diff --git a/acknowledgements.tex b/acknowledgements.tex index 9699c93..3231879 100644 --- a/acknowledgements.tex +++ b/acknowledgements.tex @@ -6,6 +6,5 @@ Ignore the following markings: \todo[] \source[] -\newresult[]z - +\newresult[] \end{abstract} \ No newline at end of file diff --git a/chapters/classical_clustering_functors.tex b/chapters/classical_clustering_functors.tex index 19e985d..5abbfa2 100644 --- a/chapters/classical_clustering_functors.tex +++ b/chapters/classical_clustering_functors.tex @@ -2,15 +2,15 @@ \chapter{Classification of Classical Clustering Functors} \label{chapter__classical} In this chapter all clustering functor are of the classical type. Our goal will ultimately be to prove a uniqueness result for the so-called \emph{Vietoris-Rips} functor. - +% We start by presenting a useful way of constructing clustering functors. Namely, we show that any \emph{excessive} clustering functor can be \emph{represented} by a family of metric spaces. This will give an alternative view on the Vietoris-Rips functor. - +% After this we are ready to tackle the task of characterizing the Vietoris-Rips functor by defining properties such as \emph{surjectivity}, \emph{spanning}, and \emph{splitting}. Depending on the setting it will turn out that these conditions are all equivalent and unique to the Vietoris-Rips functor. - +% This chapter is largely based on the results presented in \cite{Carlsson2010}. \begin{defprop}{Vietoris-Rips Functor \cite[Def.~6.1]{Carlsson2010}}{classical_vr} - Let $\delta > 0$ and $\M \in \{\iso, \inj, \gen\}$. The Vietoris-Rips functor $\Rf_\delta\colon \M \to \C$ assigns to each metric space $(X,d) \in \M$ the partition $(X,P)$ where $\sim_P$ is the equivalence generated by: + Let $\delta > 0$ and $\M \in \{\iso, \inj, \gen\}$. The Vietoris-Rips functor $\Rf_\delta\colon \M \to \C$ assigns to each metric space $(X,d) \in \M$ the partition $(X,P)$ where $\sim_P$ is the equivalence relation generated by: \begin{equation} \label{eq:vietoris_rips_equivalence_relation} \forall x,y \in X: d(x,y) \leq \delta \implies x \sim_P y. @@ -25,17 +25,17 @@ \chapter{Classification of Classical Clustering Functors} \begin{proof} It is sufficient to show that $\Rf_\delta$ is $\gen$-functorial. By Remark \ref{rem:induced_functor_by_inclusion}, functoriality over $\iso$ and $\inj$ will follow. -Let $X,Y \in \ob(\gen)$ and $(X,P) := \Rf_\delta(X)$ as well as $(Y,Q) := \Rf_\delta(Y)$. +Let $(X,d),(Y,d') \in \ob(\gen)$ and $(X,P) := \Rf_\delta(X)$ as well as $(Y,Q) := \Rf_\delta(Y)$. Take any $f \in \mor_\gen(X,Y)$, recall that $f$ is distance non-increasing. We have to show that $$P \refines_f Q.$$ Indeed, let $x,y \in X$ such that $d(x,y) \leq \delta$. -Then, we have $d(f(x), f(y)) \leq d(x,y) \leq \delta$ and therefore $f(x) \sim_{Q} f(y)$. +Then, we have $d'(f(x), f(y)) \leq d(x,y) \leq \delta$ and therefore $f(x) \sim_{Q} f(y)$. By taking the transitive closure the statement follows. \end{proof} \begin{example}{}{} -Consider the seven points $\{a,b,c,d,e,f,g\} \subset \R^2$ shown in the Figure \ref{fig:vietoris_rips_example}. Then, $\Rf_\delta$ creates the clusters $\{a,b,c\}$ and $\{d,e,f,g\}$, drawn in red and blue. +Consider the seven points $\{a,b,c,d,e,f,g\} \subset \R^2$ shown in the Figure \ref{fig:vietoris_rips_example}. $\Rf_\delta$ creates the clusters $\{a,b,c\}$ and $\{d,e,f,g\}$, drawn in red and blue. \begin{center} \begin{minipage}{\textwidth} \centering @@ -92,7 +92,6 @@ \chapter{Classification of Classical Clustering Functors} \medskip