlying simplicial complexes preserving
the order, i.e., :math:`v \leq w` implies :math:`f(v) \leq f(w)`.
.. _directed_simplicial_complex:
Directed simplicial complex
---------------------------
A *directed simplicial complex* is a pair of sets :math:`(V, X)` with
the elements of :math:`X` being tuples of elements of :math:`V`, i.e.,
elements in :math:`\bigcup_{n\geq1} V^{\times n}` such that:
#. for every :math:`v` in :math:`V`, the tuple :math:`v` is in :math:`X`
#. if :math:`x` is in :math:`X` and :math:`y` is a subtuple of
:math:`x`, then :math:`y` is in :math:`X`.
With appropriate modifications the same terminology and notation
introduced for ordered simplicial complex applies to directed simplicial
complex.
.. _chain_complex:
Chain complex
-------------
A *chain complex* of is a pair :math:`(C_*, \partial)` where
.. math:: C_* = \bigoplus_{n \in \mathbb Z} C_n \quad \mathrm{and} \quad \partial = \bigoplus_{n \in \mathbb Z} \partial_n
with :math:`C_n` a :math:`\Bbbk`-vector space and
:math:`\partial_n : C_{n+1} \to C_n` is a :math:`\Bbbk`-linear map such
that :math:`\partial_{n+1} \partial_n = 0`. We refer to :math:`\partial`
as the *boundary map* of the chain complex.
The elements of :math:`C` are called *chains* and if :math:`c \in C_n`
we say its *degree* is :math:`n` or simply that it is an
:math:`n`-chain. Elements in the kernel of :math:`\partial` are called
*cycles*, and elements in the image of :math:`\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 :math:`\Bbbk`-linear map :math:`f : C \to C'` between
chain complexes such that :math:`f(C_n) \subseteq C'_n` and
:math:`\partial f = f \partial`.
Given a chain complex :math:`(C_*, \partial)`, its linear dual
:math:`C^*` is also a chain complex with
:math:`C^{-n} = \mathrm{Hom_\Bbbk}(C_n, \Bbbk)` and boundary map
:math:`\delta` defined by :math:`\delta(\alpha)(c) = \alpha(\partial c)`
for any :math:`\alpha \in C^*` and :math:`c \in C_*`.
.. _homology_and_cohomology:
Homology and cohomology
-----------------------
Let :math:`(C_*, \partial)` be a chain complex. Its :math:`n`\ *-th
homology group* is the quotient of the subspace of :math:`n`-cycles by
the subspace of :math:`n`-boundaries, that is,
:math:`H_n(C_*) = \mathrm{ker}(\partial_n)/ \mathrm{im}(\partial_{n+1})`.
The *homology* of :math:`(C, \partial)` is defined by
:math:`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 :math:`H^n` for :math:`H_{-n}`.
A chain map :math:`f : C \to C'` induces a map between the associated
homologies.
.. _simplicial_chains_and_simplicial_homology:
Simplicial chains and simplicial homology
-----------------------------------------
Let :math:`X` be an ordered or directed simplicial complex and denote
the subset of :math:`n`-simplices by :math:`X_n`. Define its *simplicial
chain complex with* :math:`\Bbbk`\ *-coefficients* :math:`C_*(X; \Bbbk)`
by
.. math:: 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* :math:`\Bbbk`\ *-coefficients* as
the homology and cohomology of this chain complex. We use the notation
:math:`H_*(X; \Bbbk)` and :math:`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:
Cubical chains and cubical homology
-----------------------------------
Let :math:`X` be a cubical complex and denote the subset of
:math:`n`-cubes by :math:`X_n`. Define the *cubical chain complex with*
:math:`\Bbbk`\ *-coefficients* :math:`C_*(X; \Bbbk)` by
.. math:: 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 :math:`x = I_1 \times \cdots \times I_N` and :math:`s(i)` is the
dimension of :math:`I_1 \times \cdots \times I_i`. Its *homology and
cohomology with* :math:`\Bbbk`\ *-coefficients* is the homology and
cohomology of this chain complex. We use the notation
:math:`H_*(X; \Bbbk)` and :math:`H^*(X; \Bbbk)` for these.
