From 48a6b0e75f0466790727904520334df564647253 Mon Sep 17 00:00:00 2001 From: Dimitri Lozeve Date: Thu, 26 Jul 2018 11:08:18 +0100 Subject: [PATCH] Dissertation: TDA --- dissertation/dissertation.pdf | Bin 62380 -> 147984 bytes dissertation/dissertation.tex | 167 ++++++++++++++++++++++++++++++++-- dissertation/preamble.tex | 14 +++ 3 files changed, 175 insertions(+), 6 deletions(-) diff --git a/dissertation/dissertation.pdf b/dissertation/dissertation.pdf index e165a83541e1bf316e68d604abc3e6f84d831864..87e05b5bcd62b1e0068bdfded872f959611a443d 100644 GIT binary patch delta 117199 zcmZs?V~i$1w}sp8nYP{2wl&RZ+qP|6Z`*#`wr$(CZQH!xIX5@C$#;I$PAaKiPu8ln zYggq7cwEqLGBSB#Q934iHW;$m#i3Og*1v>|gm#9OFg!dA(k8ZM&gO)FIoN=H#pHl! zm5G=QM!3!snn&HykjVJnYU;Y{~^6kh1Twq<3j5+97N zogxF^(ogWvk4 z7pd@cay^T*A@NW(xLlu==Vh^zbs0TZi3}WdzW$mW3fjg5!b;ms{&_?M6Jo$Tw;^?? zA&5>aPXrN8OV~PkkvVFfK}zMxHFdS6Q60SsJJ+ZbnLK|p4CYg+4jx(mWnph1k${+W z?x7VvUO%)2nPdp5cRz=#AUfcJOIbdAtKyf6YByTG(p0_5j!wWf*MU>*p*DLYjOj>z zpI_&Sor!#KYFsuE?F6NdM+LUFm^?7{&PkzH>P1}Sm>$wKCp@F%+|DYyj@lHy$#rL9 z!2qZ%G9Ja|?vm;0sD<+NucXvhj91m@s!v`iozv1%O~wx2gnN0Lej7lV?TKA+Q=d`p`{QL0EJs{D5TGM z1^>Kb#O50?xbN*r0Xm25^Sr}i{)E-mZq9*sNJU|)Q_CoM}XD)u9}yeyZ*%>=x>}_zGveBLa%l+lZ*E#`8HQubjnS z>@!`K&2?^`TX`ZW*lS#_Mjdxg(nIzZY;=SiMV@Hy5_?^F?w+P12 zjFpV>wnG@rroo=|#8y#<68*-8cD)GF84gFdctFoZgN}d(X5>IP8!4c^IT|q3OdFIK zcxYw{QRT}E1Xh{sh!M_-YZx5WQ*o zvYRC5v&z7gU_MILpa75;4t0n{30#&eOO)?=%HQzUC=>y$07I>1#BG5agBfRuTc5UK z>urLVr1{`*UU0oplT(X`0?FKY%+J#-43ZPBuySYXRDEkdU34|PY#UL{WWwTym>BWg zaBxa27g!o)?RC)_x!<|n3}rt?=g#|YI5?Pr@76jPod1Uaa zLmQ4)Q!^+oDw1ar_OY=Kq9cU;^eF0jNg<{qK?ImDF8tY0deZi7VsqBxKHa{_?bNI8 zaKr4Mj&s}wbz5P`N=A>{F_s@&(6~_vliKN!g9Z@-fdg@lZ-E<)F)IFkcp5-ZzQw1E zR!Bg(+i71~94TFO188FK`xZz1E9-4yE4Nhhn&Qn;^gHy(`l5AhF7iCB%9pV;5VVa~QY$TFjb-mzH@wB2A^g_dR>j+=wmfS+4W|b8G*Rgu{Y#QKz#mQG2Nnz>`vuyEZ zG9%HfT5iN|(apZxn6f_;Z`S>|WO87!AYj$e(i`W|{q z_FJP+oKw$@4`jl<8S=k*XJY&x;Pue|n|CG_j{o5uStYs;28;<#_~rxs&A)1a)dpB( zL#UPxVd$X2Li|MGK~jrtXg?XAP)-Bhb=J*4qiKey59$tC2+E`xT5c+`$eA%-n^1PC zzT7wL>GQb3zSAGfbJ??G$ARfEh8aPA8GAY3+;bt7!z+SFod@KPK*M1CugeDu+g~Q2 ze=!-L32!WhrseLtTW}<8fYI0?mI92`t`K63&K(-O+0H+jaP(hJ&b}^b##R*%^l)nx zM#{Qdnx1dR?LS}Ye4bM-92}wvD44SyHrp) zg5IiE;T)rCV)FU7a(+dx`|C*xR0C$ZH8^rE+rfZk6i?5|#J;D{>}W%VOX311cxHCT z%=(dTJCs!kGcQ#cigGN|P&YDCG{S$ThLJU1SV%6x5GTgcRND{#@QT(`f`T7w^3xXsa;1dE#2svOg(*ROvR9+&) znwl+hzbco!uU7_XL{^Fabhcr6oK76g5qgo&M*KnSQWyTkM zPUU$2LX7a)D5T>59m5iXlloG=<9b@AFqD~5ttsKKb)2^F)%?QLbjhKxFxLcOm7aJA z-yM88?(A{#8ylPodwm{KpqUAr3>ylU#PrjN#6b7JLv)|A9U2tC&Q+O9eIg=j8Js$y zd#bO8Mq2#NsOv=Qf^yZ)d`(pb#0ao|0vjE^GGf z0PcgScqH~NIyD5t5GCR~PKSwa{||)5Q(IfQonmF(CF=7nFqOpsYzBfz^g;yV3m3XL zxO;_aa>wLxdm1xM0}#llf>i{@TP@jkbFj{L8LaK$6SyHh%h@?UgrLnf3M)CckAmTH=fe=IfLWMo{4+9W?S-CY(7CY(?T-%m|)7@E*}43zjfE_LEJ2Tlz>I_81D! z51_U{(`5rDbLIGv`}w=9dIHw0Q}gq|A5XQFX;rCkbae8}5prqVF^?^$8s}ATC(iRD z>KaSl2zNy$B?DIBSew2+kOv=%enbT|&nX7S+i{S0V_BYDgVU0iJgHj&Hd!!3T^@%4 z9TJc)w`;GTje9y*Ae4&P3>AF09y4-1HiZGPS403{ovVrr+G2vmk?gP<>JI(Cr0vE> zV;$a_YwLwkYPS#-iX7J_a*`dogZr#@?;y~3O|SNn1_+(?l^t9wKKVwL2|hs zwnu=jW@62nMI6K_`1JZuuw;yE19ulwja>9g0j~DC4&J=Jk+W03-P2WGnDCnQM0@Ug3y?$DKu< zg#SQLI&u=C$+DXs9|8ji&MgB^{I}k2A3Dd2yee|IKrPKK6|vzIH&WB$OD*q z4?aA&cD6(=6oXlxo3#IeHL$EQ2FuEVC7#(&za&sbp;Kn1`Ir(|QT|?D({y#HU92IC$s#6~KRlBZ2{see41}OwGa){2lP=6Rc4V#K>XXPZ4{|RUS~1Kt`m}RoL?{@e z^0hxWGaH|)4xw0bh#Dss@k zp>Dd!ATKyojVe35Y0zfWFcgP

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a/dissertation/dissertation.tex +++ b/dissertation/dissertation.tex @@ -2,6 +2,7 @@ \input{preamble} +\usepackage[firstpage]{draftwatermark} \begin{document} @@ -78,18 +79,176 @@ Thank you! \chapter{Topological Data Analysis and Persistent Homology}% \label{cha:tda-ph} +\section{Homology}% +\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 +$i$-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 +number of voids. + +Crucially, these vector spaces are robust to continuous deformation of +the underlying metric space (they are \emph{homotopy + invariant}). However, computing the homology of an arbitrary metric +space can be extremely difficult. It is necessary to approximate it in +a structure that would be both combinatorial and topological in +nature. + \section{Simplicial Complexes}% \label{sec:simplicial-complexes} -\section{Homology}% -\label{sec:homology} +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. + +The building blocks of this representation will be \emph{simplices}, +which are simply 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 simplices + defined by the subsets of $\{x_0,\ldots,x_k\}$ are called the + \emph{faces} of $\sigma$. +\end{defn} + +We then need a way to combine these basic building blocks meaningfully +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 simplices such + that: + \begin{itemize} + \item any face of a simplex of $K$ is a simplex of $K$ + \item the intersection of two simplices of $K$ is either the empty + set or a common face or both. + \end{itemize} +\end{defn} + +%% TODO figure with examples of simplicial complexes + +Using these definitions, we can define homology on simplicial +complexes. %% TODO add reference for more details/do it myself? \section{Filtrations}% \label{sec:filtrations} +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. + +%% TODO rewrite using the Cech filtration as an example? + +The ideal solution to these problems is to consider all scales at +once: this is the objective of \emph{filtered simplical complexes}. + +\begin{defn}[Filtration] + A \emph{filtered simplicial complex}, or