A clustering functor $\Cf\colon \iso \to \C$ is uniquely determined by a choice of $P_X \in \Xi_X$ for each $X \in \mathcal{I}$. \end{myremark} -Clearly, $\iso$ permits many clustering functors. In some sense, $\iso$ clustering functors can be thought of as any algorithm which ``does not consider any particular ordering'' of the data points. This is the reason that algorithms like single linkage clustering \ref{section__linkage_clustering} are not even $\iso$-functorial. @@ -107,7 +106,7 @@ \section{Excessive and Representable Clustering Functors} $$ \end{definition} -Let $\Omega$ be a family of finite non-empty metric spaces. Such $\Omega$'s can be used to construct clustering functors. We think of $\Omega$ as being a collection of \emph{patterns} and our clustering algorithm as detecting these \emph{patterns}. +Let $\Omega$ be a family of finite non-empty metric spaces. Such $\Omega$'s can be used to construct clustering functors. We think of $\Omega$ as being a collection of ``patterns'' and our clustering algorithm as detecting these ``patterns''. \begin{definition}{Representable Clustering Functors \cite[Sec.~6.2]{Carlsson2010}}{construction_of_representable_clustering_functors} Let $\M \in \{\inj, \gen\}$. We define the clustering functor \emph{represented by $\Omega$} as @@ -147,7 +146,11 @@ \section{Excessive and Representable Clustering Functors} This leads to an alternative characterization of $\Cf^\Omega$. \begin{proposition}{}{characterization_of_representable} -Let $\M \in \{\inj, \gen\}$. For any $\M$-clustering functor $\Cf$ such that $\Cf(X,d)$ is trivial for all $(X,d) \in \Omega$ we have +Let $\M \in \{\inj, \gen\}$. For any $\M$-clustering functor +$$ +\Cf\colon \M \to \C +$$ +such that $\Cf(X,d)$ is trivial for all $(X,d) \in \Omega$ we have $$ \Cf^\Omega \refines \Cf. $$ @@ -159,8 +162,6 @@ \section{Excessive and Representable Clustering Functors} Let $x,y \in X$ such that the generating condition of $P$ holds, \ie, there exists $\omega \in \Omega$ and $\phi \in \mor_\M(\omega, (X,d))$ such that $\{x,y\} \subset \image(\phi)$. Since $\Cf(\omega)$ is trivial by assumption and by functoriality of $\Cf(\omega) \refines_\phi \Cf(X,d)$, we get that $x \sim_Q y$. Taking the transitive closure gives the statement. - -\newresult[check this] \end{proof} Importantly, $\Cf^\Omega$ is the finest clustering functor such that $\Cf^\Omega(X,d)$ is trivial for all $(X,d) \in \Omega$. The existence of this minimal clustering functor follows from our initial construction in Definition \ref{def:construction_of_representable_clustering_functors}. @@ -181,21 +182,17 @@ \section{Excessive and Representable Clustering Functors} We will show that $\Cf = \Cf^\Omega$. By proposition \ref{prop:characterization_of_representable} and since $\Cf$ is by definition trivial for all $(X,d) \in \Omega$ we have that $\Cf^\Omega \refines \Cf$. So it remains to show $\Cf \refines \Cf^\Omega$. To this end let $(X,d) \in \ob(\M)$ and $(X,P) = \Cf(X,d)$ and $(X,Q) = \Cf^\Omega(X,d)$. Assume that $x,y \in X$ are such that $x \sim_P y$. But then by definition, there exists $\omega \subset X$ and $\omega \in \Omega$ such that $\{x,y\} \subset \omega$. Consider the inclusion $\iota: \omega \hookrightarrow X$ which is a morphism in $\mor_\inj(\omega, (X,d))$. Thus by definition of $\Cf^\Omega$, we have $x \sim_Q y$. - -\todo[this is a new proof (kinda), check it] \end{proof} \section{Surjective Clustering Functors} -One of the conditions in \textsc{Kleinberg}'s impossibility theorm (Theorem \ref{thm:kleinberg}) was richness (Definition \ref{def:richness}). Here we extend this notion to clustering functors. +One of the conditions in \textsc{Kleinberg}'s impossibility theorem (Theorem \ref{thm:kleinberg}) was richness (Definition \ref{def:richness}). Here we extend this notion to clustering functors. \begin{definition}{Surjective Clustering Functors}{} A classical clustering functor $\Cf\colon \M \to \C$ is called \emph{surjective} if for every finite set $X$ and every $P \in \P(X)$ there exists a metric $d$ on $X$ such that - \begin{equation*} - \Cf(X,d) = (X,P) + \Cf(X,d) = (X,P). \end{equation*} - \end{definition} \begin{proposition}{\cite[Rem.~6.1]{Carlsson2010}}{vietoris_rips_is_surjective} @@ -214,10 +211,10 @@ \section{Surjective Clustering Functors} \end{proof} \begin{example}{}{} -Let $\delta > 0$ consider $X := \{a,b,c\}$. We would like to check that the Vietoris-Rips functor is indeed able to reach any partition of $X$. Up to permuting $X$ there are three possible partitions of $X$: +Let $\delta > 0$ consider $X := \{a,b,c\}$. We would like to check that the Vietoris-Rips functor is indeed able to reach any partition of $X$. Up to permutation, there are three possible partitions of $X$: \begin{itemize} - \item $P_1 = \{\{a\}, \{b\}, \{c\}\}$, - \item $P_2 = \{\{a,b,c\}\}$, + \item $P_1= \{\{a,b,c\}\}$, + \item $P_2 = \{\{a\}, \{b\}, \{c\}\}$, \item $P_3 = \{\{a,b\}, \{c\}\}$. \end{itemize} First, let us consider $P_1$. For this we can define the metric $d_1$ on $X$ such that $d_1(i,j) = \delta$ for all $i \neq j$. Then, $\Rf_\delta(X,d_1) = (X,P_1)$. @@ -235,7 +232,7 @@ \section{Surjective Clustering Functors} \end{enumerate} \end{definition} -The concept of spanning clustering functors, a term which we introduce here, was previously discussed by \textsc{Carlsson} and \textsc{M\'emoli} as part of the assumptions of a theorem \cite[Thm.~6.4]{Carlsson2010}. +The concept of spanning clustering functors, a term which we introduced here, was previously discussed by \textsc{Carlsson} and \textsc{M\'emoli} as part of the assumptions of a theorem \cite[Thm.~6.4]{Carlsson2010}. \begin{lemma}{}{surjective_implies_eventually_discrete} Let $\M \in \{\inj, \gen\}$ and $\Cf\colon \M \to \C$ be a surjective clustering functor. Then $\Cf$ is spanning. @@ -264,13 +261,12 @@ \section{Surjective Clustering Functors} \begin{proof}[Proof of Lemma \ref{lem:surjective_implies_eventually_discrete}] Let $(X,d) \in \ob(\M)$, we assume that $|X| > 1$ otherwise the statement follows directly. Since $\Cf$ is surjective there exists metrics $d_0, d_1$ on $X$ such that $\Cf(X,d_0)$ is trivial and $\Cf(X,d_1)$ is discrete. - \begin{enumerate} \item We take $$ \lambda_0 := \frac{\sep(X,d_0)}{\diam(X,d)}. $$ - Notice that for any $\lambda \le \lambda_0$ we have that $\diam(X,\lambda \cdot d) \le \sep(X,d_0)$. Because of this the function + Notice that for any $0 < \lambda \le \lambda_0$ we have that $\diam(X,\lambda \cdot d) \le \sep(X,d_0)$. Because of this the function \begin{align*} f: (X, d_0) &\longrightarrow (X, \lambda \cdot d),\\ x &\longmapsto x @@ -287,9 +283,7 @@ \section{Surjective Clustering Functors} f: (X, \lambda \cdot d) &\longrightarrow (X, d_1),\\ x &\longmapsto x. \end{align*} - We conclude since we have $\Cf(X,\lambda \cdot d) \refines \Cf(X,d_1)$ for the same reason as above. - - \newresult[check this] + We conclude since we have $\Cf(X,\lambda \cdot d) \refines \Cf(X,d_1)$ for the same reason as above. \end{enumerate} \end{proof} @@ -389,7 +383,7 @@ \section{Uniqueness of the Vietoris-Rips Functor} $$ \lambda \mapsto \Cf(X, \lambda \cdot d), $$ -which will be piecewise constant and takes only finitely many values. +which is piecewise constant and takes only finitely many values. Regularity ensures that this function is constant on intervalls of the form $(a, b]$ for some $a < b$. This ensures compatibility with the regularity condition for dendrograms from Definition \ref{def:dendrogram}. @@ -406,6 +400,8 @@ \section{Uniqueness of the Vietoris-Rips Functor} \end{itemize} \end{theorem} +For the proof of this theorem we use the following lemma. + \begin{lemma}{}{surjective_implies_splitting} Let $\Cf\colon \M \to \C$ be a surjective regular clustering functor. Then $\Cf$ is splitting at some $\delta > 0$. \end{lemma} @@ -428,10 +424,10 @@ \section{Uniqueness of the Vietoris-Rips Functor} \R_{\geq0} &\longrightarrow \P(X), \\ \lambda &\longmapsto \Cf(\Delta_2(\lambda)) \end{align*} - is piecewise constant and can take at most two values (either discrete or trivial) but by Lemma \ref{lem:surjective_implies_eventually_discrete} this function takes exactly two values and by \eqref{eq:lemma_5_19_1} this function is also monotonically decreasing (with respect to $\le$ on $\R_{\geq0}$ and $\refines$ on $\P(X)$). + is piecewise constant and can take at most two values (either discrete or trivial), but by Lemma \ref{lem:surjective_implies_eventually_discrete} this function takes exactly two values and by \eqref{eq:lemma_5_19_1} this function is also monotonically decreasing (with respect to $\le$ on $\R_{\geq0}$ and $\refines$ on $\P(X)$). So we can find some $\delta_0 > 0$ such that $\Cf(\Delta_2(\delta))$ is trivial and $\Cf(\Delta_2(\delta'))$ is discrete for all $\delta < \delta_0 < \delta'$. - As for the value at $\delta_0$ we recall the previous Remark \ref{rem:regularity_classical_clustering_functors} and conclude that $\Cf(\Delta_2(\delta_0))$ is trivial. Therefore, $\Cf$ is splitting at $\delta_0$. + As for the value at $\delta_0$ we recall Remark \ref{rem:regularity_classical_clustering_functors} and conclude that $\Cf(\Delta_2(\delta_0))$ is trivial. Therefore, $\Cf$ is splitting at $\delta_0$. \end{proof} \begin{proof}[Proof of Theorem] @@ -479,7 +475,7 @@ \section{Scale Invariant Clustering Functors} \end{proof} In the case of $\inj$ we have a more interesting behavior. \begin{proposition}{\cite[Thm.~6.6]{Carlsson2010}}{} - Let $\Cf\colon \inj \to \C$ be a scale invariant functor then there exists a $k \in \N \sqcup \{0, \infty\}$ such that for all $(X,d) \in \ob(\inj)$: + Let $\Cf\colon \inj \to \C$ be a scale invariant functor then there exists a $k \in \N \sqcup \{\infty\}$ such that for all $(X,d) \in \ob(\inj)$: \begin{itemize} \item If $|X| > k$ then $\mathfrak{C}(X,d)$ is trivial. @@ -500,7 +496,7 @@ \section{Scale Invariant Clustering Functors} $$ On the other hand, any permutation of $\Delta_n(\delta)$ is also a morphism in $\inj$. This gives us that $\Cf(\Delta_n(\delta))$ is either discrete or trivial. - So together we get that there exists some $k \in \N \sqcup \{0, \infty\}$ such that + So together we get that there exists some $k \in \N \sqcup \{\infty\}$ such that \begin{itemize} \item $\forall n > k: \Cf(\Delta_n(\delta))$ is trivial. \item $\forall n \le k: \Cf(\Delta_n(\delta))$ is discrete. diff --git a/chapters/clustering_functors.tex b/chapters/clustering_functors.tex index 31c2857..e665716 100644 --- a/chapters/clustering_functors.tex +++ b/chapters/clustering_functors.tex @@ -298,7 +298,6 @@ \section{Clustering Functors} \end{definition} We can express the functoriality of a clustering functor $\Cf$ by the following commutative diagram. - \begin{equation*} \begin{tikzcd} {(X,d)} \arrow[r, "f"] \arrow[d, "\Cf", Rightarrow] & {(Y,d)} \arrow[d, "\Cf", Rightarrow] \\ diff --git a/chapters/hierarchical_clustering_functors.tex b/chapters/hierarchical_clustering_functors.tex index 8b7928b..c370021 100644 --- a/chapters/hierarchical_clustering_functors.tex +++ b/chapters/hierarchical_clustering_functors.tex @@ -1,7 +1,7 @@ \chapter{Hierarchical Clustering Functors} \label{chapter__hierarchical} In this chapter we first introduce the hierarchical version of the Vieotirs-Rips functor. -Next we talk about a crucial link between certain hierarchical clustering functors and regular classical clustering functors. With this, we can then tackle the modified \textsc{Kleinberg} conditions presented in \cite[Sec.~7.3.1]{Carlsson2010} and show that the Vietoris-Rips is the unique hierarchical clustering functor satisfying them. +Next we talk about a crucial link between \emph{scaling} hierarchical clustering functors and regular spanning classical clustering functors. With this, we can then tackle the modified \textsc{Kleinberg} conditions presented in \cite[Sec.~7.3.1]{Carlsson2010} and show that the Vietoris-Rips is the unique hierarchical clustering functor satisfying them. \begin{definition}{Vietoris-Rips Functor \cite[Ex.~7.1]{Carlsson2010}}{hierarchical_vr} For $\M \in \{\gen,\inj,\iso\}$ we can define the Vietoris-Rips hierarchical clustering functor by @@ -18,7 +18,7 @@ \chapter{Hierarchical Clustering Functors} Similar to classical clustering functors, we also want to define some properties for hierarchical clustering functors. Moreover, from now on we will again be working with $\M \in \{\gen, \inj\}$ unless otherwise stated. \begin{definition}{}{} - A hierarchical clustering functor $\Hf$ is said to be \emph{surjective} if for every $(X, \theta_X) \in \ob(\H)$ there exists a metric $d$ on $X$ such that $\Hf(X,d) = (X, \theta_X)$. + A hierarchical clustering functor $\Hf$ is said to be \emph{surjective} if for every ${(X, \theta) \in \ob(\H)}$ there exists a metric $d$ on $X$ such that $\Hf(X,d) = (X, \theta)$. \end{definition} \begin{myremark}{}{hierarchical_to_classical_surjective} @@ -43,7 +43,7 @@ \chapter{Hierarchical Clustering Functors} Instead of scale invariance we can ask that this shift functor behaves nicely with scaling of the metric. \begin{definition}{}{} -Let $\M \in \{\iso,\inj,\gen\}$. A hierarchical clustering functor $\Hf\colon \M \to \H$ for is called \emph{scaling} if for all ${(X,d) \in \ob(\M)}$ and $\lambda > 0$ we have +Let $\M \in \{\iso,\inj,\gen\}$. A hierarchical clustering functor $\Hf\colon \M \to \H$ is called \emph{scaling} if for all ${(X,d) \in \ob(\M)}$ and $\lambda > 0$ we have $$ \Hf(X, \lambda \cdot d) = s_\lambda \Hf(X,d). $$ @@ -72,11 +72,11 @@ \chapter{Hierarchical Clustering Functors} -Extending the Vietoris-Rips functor to a hierarchical clustering functor can be done more generally for any regular classical clustering functor. +Extending the Vietoris-Rips functor to a hierarchical clustering functor can be done more generally for any regular spanning classical clustering functor. \begin{proposition}{}{scaling_extension_correspondence} Let $\M \in \{\gen,\inj\}$. Then there exists a one to one correspondence between -scaling hierarchical clustering functors and spanning\footnote{Recall definitions \ref{def:regular} and \ref{def:spanning}.