Persistence
===========
.. _filtered_complex:
Filtered complex
----------------
A *filtered complex* is a collection of simplicial or cubical complexes
:math:`\{X_s\}_{s \in \mathbb R}` such that :math:`X_s` is a subcomplex
of :math:`X_t` for each :math:`s \leq t`.
.. _cellwise_filtration:
Cellwise filtration
-------------------
A *cellwise filtration* is a simplicial or cubical complex :math:`X`
together with a total order :math:`\leq` on its simplices or elementary
cubes such that for each :math:`y \in X` the set
:math:`\{x \in X\ :\ x \leq y\}` is a subcomplex of :math:`X`. A
cellwise filtration can be naturally thought of as a filtered complex.
.. _clique_and_flag_complexes:
Clique and flag complexes
-------------------------
Let :math:`G` be a :math:`1`-dimensional abstract (resp. directed)
simplicial complex. The abstract (resp. directed) simplicial complex
:math:`\langle G \rangle` has the same set of vertices as :math:`G` and
:math:`\{v_0, \dots, v_n\}` (resp. :math:`(v_0, \dots, v_n)`) is a
simplex in :math:`\langle G \rangle` if an only if :math:`\{v_i, v_j\}`
(resp. :math:`(v_i, v_j)`) is in :math:`G` for each pair of vertices
:math:`v_i, v_j`.
An abstract (resp. directed) simplicial complex :math:`X` is a *clique
(resp. flag) complex* if :math:`X = \langle G \rangle` for some
:math:`G`.
Given a function
.. math:: w : G \to \mathbb R \cup \{\infty\}
consider the extension
.. math:: w : \langle G \rangle \to \mathbb R \cup \{\infty\}
defined respectively by
.. math::
\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}
and define the filtered complex
:math:`\{\langle G \rangle_{s}\}_{s \in \mathbb R}` by
.. math:: \langle G \rangle_s = \{\sigma \in \langle G \rangle\ |\ w(\sigma) \leq s\}.
A filtered complex :math:`\{X_s\}_{s \in \mathbb R}` is a *filtered
clique (resp. flag) complex* if :math:`X_s = \langle G \rangle_s` for
some :math:`(G,w)`.
.. _persistence_module:
Persistence module
------------------
A *persistence module* is a collection containing a :math:`\Bbbk`-vector
spaces :math:`V(s)` for each real number :math:`s` together with
:math:`\Bbbk`-linear maps :math:`f_{st} : V(s) \to V(t)`, referred to as
*structure maps*, for each pair :math:`s \leq t`, satisfying naturality,
i.e., if :math:`r \leq s \leq t`, then
:math:`f_{rt} = f_{st} \circ f_{rs}` and tameness, i.e., all but
finitely many structure maps are isomorphisms.
A *morphism of persistence modules* :math:`F : V \to W` is a collection
of linear maps :math:`F(s) : V(s) \to W(s)` such that
:math:`F(t) \circ f_{st} = f_{st} \circ F(s)` for each par of reals
:math:`s \leq t`. We say that :math:`F` is an *isomorphisms* if each
:math:`F(s)` is.
.. _persistent_simplicial_(co)homology:
Persistent simplicial (co)homology
----------------------------------
Let :math:`\{X(s)\}_{s \in \mathbb R}` be a set of ordered or directed
simplicial complexes together with simplicial maps
:math:`f_{st} : X(s) \to X(t)` for each pair :math:`s \leq t`, such that
.. math:: 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* :math:`\Bbbk`\ *-coefficients* is the persistence module
.. math:: H_*(X(s); \Bbbk)
with structure maps
:math:`H_*(f_{st}) : H_*(X(s); \Bbbk) \to H_*(X(t); \Bbbk)` induced form
the maps :math:`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* :math:`\Bbbk`\ *-coefficients* is defined
analogously.
