work on chapters 4-5

acknowledgements.tex

@@ -6,6 +6,5 @@
 Ignore the following markings:
 \todo[]
 \source[]
-\newresult[]z
-
+\newresult[]
 \end{abstract}

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.

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] \\

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: