dimitri/tda-networks
1
0
Fork 0
Code Issues Pull requests Projects Releases Packages Wiki Activity Actions

Dissertation: Mason's remarks

Browse source
This commit is contained in:
Dimitri Lozeve 2018-07-31 15:23:15 +01:00
parent 89536b749e
commit 79b8e09e00
5 changed files with 389 additions and 183 deletions
Download patch file Download diff file Expand all files Collapse all files

450
dissertation/dissertation.tex
View file

@ -63,9 +63,9 @@ Thank you!
\cleardoublepage%
\tableofcontents*
\listoffigures*
\listoftables*
\tableofcontents
\listoffigures
% \listoftables
\clearpage
@ -81,52 +81,70 @@ Thank you!
\section{Definition and basic properties}%
\label{sec:defin-basic-prop}
In this section, we will introduce the notion of temporal networks or
graphs. This is a complex notion, with many concurrent definitions and
interpretations. First, we restate the standard definition of a
non-temporal, static graph.
In this section, we introduce the notion of temporal networks (or
temporal graphs). This is a complex notion, with many concurrent
definitions and interpretations.
After clarifying the notations, we restate the standard definition of
a non-temporal graph.
\begin{notation}
\begin{itemize}
\item $\mathbb{N}$ is the set of non-negative natural numbers
$0,1,2,\ldots$
\item $\mathbb{N}^*$ is the set of positive integers $1,2,\ldots$
\item $\mathbb{R}$ is the set of real numbers.
$\mathbb{R}_+ = \{x\in\mathbb{R} \;|\; x\geq 0\}$, and
$\mathbb{R}_+^* = \{x\in\mathbb{R} \;|\; x>0\}$.
\end{itemize}
\end{notation}
\begin{defn}[Graph]
A \emph{graph} is a couple $G = (V, E)$, where $V$ is a finite set
of \emph{nodes} (or \emph{vertices}), and $E \subseteq V\times V$ is
a set of \emph{edges}. A \emph{weighted graph} is defined by
$G = (V, E, w)$, where $w : E\mapsto \mathbb{R}_+$ is called the
A \emph{graph} is a couple $G = (V, E)$, where $V$ is a set of
\emph{nodes} (or \emph{vertices}), and $E \subseteq V\times V$ is a
set of \emph{edges}. A \emph{weighted graph} is defined by
$G = (V, E, w)$, where $w : E\mapsto \mathbb{R}_+^*$ is called the
\emph{weight function}.
\end{defn}
We also define some basic concepts that will be needed later on to
build simplicial complexes on graphs.
We also define some basic concepts that we will need later to build
simplicial complexes on graphs.
\begin{defn}[Clique]
A \emph{clique} is a set of nodes where each pair is connected. That
A \emph{clique} is a set of nodes where each pair is adjacent. That
is, a clique $C$ of a graph $G = (V,E)$ is a subset of $V$ such that
$\forall i,j\in C, i \neq j \implies (i,j)\in E$. A clique is said
to be \emph{maximal} if it cannot be augmented by any node.
for all $i,j\in C, i \neq j \implies (i,j)\in E$. A clique is said
to be \emph{maximal} if it cannot be augmented by any node, such
that the resulting set of nodes is itself a clique.
\end{defn}
Temporal networks are defined in the more general framework of
Temporal networks can be defined in the more general framework of
\emph{multilayer networks}~\cite{kivela_multilayer_2014}. However,
this definition is much too general for our simple applications, and
we restrict ourselves to edge-centric time-varying
graphs~\cite{casteigts_time-varying_2012}. In this model, the set of
nodes is fixed and doesn't change over time, whereas edges can appear
or disappear at different timestamps.
nodes is fixed, but edges can appear or disappear at different times.
In this study, we restrict ourselves to discrete time stamps. Each
interaction is taken to be instantaneous.