.. _vietoris-rips_complex_and_vietoris-rips_persistence:
Vietoris-Rips complex and Vietoris-Rips persistence
---------------------------------------------------
Let :math:`(X, d)` be a finite metric space. Define the Vietoris-Rips
complex of :math:`X` as the filtered complex :math:`VR_s(X)` that
contains a subset of :math:`X` as a simplex if all pairwise distances in
the subset are less than or equal to :math:`s`, explicitly
.. math:: 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 :math:`(X, d)` is the persistent
simplicial (co)homology of :math:`VR_s(X)`.
A more general version is obtained by replacing the distance function
with an arbitrary function
.. math:: w : X \times X \to \mathbb R \cup \{\infty\}
and defining :math:`VR_s(X)` as the filtered clique complex associated
to :math:`(X \times X ,w)`.
.. _cech_complex_and_cech_persistence:
Čech complex and Čech persistence
---------------------------------
Let :math:`(X, d)` be a point cloud. Define the Čech complex of
:math:`X` as the filtered complex :math:`\check{C}_s(X)` that is empty
if :math:`s<0` and, if :math:`s \geq 0`, contains a subset of :math:`X`
as a simplex if the balls of radius :math:`s` with centers in the subset
have a non-empty intersection, explicitly
.. math:: \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 :math:`(X,d)` is the persistent
simplicial (co)homology of :math:`\check{C}_s(X)`.
Multiset
--------
A *multiset* is a pair :math:`(S, \phi)` where :math:`S` is a set and
:math:`\phi : S \to \mathbb N \cup \{+\infty\}` is a function attaining
positive values. For :math:`s \in S` we refer to :math:`\phi(s)` as its
*multiplicity*. The *union* of two multisets
:math:`(S_1, \phi_1), (S_2, \phi_2)` is the multiset
:math:`(S_1 \cup S_2, \phi_1 \cup \phi_2)` with
.. math::
(\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}
.. _persistence_diagram:
Persistence diagram
-------------------
A *persistence diagram* is a multiset of points in
.. math:: \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 :math:`s,t` the
number counted with multiplicity of points :math:`(b,d)` in the
multiset, satisfying :math:`b \leq s \leq t < d` is equal to the rank of
:math:`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.
.. _reference-1:
Reference:
~~~~~~~~~~
(Zomorodian and Carlsson 2005)
Earlier work:
~~~~~~~~~~~~~
(Morse 1940; Gabriel 1972; Frosini 1990; Barannikov 1994; Robins 1999;
Edelsbrunner, Letscher, and Zomorodian 2002)
.. _wasserstein_and_bottleneck_distance:
Wasserstein and bottleneck distance
-----------------------------------
The :math:`p`\ *-Wasserstein distance* between two persistence diagrams
:math:`D_1` and :math:`D_2` is the infimum over all bijections
:math:`\gamma: D_1 \cup \Delta \to D_2 \cup \Delta` of
.. math:: \Big(\sum_{x \in D_1 \cup \Delta} ||x - \gamma(x)||_\infty^p \Big)^{1/p}
where :math:`||-||_\infty` is defined for :math:`(x,y) \in \mathbb R^2`
by :math:`\max\{|x|, |y|\}`.
The limit :math:`p \to \infty` defines the *bottleneck distance*. More
explicitly, it is the infimum over the same set of bijections of the
value
.. math:: \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-2:
Reference:
~~~~~~~~~~
(Kerber, Morozov, and Nigmetov 2017)
.. _persistence_landscape:
Persistence landscape
---------------------
Let :math:`\{(b_i, d_i)\}_{i \in I}` be a persistence diagram. Its
*persistence landscape* is the set
:math:`\{\lambda_k\}_{k \in \mathbb N}` of functions
.. math:: \lambda_k : \mathbb R \to \overline{\mathbb R}
defined by letting :math:`\lambda_k(t)` be the :math:`k`-th largest
value of the set :math:`\{\Lambda_i(t)\}_ {i \in I}` where
.. math:: \Lambda_i(t) = \left[ \min \{t-b_i, d_i-t\}\right]_+
and :math:`c_+ := \max(c,0)`. The function :math:`\lambda_k` is referred
to as the :math:`k`-*layer of the persistence landscape*.