simply a \emph{filtration}, + $K$ is a sequence ${(K_i)}_{i\in I}$ of simplicial complexes such + that: + \begin{itemize} + \item for any $i, j \in I$, if $i < j$ then $K_i \subseteq K_j$, + \item $\bigcup_{i\in I} K_i = K$. + \end{itemize} +\end{defn} + +\section{Persistent Homology}% +\label{sec:persistent-homology} + +We can now compute the homology for each step in a filtration. This +leads to the notion of \emph{persistent homology}, which gives us all +the information necessary to establish the topological structure of +the metric space at multiple scales. + +\begin{defn}[Persistent homology] + The \emph{$p$-th persistent homology} of a simplicial complex + $K = {(K_i)}_{i\in I}$ is the pair + $(\{H_p(K_i)\}_{i\in I}, \{f_{i,j}\}_{i,j\in I, i\leq j})$, where + for all $i\leq j$, $f_{i,j} : H_p(K_i) \mapsto H_p(K_j)$ is induced + 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 + \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 a similarity +measure between barcodes and a way to evaluate the statistical +significance of the results. + +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. + +\begin{defn}[Persistence diagrams] + A \emph{persistence diagram} is the union of a finite multiset of + points in $\bar{\mathbb{R}}^2$ zith 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. + +We can now define several distances on the space of persistence +diagrams. + +\begin{defn}[Wasserstein distance] + 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 + \[ W_\infty[d](X, Y) = \inf_{\phi:X\mapsto Y} \sup_{x\in X} d\left(x, + \phi(x)\right) \] for $p = \infty$, where $d$ is a distance on + $\mathbb{R}^2$ and $\phi$ ranges over all bijections from $X$ to + $Y$. +\end{defn} + +\begin{defn}[Bottleneck distance] + The \emph{bottleneck distance} is defined as the infinite + Wasserstein distance with $d$ 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 $\Delta$. + \section{Stability}% \label{sec:stability} @@ -97,10 +256,6 @@ Thank you! \chapter{Temporal Networks}% \label{cha:temporal-networks} - - - - \backmatter% \nocite{*} diff --git a/dissertation/preamble.tex b/dissertation/preamble.tex index df6c935..2054723 100644 --- a/dissertation/preamble.tex +++ b/dissertation/preamble.tex @@ -16,6 +16,20 @@ \usepackage{lettrine} \usepackage{amssymb, amsmath} +\usepackage{amsthm} + +\theoremstyle{plain} +\newtheorem{thm}{Theorem}[chapter] +\newtheorem{lem}[thm]{Lemma} +\newtheorem{cor}[thm]{Corollary} +\newtheorem{prop}[thm]{Proposition} +\theoremstyle{definition} +\newtheorem{defn}{Definition}[chapter] +\newtheorem{expl}{Example}[chapter] +\theoremstyle{remark} +\newtheorem*{rem}{Remark} +\newtheorem*{note}{Note} +\newtheorem*{notation}{Notation} \usepackage{pdfpages}