%% TODO note about data collection, oversampling,
%% duration of interactions
\begin{defn}[Temporal network]
A \emph{temporal network} (or graph) is a tuple
A \emph{temporal network} is a tuple
$G = (V, E, \mathcal{T}, \rho)$, where:
\begin{itemize}
\item $V$ is a finite set of nodes,
\item $V$ is a set of nodes,
\item $E\subseteq V\times V$ is a set of edges,
\item $\mathbb{T}$ is the \emph{temporal domain} (often taken as
$\mathbb{N}$ or $\mathbb{R}_+$), and
$\mathbb{N}$ or any other countable set), and
$\mathcal{T}\subseteq\mathbb{T}$ is the \emph{lifetime} of the
network,
\item $\rho: E\times\mathcal{T}\mapsto\{0,1\}$ is the \emph{presence
function}, which determines whether an edge is present in the
network at each timestamp.
network at each time stamp.
\end{itemize}
The \emph{available dates} of an edge are the set
The \emph{available times} of an edge are the set
$\mathcal{I}(e) = \{t\in\mathcal{T}: \rho(e,t)=1\}$.
\end{defn}
@ -135,13 +153,19 @@ possible to have constant weights (edges can only appear or disappear
over time, and always have the same weight), or time-varying
weights. In the latter case, we can set the domain of the presence
function to be $\mathbb{R}_+$ instead of $\{0,1\}$, where by
convention a zero weight corresponds to an absent edge.
convention a 0 weight corresponds to an absent edge.
\begin{defn}[Additive temporal network]
\begin{defn}[Additive and dismantling temporal
networks]\label{defn:additive}
A temporal network is said to be \emph{additive} if for all $e\in E$
and $t\in\mathcal{T}$, if $\rho(e,t)=1$, then
$\forall t'>t, \rho(e, t') = 1$. Edges can only be added to the
network, never removed.
and $t\in\mathcal{T}$, if $\rho(e,t)=1$, then for all
$t'>t, \rho(e, t') = 1$. An additive network can only gain edges
over time.
A temporal network is said to be \emph{dismantling} if for all
$e\in E$ and $t\in\mathcal{T}$, if $\rho(e,t)=0$, then for all
$t'>t, \rho(e, t') = 0$. An dismantling network can only lose edges
over time.
\end{defn}
\section{Examples of applications}%
@ -152,6 +176,8 @@ convention a zero weight corresponds to an absent edge.
\section{Network partitioning}%
\label{sec:network-partitioning}
%% TODO clarify, organise, references
Temporal networks are a very active research subject, leading to
multiple interesting problems. The additional time dimension adds a
significant layer of complexity that cannot be adequately treated by
@ -195,6 +221,8 @@ outliers, or even maximise temporal communities.
\chapter{Topological Data Analysis and Persistent Homology}%
\label{cha:tda-ph}
%% TODO references
\section{Basic constructions}%
\label{sec:basic-constructions}
@ -202,9 +230,9 @@ outliers, or even maximise temporal communities.
\label{sec:homology}
Our goal is to understand the topological structure of a metric
space. For this, we can use \emph{homology}, which consists in
associating for a metric space $X$ and a dimension $i$ a vector space
$H_i(X)$. The dimension of $H_i(X)$ will give us the number of
space. For this, we can use \emph{homology}, which consists of
associating a vector space $H_i(X)$ to a metric space $X$ and a
dimension $i$. The dimension of $H_i(X)$ gives us the number of
$i$-dimensional components in $X$: the dimension of $H_0(X)$ is the
number of path-connected components in $X$, the dimension of $H_1(X)$
is the number of holes in $X$, and the dimension of $H_2(X)$ is the
@ -217,28 +245,27 @@ space can be extremely difficult. It is necessary to approximate it in
a structure that would be both combinatorial and topological in
nature.
\subsection{Simplicial Complexes}%
\subsection{Simplicial complexes}%
\label{sec:simplicial-complexes}
In order to understand the topological structure of a metric space, we
need a way to decompose it in smaller pieces which, when assembled,
conserve the overall organisation of the space. For this, we use a
structure called a \emph{simplicial complex}, which is a kind of
higher-dimensional generalization of graphs.
To understand the topological structure of a metric space, we need a
way to decompose it in smaller pieces that, when assembled, conserve
the overall organisation of the space. For this, we use a structure
called a \emph{simplicial complex}, which is a kind of
higher-dimensional generalization of a graph.
The building blocks of this representation will be \emph{simplexes},
which are simply the convex hull of an arbitrary set of
points. Examples of simplexes include single points, segments,
triangles, and tetrahedrons (in dimensions 0, 1,, 2, and 3
respectively).
The building blocks of this representation is the \emph{simplex},
which is the convex hull of an arbitrary set of points. Examples of
simplices include single points, segments, triangles, and tetrahedrons
(in dimensions 0, 1,, 2, and 3 respectively).
\begin{defn}[Simplex]
The \emph{$k$-dimensional simplex} $\sigma = [x_0,\ldots,x_k]$ is
the convex hull of the set $\{x_0,\ldots,x_k\} \in \mathbb{R}^d$,
where $x_0,\ldots,x_k$ are affinely independent. $x_0,\ldots,x_k$
are called the \emph{vertices} of $\sigma$, and the simplexes
defined by the subsets of $\{x_0,\ldots,x_k\}$ are called the
\emph{faces} of $\sigma$.
A \emph{$k$-dimensional simplex} $\sigma = [x_0,\ldots,x_k]$ is the
convex hull of the set $\{x_0,\ldots,x_k\} \in \mathbb{R}^d$, where
$x_0,\ldots,x_k$ are affinely independent. $x_0,\ldots,x_k$ are
called the \emph{vertices} of $\sigma$, and the simplices defined by
the subsets of $\{x_0,\ldots,x_k\}$ are called the \emph{faces} of
$\sigma$.
\end{defn}
\begin{figure}[ht]
@ -284,22 +311,22 @@ respectively).
\caption{Triangle}
\end{subfigure}%
%
\caption{Examples of simplexes}%
\caption{Examples of simplices}%
\label{fig:simplex}
\end{figure}
We then need a way to combine these basic building blocks meaningfully
We then need a way to meaningfully combine these basic building blocks
so that the resulting object can adequately reflect the topological
structure of the metric space.
\begin{defn}[Simplicial complex]
A \emph{simplicial complex} is a collection $K$ of simplexes such
A \emph{simplicial complex} is a collection $K$ of simplices such
that:
\begin{itemize}
\item any face of a simplex of $K$ is a simplex of $K$
\item the intersection of two simplexes of $K$ is either the empty
set or a common face or both.
\item the intersection of two simplices of $K$ is either the empty
set, or a common face, or both.
\end{itemize}
\end{defn}
@ -340,13 +367,13 @@ structure of the metric space.
\draw (5) -- (6) -- (4) -- (7);
\end{scope}
\end{tikzpicture}
\caption{Example of a simplicial complex, with two connected
components, two 3-simplexes, and one 5-simplex.}%
\caption{Example of a simplicial complex that has two connected
components, two 3-simplices, and one 5-simplex.}%
\label{fig:simplical-complex}
\end{figure}
The notion of simplicial complex is closely related to that of a
hypergraph. The important distinction lies in the fact that a subset
hypergraph. One important distinction lies in the fact that a subset
of a hyperedge is not necessarily a hyperedge itself.
Using these definitions, we can define homology on simplicial
@ -355,20 +382,21 @@ complexes. %% TODO add reference for more details/do it myself?
\subsection{Filtrations}%
\label{sec:filtrations}
%% TODO rewrite it using the Cech complex as an introductory example,
%% to understand the problem with scale
If we consider that a simplicial complex is a kind of
``discretization'' of a metric space, we realise that there must be an
issue of \emph{scale}. For our analysis to be invariant under small
perturbations in the data, we need a way to find the optimal scale
parameter to capture the adequate topological structure, without
taking into account some small perturbations, nor ignoring some
important smaller features.
``discretization'' of a subset of a metric space, we realise that
there must be an issue of \emph{scale}. For our analysis to be
invariant under small perturbations in the data, we need a way to find
the optimal scale parameter to capture the adequate topological
structure, without taking into account some small perturbations, nor
ignoring some important smaller features.