We describe the graph of each :math:`\lambda_k` intuitively. For each
:math:`i \in I`, draw an isosceles triangle with base the interval
:math:`(b_i, d_i)` on the horizontal :math:`t`-axis, and sides with
slope 1 and :math:`-1`. This subdivides the plane into a number of
polygonal regions. Label each of these regions by the number of
triangles containing it. If :math:`P_k` is the union of the polygonal
regions with values at least :math:`k`, then the graph of
:math:`\lambda_k` is the upper contour of :math:`P_k`, with
:math:`\lambda_k(a) = 0` if the vertical line :math:`t=a` does not
intersect :math:`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 :math:`\mathbb N \times \mathbb R`. For any
:math:`p = 1,\dots,\infty` we can restrict attention to persistence
diagrams :math:`D` whose associated persistence landscape
:math:`\lambda` is :math:`p`-integrable, that is to say,
.. math::
\label{equation:persistence_landscape_norm}
||\lambda||_p = \left( \sum_{i \in \mathbb N} ||\lambda_i||^p_p \right)^{1/p}
where
.. math:: ||\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] <#equation:persistence_landscape_norm>`__
as the :math:`p`-*landscape norm* of :math:`D` and, for :math:`p = 2`,
define the value of the *landscape kernel* on two persistence diagrams
:math:`D` and :math:`E` as
.. math:: \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 :math:`\lambda` and :math:`\mu` are their associated persistence
landscapes.
References:
~~~~~~~~~~~
(Bubenik 2015)
.. _weighted_silhouette:
Weighted silhouette
-------------------
Let :math:`D = \{(b_i, d_i)\}_{i \in I}` be a persistence diagram and
:math:`w = \{w_i\}_{i \in I}` a set of positive real numbers. The
*silhouette of* :math:`D` *weighted by* :math:`w` is the function
:math:`\phi : \mathbb R \to \mathbb R` defined by
.. math:: \phi(t) = \frac{\sum_{i \in I}w_i \Lambda_i(t)}{\sum_{i \in I}w_i},
where
.. math:: \Lambda_i(t) = \left[ \min \{t-b_i, d_i-t\}\right]_+
and :math:`c_+ := \max(c,0)`. When :math:`w_i = \vert d_i - b_i \vert^p`
for :math:`0 < p \leq \infty` we refer to :math:`\phi` as the
:math:`p`-*power-weighted silhouette* of :math:`D`. The silhouette
construction defines a vectorization of the set of persistence diagrams
with target the vector space of continuous real-valued functions on
:math:`\mathbb R`.
.. _references-1:
References:
~~~~~~~~~~~
(Chazal et al. 2014)
.. _heat_vectorization:
Heat vectorizations
-------------------
Considering the points in a persistence diagram as the support of Dirac
deltas one can construct, for any :math:`t > 0`, two vectorizations of
the set of persistence diagrams to the set of continuous real-valued
function on the first quadrant :math:`\mathbb{R}^2_{>0}`. The *symmetry
heat vectorization* is constructed for every persistence diagram
:math:`D` by solving the heat equation
.. math::
\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}
where :math:`\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] <#equation: heat equation>`__ with the change
of coordinates :math:`(x_1, x_2) \mapsto (x_2, x_1)`, and defining the
image of :math:`D` to be the difference between these two solutions at
the chosen time :math:`t`.
Similarly, the *rotation heat vectorization* is defined by sending
:math:`D` to the solution, evaluated at time :math:`t`, of the equation
obtained by precomposing the data of
`[equation: heat equation] <#equation: heat equation>`__ with the change
of coordinates :math:`(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 :math:`p \in \mathbb R^2` is
.. math:: \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
:math:`\sigma = \sqrt{2t}`.
.. _references-2:
References:
~~~~~~~~~~~
(Reininghaus et al. 2015; Adams et al. 2017)
.. _persistence_entropy:
Persistence entropy
-------------------
Intuitively, this is a measure of the entropy of the points in a
persistence diagram. Precisely, let :math:`D = \{(b_i, d_i)\}_{i \in I}`
be a persistence diagram with each :math:`d_i < +\infty`. The
*persistence entropy* of :math:`D` is defined by
.. math:: E(D) = - \sum_{i \in I} p_i \log(p_i)
where
.. math:: p_i = \frac{(d_i - b_i)}{L_D} \qquad \text{and} \qquad L_D = \sum_{i \in I} (d_i - b_i) .
.. _references-3:
References:
~~~~~~~~~~~
(Rucco et al. 2016)
.. _betti_curve:
Betti curve
-----------
Let :math:`D` be a persistence diagram. Its *Betti curve* is the
function :math:`\beta_D : \mathbb R \to \mathbb N` whose value on
:math:`s \in \mathbb R` is the number, counted with multiplicity, of
points :math:`(b_i,d_i)` in :math:`D` such that :math:`b_i \leq s 0` the map
.. math:: M \to \mathbb R^{2n+1}
defined by
.. math:: x \mapsto\big( f(x), f(x_1), f(x_2), \dots, f(x_{2n}) \big)
where
.. math:: x_i = \varphi(i \cdot \tau, x)
is an injective map with full rank.
.. _reference-3:
Reference:
~~~~~~~~~~
(Takens 1981)
Manifold
--------
Intuitively, a manifold of dimension :math:`n` is a space locally
equivalent to :math:`\mathbb R^n`. Formally, a subset :math:`M` of
:math:`\mathbb R^d` is an :math:`n`-dimensional manifold if for each
:math:`x \in M` there exists an open ball
:math:`B(x) = \{ y \in M\,;\ d(x,y) < \epsilon\}` and a smooth function
with smooth inverse
.. math:: \phi_x : B(x) \to \{v \in \mathbb R^n\,;\ ||v||<1\}.
.. _references-4:
References:
~~~~~~~~~~~
(Milnor and Weaver 1997; Guillemin and Pollack 2010)
.. _compact_subset:
Compact subset
--------------
A subset :math:`K` of a metric space :math:`(X,d)` is said to be
*bounded* if there exist a real number :math:`D` such that for each pair
of elements in :math:`K` the distance between them is less than
:math:`D`. It is said to be *complete* if for any :math:`x \in X` it is
the case that :math:`x \in K` if for any :math:`\epsilon > 0` the
intersection between :math:`K` and :math:`\{y \,;\ d(x,y) < \epsilon \}`
is not empty. It is said to be *compact* if it is both bounded and
complete.
Bibliography
============
.. container:: references csl-bib-body hanging-indent
:name: refs
.. container:: csl-entry
:name: ref-adams2017persistence
Adams, Henry, Tegan Emerson, Michael Kirby, Rachel Neville, Chris
Peterson, Patrick Shipman, Sofya Chepushtanova, Eric Hanson,
Francis Motta, and Lori Ziegelmeier. 2017. “Persistence Images: A
Stable Vector Representation of Persistent Homology.” *The Journal
of Machine Learning Research* 18 (1): 218–52.
.. container:: csl-entry
:name: ref-barannikov1994morse
Barannikov, S. A. 1994. “The Framed Morse Complex and Its
Invariants.” In *Singularities and Bifurcations*, 21:93–115. Adv.
Soviet Math. Amer. Math. Soc., Providence, RI.
.. container:: csl-entry
:name: ref-bubenik2015statistical
Bubenik, Peter. 2015. “Statistical Topological Data Analysis Using
Persistence Landscapes.” *The Journal of Machine Learning
Research* 16 (1): 77–102.
.. container:: csl-entry
:name: ref-chazal2014stochastic
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.
.. container:: csl-entry
:name: ref-cohen-steiner2007stability
Cohen-Steiner, David, Herbert Edelsbrunner, and John Harer. 2007.
“Stability of Persistence Diagrams.” *Discrete Comput. Geom.* 37
(1): 103–20. https://doi.org/10.1007/s00454-006-1276-5.
.. container:: csl-entry
:name: ref-edelsbrunner2002simplification
Edelsbrunner, Herbert, David Letscher, and Afra Zomorodian. 2002.
“Topological Persistence and Simplification.” In *Discrete Comput.
Geom.*, 28:511–33. 4. https://doi.org/10.1007/s00454-002-2885-2.
.. container:: csl-entry
:name: ref-frosini1990shapes
Frosini, Patrizio. 1990. “A Distance for Similarity Classes of
Submanifolds of a Euclidean Space.” *Bull. Austral. Math. Soc.* 42
(3): 407–16. https://doi.org/10.1017/S0004972700028574.
.. container:: csl-entry
:name: ref-gabriel72decomposition
Gabriel, Peter. 1972. “Unzerlegbare Darstellungen. I.”
*Manuscripta Math.* 6: 71–103; correction, ibid. 6 (1972), 309.
https://doi.org/10.1007/BF01298413.
.. container:: csl-entry
:name: ref-guillemin2010differential
Guillemin, Victor, and Alan Pollack. 2010. *Differential
Topology*. Vol. 370. American Mathematical Soc.
.. container:: csl-entry
:name: ref-mischaikow04computational
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.
.. container:: csl-entry
:name: ref-kerber2017geometry
Kerber, Michael, Dmitriy Morozov, and Arnur Nigmetov. 2017.
“Geometry Helps to Compare Persistence Diagrams.” *Journal of
Experimental Algorithmics (JEA)* 22: 1–4.
.. container:: csl-entry
:name: ref-lee2003nonlinear
Lee, Ann B, Kim S Pedersen, and David Mumford. 2003. “The
Nonlinear Statistics of High-Contrast Patches in Natural Images.”
*International Journal of Computer Vision* 54 (1): 83–103.
.. container:: csl-entry
:name: ref-milnor1997topology
Milnor, John Willard, and David W Weaver. 1997. *Topology from the
Differentiable Viewpoint*. Princeton university press.
.. container:: csl-entry
:name: ref-morse1927rank
Morse, Marston. 1940. “Rank and Span in Functional Topology.”
*Ann. Of Math. (2)* 41: 419–54. https://doi.org/10.2307/1969014.
.. container:: csl-entry
:name: ref-reininghaus2015stable
Reininghaus, Jan, Stefan Huber, Ulrich Bauer, and Roland Kwitt.
2015. “A Stable Multi-Scale Kernel for Topological Machine
Learning.” In *Proceedings of the IEEE Conference on Computer
Vision and Pattern Recognition*, 4741–48.
.. container:: csl-entry
:name: ref-robins1999approximations
Robins, V. 1999. “Towards Computing Homology from Finite
Approximations.” In *Proceedings of the 14th Summer Conference on
General Topology and Its Applications (Brookville, NY, 1999)*,
24:503–532 (2001). Summer.
.. container:: csl-entry
:name: ref-rucco2016characterisation
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.
.. container:: csl-entry
:name: ref-takens1981detecting
Takens, Floris. 1981. “Detecting Strange Attractors in
Turbulence.” In *Dynamical Systems and Turbulence, Warwick 1980*,
366–81. Springer.
.. container:: csl-entry
:name: ref-verri1993use
Verri, Alessandro, Claudio Uras, Patrizio Frosini, and Massimo
Ferri. 1993. “On the Use of Size Functions for Shape Analysis.”
*Biological Cybernetics* 70 (2): 99–107.
.. container:: csl-entry
:name: ref-vietoris1927hoheren
Vietoris, Leopold. 1927. “Über Den höheren Zusammenhang Kompakter
räume Und Eine Klasse von Zusammenhangstreuen Abbildungen.”
*Mathematische Annalen* 97 (1): 454–72.
.. container:: csl-entry
:name: ref-zomorodian2005computing
Zomorodian, Afra, and Gunnar Carlsson. 2005. “Computing Persistent
Homology.” *Discrete & Computational Geometry* 33 (2): 249–74.