} classical clustering functors. +scaling hierarchical clustering functors and regular spanning\footnote{Recall definitions \ref{def:regular} and \ref{def:spanning}.} classical clustering functors. \medskip More precisely given a regular spanning classical clustering functor $\Cf\colon \M \to \C$ there exists a unique scaling hierarchical clustering functor $\Hf_\Cf\colon \M \to \H$ such that we have @@ -88,7 +88,7 @@ \chapter{Hierarchical Clustering Functors} \end{proposition} \begin{proof} -Given a regular classical clustering functor $\Cf\colon \M \to \C$ we can define +Given a regular spanning classical clustering functor $\Cf\colon \M \to \C$ we can define $$ \Hf_\Cf(X,d; r) := \Cf(X, r^{-1} \cdot d) $$ @@ -115,7 +115,7 @@ \chapter{Hierarchical Clustering Functors} \Hf_\Cf(X, \lambda \cdot d; r) = \Cf(X, r^{-1} \lambda \cdot d) = \Hf_\Cf(X,d;r \lambda^{-1}). $$ -On the other hand, given any scaling hierarchical clustering functor $\Hf\colon \M \to \H$ we can take $\Cf(X,d) := \Hf(X,d; 1)$ which is a regular classical clustering functor such that $\Hf = \Hf_\Cf$. +On the other hand, given any scaling hierarchical clustering functor $\Hf\colon \M \to \H$ we can take $\Cf(X,d) := \Hf(X,d; 1)$ which is a regular spanning classical clustering functor such that $\Hf = \Hf_\Cf$. \end{proof} \begin{example}{}{shift_of_vietoris_rips} @@ -131,7 +131,7 @@ \chapter{Hierarchical Clustering Functors} \section{Kleinberg's Conditions} -\textsc{Carlsson} and \textsc{M\'emoli} mention that \textsc{Kleinberg}'s impossibility conditions can be interpreted in the context of hierarchical clustering functors \cite[Sec.~7.3.1]{Carlsson2010}. +\textsc{Carlsson} and \textsc{M\'emoli} noticed that \textsc{Kleinberg}'s impossibility conditions can be interpreted in the context of hierarchical clustering functors \cite[Sec.~7.3.1]{Carlsson2010}. \begin{definition}{Modified Kleinberg Conditions \cite[Sec.~7.3.1]{Carlsson2010}}{} We say that a hierarchical clustering $\Hf\colon \M \to \H$ functor fulfills the \emph{modified Kleinberg conditions} if all the following holds: diff --git a/chapters/notation.tex b/chapters/notation.tex index 337e566..1fbdfae 100644 --- a/chapters/notation.tex +++ b/chapters/notation.tex @@ -6,7 +6,7 @@ \chapter{Notation} $\sep(X)$ & the separation of a metric space $X$\\ $\diam(X)$ & the diameter of a metric space $X$\\ -$\P(X)$ & the containing all partitions of $X$\\ +$\P(X)$ & the set containing all partitions of $X$\\ $P \in \P(X)$ & a partition of a set $X$\\ $\sim_P$ & equivalence relation corresponding to the partition $P \in \P(X)$\\ @@ -19,11 +19,10 @@ \chapter{Notation} $\H$ & the category of outputs of hierarchical clustering functors\\ $\Cf$ & a classical clustering functor $\Cf\colon \M \to \C$\\ -$\Rf_\delta$ & the Vietoris-Rips clustering functor at scale $\delta>0$\\ +$\Rf_\delta$ & the Vietoris-Rips classical clustering functor at scale $\delta>0$\\ $\Cf^\Omega$ & the classical clustering functor represented by $\Omega$\\ $\Hf$ & a hierarchical clustering functor $\Hf\colon \M \to \H$\\ -$\Hf_\Cf$ & the hierarchical clustering functor corresponding to $\Cf$\\ +$\Hf_\Cf$ & the scaling hierarchical clustering functor corresponding to $\Cf$\\ $\Rf$ & the Vietoris-Rips hierarchical clustering functor\\ - \end{tabular} \ No newline at end of file diff --git a/main.tex b/main.tex index 85ab534..d6cd11f 100644 --- a/main.tex +++ b/main.tex @@ -32,5 +32,4 @@ \bibliographystyle{alpha} \bibliography{bib} - \end{document}