%% TODO rewrite using the Cech filtration as an example?
One possible solution to these problems is to consider all scales at
once. This is the objective of \emph{filtered simplicial complexes}.
The ideal solution to these problems is to consider all scales at
once: this is the objective of \emph{filtered simplicial complexes}.
\begin{defn}[Filtration]
\begin{defn}[Filtration]\label{defn:filt}
A \emph{filtered simplicial complex}, or simply a \emph{filtration},
$K$ is a sequence ${(K_i)}_{i\in I}$ of simplicial complexes such
that:
@ -384,8 +412,8 @@ once: this is the objective of \emph{filtered simplicial complexes}.
We can now compute the homology for each step in a filtration. This
leads to the notion of \emph{persistent
homology}~\cite{carlsson_topology_2009,zomorodian_computing_2005},
which gives us all the information necessary to establish the
topological structure of the metric space at multiple scales.
which gives all the information necessary to establish the topological
structure of a metric space at multiple scales.
\begin{defn}[Persistent homology]
The \emph{$p$-th persistent homology} of a simplicial complex
@ -395,44 +423,85 @@ topological structure of the metric space at multiple scales.
by the inclusion map $K_i \mapsto K_j$.
\end{defn}
The functions $f_{i,j}$ allow us to link generators in each successive
homology space in the filtration. Since each generator correspond to a
topological feature (connected component, hole, void, etc, depending
on the dimension $p$), we can determine whether it survives in the
next step of the filtration. We can now determine when each feature is
born and when it dies (if it dies at all). This representation will be
dependent on the choice of basis for each homology space
$H_p(K_i)$. However, by the Fundamental Theorem of Persistent
Homology, we can choose base vectors in each homology space such that
the collection of half-open intervals is well-defined and unique. This
construction is called a \emph{barcode}.
%% TODO references for the Fundamental Theorem
The functions $f_{i,j}$ allow one to link generators in each
successive homology space in a filtration. Because each generator
corresponds to a topological feature (connected component, hole, void,
and so on, depending on the dimension $p$), we can determine whether
it survives in the next step of the filtration. We can also determine
when each feature is born and when it dies (if it dies at all). The
couples of intervals (birth time, death time) depends on the choice of
basis for each homology space $H_p(K_i)$. However, by the Fundamental
Theorem of Persistent Homology~\cite{zomorodian_computing_2005}, we
can choose basis vectors in each homology space such that the
collection of half-open intervals is well-defined and unique. This
construction is called a \emph{barcode}~\cite{carlsson_topology_2009}.
\section{Topological summaries: barcodes and persistence diagrams}%
\label{sec:topol-summ}
In order to interpret the results of the persistent homology
computation, we need to compare the output for a particular data set
to a suitable null model. For this, we need some kind of similarity
measure between barcodes and a way to evaluate the statistical
significance of the results.
%% TODO need more context
One possible approach for this is to define a space in which we can
project barcodes and study their geometric
properties. \emph{Persistence diagrams} are an example of such a
space.
To interpret the results of the persistent-homology computation, we
need to compare the output for a particular data set to a suitable
null model. For this, we need some kind of similarity measure between
barcodes and a way to evaluate the statistical significance of the
results.
One possible approach is to define a space in which we can project
barcodes and study their geometric properties. One such space is the
space of \emph{persistence
diagrams}~\cite{edelsbrunner_computational_2010}.
\begin{defn}[Multiset]
A \emph{multiset} $M$ is the couple $(A, m)$, where $A$ is the
\emph{underlying set} of $M$, formed by its distinct elements, and
$m : A\mapsto\mathbb{N}^*$ is the \emph{multiplicity function}
giving the number of occurrences of each element of $A$ in $M$.
\end{defn}
\begin{defn}[Persistence diagrams]
A \emph{persistence diagram} is the union of a finite multiset of
points in $\overline{\mathbb{R}}^2$ zith the diagonal
points in $\overline{\mathbb{R}}^2$ with the diagonal
$\Delta = \{(x,x) \;|\; x\in\mathbb{R}^2\}$, where every point of
$\Delta$ has infinite multiplicity.
\end{defn}
The diagonal $\Delta$ is added to facilitate comparisons between
diagrams, as points near the diagonal correspond to short-lived
topological feature, thus likely to be caused by small perturbations
in the data.
One adds the diagonal $\Delta$ for technical reasons. It is convenient
to compare persistence diagrams by using bijections between them, so
persistence diagrams must have the same cardinality.
In some cases, the diagonal in the persistence diagrams can also
facilitate comparisons between diagrams, as points near the diagonal
correspond to short-lived topological features, so they are likely to
be caused by small perturbations in the data.
One can build a persistence diagram from a barcode by taking the union
of the multiset of (birth, death) couples with the diagonal
$\Delta$. \autoref{fig:pipeline} summarises the entire pipeline.
\begin{figure}[ht]
\centering
\begin{tikzpicture}
\tikzstyle{pipelinestep}=[rectangle,thick,draw=black,inner sep=5pt,minimum size=15pt]
\node (data)[pipelinestep] {Data};
\node (filt)[pipelinestep,right=1cm of data] {Filtered complex};
%% \node (barcode)[pipelinestep,right=1cm of filt] {Barcodes};
\node (dgm)[pipelinestep,right=1cm of filt] {Persistence diagram};
\node (interp)[pipelinestep,right=1cm of dgm] {Interpretation};
\draw[->] (data.east) -- (filt.west);
%% \draw[->] (filt.east) -- (barcode.west);
\draw[->] (filt.east) -- (dgm.west);
\draw[->] (dgm.east) -- (interp.west);
\end{tikzpicture}
\caption{Persistent homology pipeline}%
\label{fig:pipeline}
\end{figure}
One can define an operator $\dgm$ as the first two steps in the
pipeline. It constructs a persistence diagram from a subset of a
metric space, via persistent homology on a filtered complex.
We can now define several distances on the space of persistence
diagrams.
@ -441,7 +510,7 @@ diagrams.
The \emph{$p$-th Wasserstein distance} between two diagrams $X$ and
$Y$ is
\[ W_p[d](X, Y) = \inf_{\phi:X\mapsto Y} \left[\sum_{x\in X} {d\left(x, \phi(x)\right)}^p\right] \]
for $p\in [1,\infty)$, and
for $p\in [1,\infty)$, and:
\[ W_\infty[d](X, Y) = \inf_{\phi:X\mapsto Y} \sup_{x\in X} d\left(x,
\phi(x)\right) \] for $p = \infty$, where $d$ is a distance on
$\mathbb{R}^2$ and $\phi$ ranges over all bijections from $X$ to
@ -450,25 +519,27 @@ diagrams.
\begin{defn}[Bottleneck distance]
The \emph{bottleneck distance} is defined as the infinite
Wasserstein distance with $d$ the uniform norm:
Wasserstein distance where $d$ is the uniform norm:
$d_B = W_\infty[L_\infty]$.
\end{defn}
Since the bottleneck distance is by far the most commonly used, we
will focus on it in the following. It is symmetric, non-negative, and
satisfies the triangle inequality. However, it is not a true distance,
as it is fairly straightforward to come up with two distinct diagrams
at bottleneck distance zero, even on multisets not touching the
diagonal $\Delta$.
The bottleneck distance is symmetric, non-negative, and satisfies the
triangle inequality. However, it is not a true distance, as one can
come up with two distinct diagrams with bottleneck distance 0, even
on multisets that do not touch the diagonal $\Delta$.
\section{Stability}%
\label{sec:stability}
One of the most important aspects of Topological Data Analysis is that
One of the most important aspects of topological data analysis is that
it is \emph{stable} with respect to small perturbations in the
data. In fact, the persistence diagram operator is Lipschitz with
respect to the bottleneck distance. First, we define a distance
between subsets of a metric space.
data. More precisely, the second step of the pipeline
in~\autoref{fig:pipeline} is Lipschitz with respect to a suitable
metric on filtered complexes and the bottleneck distance on
persistence
diagrams~\cite{cohen-steiner_stability_2007,chazal_persistence_2014}. First,
we define a distance between subsets of a metric
space~\cite{oudot_persistence_2015}.
\begin{defn}[Hausdorff distance]
Let $X$ and $Y$ be subsets of a metric space $(E, d)$. The
@ -477,7 +548,8 @@ between subsets of a metric space.
\sup_{y\in Y} \inf_{x\in X} d(x,y) \right]. \]
\end{defn}
We can now give the proper stability property.
We can now give an appropriate stability
property~\cite{cohen-steiner_stability_2007,chazal_persistence_2014}.
\begin{prop}
Let $X$ and $Y$ be subsets in a metric space. We have
@ -490,6 +562,14 @@ We can now give the proper stability property.
%% TODO
\cite{morozov_dionysus:_2018,bauer_ripser:_2018,reininghaus_dipha_2018,maria_gudhi_2014}
\section{Discussion}%
\label{sec:discussion}
%% TODO
%% information thrown away in filtrations and in PH
\chapter{Topological Data Analysis on Networks}%
\label{cha:topol-data-analys}
@ -498,49 +578,54 @@ We can now give the proper stability property.
We now consider the problem of applying persistent homology to network
data. An undirected network is already a simplicial complex of
dimension 1. However, this will not be sufficient to capture enough
topological information: we need to introduce higher-dimensional
simplexes. The first possible method is to project the network on a
metric space~\cite{otter_roadmap_2017}, thus transforming the network
data into a point cloud data. For this, we need to compute the
distance between each pair of nodes in the network (with the shortest
path distance for instance). This also requires the network to be
connected.
dimension 1. However, this is not sufficient to capture enough
topological information; we need to introduce higher-dimensional
simplices. One method is to project the nodes of a network onto a
metric space~\cite{otter_roadmap_2017}, thereby transforming the
network data into a point-cloud data. For this, we need to compute the
distance between each pair of nodes in the network (e.g.\ with the
shortest-path distance). This also requires the network to be
connected. %% TODO defn of connected?
Another usual method for weighted networks is called the \emph{weight
rank clique filtration} (WRCF)~\cite{petri_topological_2013}, which
filters the network based on weights. The procedure works as follows:
Another common method, for weighted networks, is called the
\emph{weight rank-clique filtration}
(WRCF)~\cite{petri_topological_2013}, which filters a network based
on weights. The procedure works as follows:
\begin{enumerate}
\item Set the set of all nodes, without any edge, as filtration
step~0.
\item Consider the set of all nodes, without any edge, to be
filtration step~0.
\item Rank all edge weights in decreasing order $\{w_1,\ldots,w_n\}$.
\item At filtration step $t$, keep only the edges whose weights are
less than $w_t$, thus creating an unweighted graph.
larger than or equal to $w_t$, thereby creating an unweighted graph.
\item Define the maximal cliques of the resulting graph to be
simplexes.
simplices.
\end{enumerate}
At each step of the filtration, we construct a simplicial complex
based on cliques: this is called a \emph{clique complex}. It is
necessarily valid since a subset of a clique is necessarily a clique
itself, and the same is true for the intersection of two cliques.
based on cliques; this is called a \emph{clique
complex}~\cite{zomorodian_tidy_2010}. The result of the algorithm is
itself a filtered simplicial complex (\autoref{defn:filt}), because a
subset of a clique is necessarily a clique itself, and the same is
true for the intersection of two cliques.
This leads to a first possibility for applying persistent homology to
temporal networks. It is possible to segment the lifetime of the
network into sliding windows, creating a static graph on each window
temporal networks. It is possible to segment the lifetime of a network
into sliding windows, creating a time-independent graph on each window
by retaining only the edges available during the time interval. We can
then apply WRCF on each static graph in the sequence, obtaining a
filtered complex for each window, to which we can then apply
persistent homology.
then apply WRCF on each graph in the sequence, obtaining a filtered
complex for each window, to which we can then apply persistent
homology.
This method can quickly become very computationally expensive, as
finding all maximal cliques (using the Bron-Kerbosch algorithm for
example) is a complicated problem in itself. In practice, we often
restrict the search to cliques of dimension lower than a certain bound
$d_M$. With this restriction, the new simplicial complex is
homologically equivalent to the original: they have the same homology
groups up to $H_{d_M-1}$.
finding all maximal cliques (e.g.\ using the Bron--Kerbosch algorithm)
is a complicated problem, with an optimal computational complexity of
$\mathcal{O}\big(3^{n/3}\big)$~\cite{tomita_worst-case_2006}. In
practice, one often restrict the search to cliques of dimension less
than or equl to a certain bound $d_M$. With this restriction, the new
simplicial complex is homologically equivalent to the original: they
have the same homology groups up to $H_{d_M-1}$.
%% TODO rewrite this paragraph
This method is sensitive to the choice of sliding windows on the time
scale. The width and the overlap of the windows can completely change
the networks created and their topological features. Too small a
@ -552,24 +637,24 @@ in the evolution of the network over time.
\label{sec:zigzag-persistence}
The standard algorithm to compute persistent homology
(\autoref{sec:persistent-homology}) only works for filtrations which
are nested sequences of simplicial complexes:
(see~\autoref{sec:persistent-homology}) relies on the fact that
filtrations (see~\autoref{defn:filt}) are nested sequences of
simplicial complexes:
\[ \cdots \subseteq K_{i-1} \subseteq K_i \subseteq K_{i+1} \subseteq
\cdots \]
When studying temporal networks, we have two possibilities:
\begin{itemize}
\item Create an independent filtration (e.g.\ WRCF) from each time
step. The issue is that the topological features will be completely
disconnected from the time dimension.
\item Create a filtration along the time dimension. The issue in this
case is that the sequence is no longer nested (except for additive
temporal networks, ie when edges are never deleted).
\end{itemize}
One can now create an independent filtration (e.g.\ with WRCF) for
each time step. The issue is that the topological features will be
orthogonal to the time dimension.
Another possibility is to create a filtration along the time
dimension. The issue in this case is that the sequence is no longer
nested (except for additive or dismantling temporal networks,
see~\autoref{defn:additive}).
The solution to consider the time dimension is provided by
\emph{zigzag persistence}~\cite{carlsson_zigzag_2009}, which allows to
compute persistence on alternating nested sequences:
\emph{zigzag persistence}~\cite{carlsson_zigzag_2009}, which allows
one to compute persistence on alternating nested sequences:
\[ \cdots \supseteq K_{i-1} \subseteq K_i \supseteq K_{i+1} \subseteq
\cdots \]
@ -580,22 +665,23 @@ constructing an alternating sequence.
Zigzag persistence is a special case of the more general concept of
\emph{multi-parameter
persistence}~\cite{carlsson_theory_2009,dey_computing_2014}, where
filtrations can span across multiple parameters.
filtrations can encompass multiple parameters.
%% Note about libraries implementing zigzag persistence: Dionysus
\chapter{Persistent Homology for Machine Learning applications}%
\chapter{Persistent Homology for Machine-Learning Applications}%
\label{cha:pers-homol-mach}
The output of persistent homology is not directly usable by most
statistical methods. Barcodes and persistence diagrams, being a
multiset of points in $\overline{\mathbb{R}}^2$, are not elements of a
metric space in which we could perform statistical computations.
statistical methods. For example, barcodes and persistence diagrams,
which are multisets of points in $\overline{\mathbb{R}}^2$, are not
elements of a metric space in which one can perform statistical
computations.
The distances between persistence diagrams defined
in~\autoref{sec:topol-summ} allow us to compare different
in~\autoref{sec:topol-summ} allow one to compare different
outputs. From a statistical perspective, it is possible to use a
generative model of simplicial complexes, and use a distance between
generative model of simplicial complexes and to use a distance between
persistence diagrams to measure the similarity of our observations
with this null model~\cite{adler_persistent_2010}. This would
effectively define a metric space of persistence diagrams. It is even
@ -603,8 +689,10 @@ possible to define some statistical summaries (means, medians,
confidence intervals) on these
spaces~\cite{turner_frechet_2014,munch_probabilistic_2015}.
%% TODO REFERENCES
The issue with this approach is that metric spaces do not offer enough
algebraic structure to be amenable to most machine learning
algebraic structure to be amenable to most machine-learning
techniques. One of the most recent development in the study of
topological summaries has been to find mappings between the space of
persistence diagrams and Banach spaces.
@ -616,38 +704,37 @@ persistence diagrams and Banach spaces.
\subsection{Persistence landscapes}
Persistence landscapes~\cite{bubenik_statistical_2015} are a mean to
project the barcodes in a space where it will be possible to add them
meaningfully. It would thus be possible to define means of persistence
diagrams, along other summary statistics.
Persistence landscapes~\cite{bubenik_statistical_2015} give a way to
project barcodes to a space where it is possible to add them
meaningfully. It is then possible to define means of persistence
diagrams, as well as other summary statistics.
As all the other vectorization techniques mentioned here, this
approach is \emph{injective}, but not surjective, and no explicit
inverse exists to go back from a persistence landscape to the
corresponding persistence diagram. Moreover, a mean of persistence
landscapes do not necessarily have a corresponding persistence
diagram.
The function mapping a persistence diagram to a persistence landscape
is \emph{injective}, but no explicit inverse exists to go back from a
persistence landscape to the corresponding persistence
diagram. Moreover, a mean of persistence landscapes does not
necessarily have a corresponding persistence diagram.
\begin{defn}[Persistence landscape]
The persistence landscape of a diagram $D = \{(b_i,d_i)\}_{i=1}^n$
is the set of functions $\lambda_k: \mathbb{R} \mapsto \mathbb{R}$,
for $k\in\mathbb{N}$ such that
for $k\in\mathbb{N}$, such that
\[ \lambda_k(x) = k\text{-th largest value of } \{f_{(b_i,
d_i)}(x)\}_{i=1}^n, \] (or zero if the $k$-th largest value does
not exist), where $f_{(b,d)}$ is a piecewise linear function defined by:
d_i)}(x)\}_{i=1}^n, \] (and $\lambda_k(x) = 0$ if the $k$-th
largest value does not exist), where $f_{(b,d)}$ is a
piecewise-linear function defined by:
\[ f_{(b,d)} =
\begin{cases}
0& \text{if }x \notin (b,d)\\
x-b& \text{if }x\in (b,\frac{b+d}{2})\\
-x+d& \text{if }x\in (\frac{b+d}{2},d).
0,& \text{if }x \notin (b,d),\\
x-b,& \text{if }x\in (b,\frac{b+d}{2}),\\
-x+d,& \text{if }x\in (\frac{b+d}{2},d)\,.
\end{cases}
\]
\end{defn}
The persistence landscape is thus a kind of superposition of piecewise
linear functions. Moreover, one can show that persistence landscapes
are stable with respect to the $L^p$ distance, and that the
Wasserstein and bottleneck distances are bounded by the $L^p$
Moreover, one can show that persistence landscapes are stable with
respect to the $L^p$ distance, and that the Wasserstein and bottleneck
distances are bounded by the $L^p$
distance~\cite{bubenik_statistical_2015}. We can thus view the
landscapes as elements of a Banach space in which we can perform the
statistical computations.
@ -667,7 +754,7 @@ statistical computations.
\cite{reininghaus_stable_2015,kwitt_statistical_2015}
\subsection{Persistence weighted Gaussian kernel}
\subsection{Persistence weighted-Gaussian kernel}
\cite{kusano_kernel_2017}
@ -678,6 +765,9 @@ statistical computations.
\section{Comparison}%
\label{sec:comparison}
\chapter{Conclusions}%
\label{cha:conclusions}