From 926b57a3052865b35e22abcca0433aa5790712fe Mon Sep 17 00:00:00 2001 From: ammedmar Date: Fri, 13 Mar 2020 10:40:52 +0100 Subject: [PATCH 01/11] Update P landscapes Signed-off-by: ammedmar --- doc/theory/glossary.tex | 18 +++++++++--------- 1 file changed, 9 insertions(+), 9 deletions(-) diff --git a/doc/theory/glossary.tex b/doc/theory/glossary.tex index 7bfab9d17..5a49f4d10 100644 --- a/doc/theory/glossary.tex +++ b/doc/theory/glossary.tex @@ -26,6 +26,7 @@ \begin{tabular}{ l l l} $\Bbbk$ & : & An arbitrary field. \\ $\mathbb R$ & : & The field of real numbers. \\ + $\overline{\mathbb R}$ & : & The two point compactification $[-\infty, +\infty]$ of the real numbers. \\ $\mathbb N$ & : & The counting numbers $0,1,2, \ldots$ as a subset of $\mathbb R$. \\ $\mathbb R^d$ & : & The vector space of $d$-tuples of real numbers. \\ $\Delta$ & : & The \hyperref[multiset]{multiset} $\{(s,s)\,;\ s \in \mathbb R\}$ with multiplicity $(s,s) \mapsto +\infty$. @@ -260,32 +261,31 @@ \subsection*{Persistence landscape} \label{persistence landscape} - A \textit{persistence landscape} is a continuous function - + A \textit{persistence landscape} isa set $\{\lambda_k\}_{k \in \mathbb N}$ of functions \begin{equation*} - \lambda : \mathbb N \times \mathbb R \to \mathbb R \cup \{+\infty\} + \lambda : \mathbb R \to \overline{\mathbb R} \end{equation*} - and the function $\lambda_k(s) = \lambda(k,s)$ is refered to as the \textit{$k$-layer of the persistence diagram}. + where $\lambda_k$ is referred to as the \textit{$k$-layer of the persistence landscape}. - Let ${(b_i, d_i)}{i \in I}$ be a \hyperref[persistence diagram] {persistence diagram}. Its \textit{associated persistence landscape} $\lambda$ is defined by letting $\lambda_k(t)$ be the $k$-th largest value of the set $\{\Lambda_i(t)\}_ {i \in I}$ where + Let $\{(b_i, d_i)\}_{i \in I}$ be a \hyperref[persistence diagram] {persistence diagram}. Its \textit{associated persistence landscape} $\lambda$ is defined by letting $\lambda_k$ be the $k$-th largest value of the set $\{\Lambda_i(t)\}_ {i \in I}$ where \begin{equation*} \Lambda_i(t) = \left[ \min \{t-b_i, d_i-t\}\right]_+ \end{equation*} and $c_+ := \max(c,0)$. - Intuitively, we can describe the graph of this persistence landscape by first joining each of the points in the multiset to the diagonal via a horizontal as well as a vertical line, then rotating the figure 45 degrees clockwise, and rescaling by $1/\sqrt{2}$. + Intuitively, we can describe the set of graphs of a persistence landscape by first joining each of the points in the multiset to the diagonal via a horizontal as well as a vertical line, then clockwise rotating the figure 45 degrees and rescaling it by $1/\sqrt{2}$. \paragraph{\\ Reference:} \cite{bubenik2015statistical} \subsection*{Persistence landscape norm} \label{persistence landscape norm} - Given a function $f : \mathbb R \to \overline{\mathbb R} = [-\infty, +\infty]$ define + Given a function $f : \mathbb R \to \overline{\mathbb R}$ define \begin{equation*} ||f||_p = \left( \int_{\mathbb R} f^p(x)\, dx \right)^{1/p} \end{equation*} whenever the right hand side exists and is finite. - The \textit{persistence landscape $p$-norm} of a \hyperref[persistence landscape]{persistence landscape} $\lambda : \mathbb N \times \mathbb R \to \overline{\mathbb R}$ is defined to be + The \textit{$p$-norm} of a \hyperref[persistence landscape]{persistence landscape} $\lambda = \{\lambda_k\}_{k \in \mathbb N}$ is defined to be \begin{equation*} ||\lambda||_p = \left( \sum_{i \in \mathbb N} ||\lambda_i||^p_p \right)^{1/p} @@ -304,7 +304,7 @@ \begin{equation*} \label{equation: lambda for persistence landscapes} \Lambda_i(t) = \left[ \min \{t-b_i, d_i-t\}\right]_+ \end{equation*} - and $c_+ := \max(c,0)$. The particular choice $w_i = \vert d_i - b_i \vert^p$ for $0 < p \leq \infty$ is referred to as \textit{power-weighted silhouettes}. + with $c_+ := \max(c,0)$. The particular choice $w_i = \vert d_i - b_i \vert^p$ for $0 < p \leq \infty$ is referred to as \textit{power-weighted silhouettes}. \paragraph{\\ References:} \cite{chazal2014stochastic} From c8fbb21ad40cfe6a364528a3a6eeaa3424e8a7d1 Mon Sep 17 00:00:00 2001 From: ammedmar Date: Tue, 17 Mar 2020 12:18:09 +0100 Subject: [PATCH 02/11] Add distances, inner products and kernels glossary entry Signed-off-by: ammedmar --- doc/theory/glossary.tex | 48 +++++++++++++++++++++++++++++++++-------- 1 file changed, 39 insertions(+), 9 deletions(-) diff --git a/doc/theory/glossary.tex b/doc/theory/glossary.tex index 5a49f4d10..464910173 100644 --- a/doc/theory/glossary.tex +++ b/doc/theory/glossary.tex @@ -331,13 +331,13 @@ The name is inspired from the case when the persistence diagram comes from persistent homology. - \subsection*{Metric space} \label{metric space} + \subsection*{Distances, inner products and kernels} \label{metric, inner product and kernel} - A pair $(X, d)$ where $X$ is a set and $d$ is a function + A set $X$ with a function \begin{equation*} d : X \times X \to \mathbb R \end{equation*} - attaining non-negative values is called a \textit{metric space} if + is called a \textit{metric space} if the values of $d$ are all non-negative and for all $x,y,z \in X$ \begin{equation*} d(x,y) = 0\ \Leftrightarrow\ x = y \end{equation*} @@ -345,19 +345,49 @@ d(x,y) = d(y,x) \end{equation*} \begin{equation*} - d(x,z) \leq d(x,y) + d(y, z) + d(x,z) \leq d(x,y) + d(y, z). \end{equation*} + In this case the $d$ is referred to as the \textit{metric} or the \textit{distance function}. - In this case, the function $d$ is refer to as the \textit{metric} and the value $d(x,y)$ is called the \textit{distance} between $x$ and $y$. + A vector space $V$ together with a function + \begin{equation*} + \langle -, - \rangle : V \times V \to \mathbb R + \end{equation*} + is called and \textit{inner product space} if for all $u,v,w \in V$ + \begin{equation*} + u \neq 0\ \Rightarrow\ \langle u, u \rangle > 0 + \end{equation*} + \begin{equation*} + \langle u, v\rangle = \langle v, u\rangle + \end{equation*} + \begin{equation*} + \langle au+v, w \rangle = a\langle u, w \rangle + \langle v, w \rangle. + \end{equation*} + In this case the function $\langle -, - \rangle$ is referred to as the \textit{inner product} and the function given by + \begin{equation*} + ||u|| = \sqrt{\langle u, u \rangle} + \end{equation*} + as its associated \textit{norm}. An inner product space is naturally a metric space with distance function + \begin{equation*} + d(u,v) = ||u-v||. + \end{equation*} + + A \textit{kernel} on a set $X$ is a function + \begin{equation*} + k : X \times X + \end{equation*} + for which there exists a function $\phi : X \to V$ to an inner product space such that + \begin{equation*} + k(x, y) = \langle \phi(x), \phi(y) \rangle. + \end{equation*} \subsection*{Euclidean distance and norm} \label{euclidean distance and norm} - The set $\mathbb R^n$ defines a metric space with euclidean distance + The vector space $\mathbb R^n$ is an \hyperref[metric, inner product and kernel]{inner product space} with inner product \begin{equation*} - d(x,y) = \sqrt{(x_1-y_1)^2 + \cdots + (x_n-y_n)^2}. + \langle x, y \rangle = (x_1-y_1)^2 + \cdots + (x_n-y_n)^2. \end{equation*} - - The norm $||x||$ of a vector $x$ is defined as its distance to the $0$ vector. + The associated norm and distance function are referred to as \textit{euclidean norm} and \textit{euclidean distance}. \subsection*{Finite metric spaces and point clouds} \label{finite metric spaces and point clouds} From 082e06483f9f41849e50ebbaf2cb091e0e9d14c8 Mon Sep 17 00:00:00 2001 From: ammedmar Date: Mon, 23 Mar 2020 16:12:48 +0100 Subject: [PATCH 03/11] Remake vectorization changes Signed-off-by: ammedmar --- doc/theory/glossary.tex | 441 +++++++++++++++++++++++++--------------- 1 file changed, 275 insertions(+), 166 deletions(-) diff --git a/doc/theory/glossary.tex b/doc/theory/glossary.tex index 464910173..4879c8176 100644 --- a/doc/theory/glossary.tex +++ b/doc/theory/glossary.tex @@ -3,8 +3,8 @@ \usepackage{enumerate} \usepackage{tikz-cd} \usepackage{bm} -\usepackage[breaklinks=true, bookmarks=true, -bookmarksnumbered=true, breaklinks=true, +\usepackage[bookmarks=true, +bookmarksnumbered=true, pdfstartview=FitH, hyperfigures=false, plainpages=false, naturalnames=true, colorlinks=true, pagebackref=true, @@ -15,26 +15,27 @@ linkcolor=blue, urlcolor=blue} - \begin{document} - - \title{Glossary} + + \title{Theory Glossary} \maketitle - + \section{Symbols} - \begin{tabular}{ l l l} - $\Bbbk$ & : & An arbitrary field. \\ - $\mathbb R$ & : & The field of real numbers. \\ - $\overline{\mathbb R}$ & : & The two point compactification $[-\infty, +\infty]$ of the real numbers. \\ - $\mathbb N$ & : & The counting numbers $0,1,2, \ldots$ as a subset of $\mathbb R$. \\ - $\mathbb R^d$ & : & The vector space of $d$-tuples of real numbers. \\ - $\Delta$ & : & The \hyperref[multiset]{multiset} $\{(s,s)\,;\ s \in \mathbb R\}$ with multiplicity $(s,s) \mapsto +\infty$. + \begin{tabular}{ l l} + $\Bbbk$ & An arbitrary field. \\ + $\mathbb R$ & The field of real numbers. \\ + $\overline{\mathbb R}$ & The two point compactification $[-\infty, +\infty]$ of the real numbers. \\ + $\mathbb N$ & The counting numbers $0,1,2, \ldots$ as a subset of $\mathbb R$. \\ + $\mathbb R^d$ & The vector space of $d$-tuples of real numbers. \\ + $\Delta$ & The + %\hyperref[multiset]{multiset} + multiset $ \lbrace (s, s) \mid s \in \mathbb{R} \rbrace $ with multiplicity $ ( s,s ) \mapsto +\infty$. \end{tabular} \section{Homology} - \subsection*{Cubical complex} \label{cubical complex} + \subsection*{Cubical complex} \label{cubical_complex} An \textit{elementary interval} $I_a$ is a subset of $\mathbb{R}$ of the form $[a, a+1]$ or $[a,a] = \{a\}$ for some $a \in \mathbb{R}$. These two types are called respectively \textit{non-degenerate} and \textit{degenerate}. To a non-degenerate elementary interval we assign two degenerate elementary intervals \begin{equation*} @@ -57,21 +58,23 @@ \paragraph{\\ Reference:} \cite{mischaikow04computational} - \subsection*{Simplicial complex} \label{simplicial complex} + \subsection*{Simplicial complex} \label{simplicial_complex} A set $\{v_0, \dots, v_n\} \subset \mathbb{R}^N$ is said to be \textit{geometrically independent} if the vectors $\{v_0-v_1, \dots, v_0-v_n\}$ are linearly independent. In this case, we refer to their convex closure as a \textit{simplex}, explicitly \begin{equation*} - [v_0,\dots ,v_n] = \left\{ \sum c_i (v_0 - v_i)\ \big|\ c_1+\dots+c_n = 1,\ c_i \geq 0 \right\} + [v_0, \dots , v_n] = \left\{ \sum c_i (v_0 - v_i)\ \big|\ c_1+\dots+c_n = 1,\ c_i \geq 0 \right\} \end{equation*} - and to $n$ as its \textit{dimension}. The \textit{$i$-th face} of $[v_0, \dots, v_n]$ is defined by + and to $n$ as its \textit{dimension}. The $i$\textit{-th face} of $[v_0, \dots, v_n]$ is defined by \begin{equation*} d_i[v_0, \dots, v_n] = [v_0, \dots, \widehat{v}_i, \dots, v_n] \end{equation*} where $\widehat{v}_i$ denotes the absence of $v_i$ from the set. - A \textit{simplicial complex} $X$ is a finite union of simplices in $\mathbb{R}^N$ satisfying that every face of a simplex in $X$ is in $X$ and that the non-empty intersection of two simplices in $X$ is a face of each. Every simplicial complex defines an \hyperref[abstract simplicial complex]{abstract simplicial complex}. + A \textit{simplicial complex} $X$ is a finite union of simplices in $\mathbb{R}^N$ satisfying that every face of a simplex in $X$ is in $X$ and that the non-empty intersection of two simplices in $X$ is a face of each. Every simplicial complex defines an + % \hyperref[abstract_simplicial_complex]{abstract simplicial complex} + abstract simplicial complex. - \subsection*{Abstract simplicial complex} \label{abstract simplicial complex} + \subsection*{Abstract simplicial complex} \label{abstract_simplicial_complex} An \textit{abstract simplicial complex} is a pair of sets $(V, X)$ with the elements of $X$ being subsets of $V$ such that: \begin{enumerate} @@ -82,21 +85,27 @@ The elements of $X$ are called \textit{simplices} and the \textit{dimension} of a simplex $x$ is defined by $|x| = \# x - 1$ where $\# x$ denotes the cardinality of $x$. Simplices of dimension $d$ are called $d$-simplices. We abuse terminology and refer to the elements of $V$ and to their associated $0$-simplices both as \textit{vertices}. - The \textit{$k$-skeleton }$X_k$ of a simplicial complex $X$ is the subcomplex containing all simplices of dimension at most $k$. A simplicial complex is said to be \textit{$d$-dimensional} if $d$ is the smallest integer satisfying $X = X_d$. + The $k$\textit{-skeleton} $X_k$ of a simplicial complex $X$ is the subcomplex containing all simplices of dimension at most $k$. A simplicial complex is said to be $d$\textit{-dimensional} if $d$ is the smallest integer satisfying $X = X_d$. A \textit{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 $X$ is a \textit{subcomplex} of a simplicial complex $Y$ if every simplex of $X$ is a simplex of $Y$. - Given a finite abstract simplicial complex $X = (V, X)$ we can choose a bijection from $V$ to a geometrically independent subset of $\mathbb R^N$ and associate a \hyperref[simplicial complex]{simplicial complex} to $X$ called its \textit{geometric realization}. + Given a finite abstract simplicial complex $X = (V, X)$ we can choose a bijection from $V$ to a geometrically independent subset of $\mathbb R^N$ and associate a + %\hyperref[simplicial_complex]{simplicial complex} + simplicial complex + to $X$ called its \textit{geometric realization}. - \subsection*{Ordered simplicial complex} \label{ordered simplical complex} + \subsection*{Ordered simplicial complex} + \label{ordered_simplical_complex} - An \textit{ordered simplicial complex} is an \hyperref[abstract simplicial complex]{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 $n$-simplex using its ordered vertices by $[v_0, \dots, v_n]$. + An \textit{ordered simplicial complex} is an + % \hyperref[abstract_simplicial_complex]{abstract simplicial complex} + 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 $n$-simplex using its ordered vertices by $[v_0, \dots, v_n]$. A \textit{simplicial map} between ordered simplicial complexes is a simplicial map $f$ between their underlying simplicial complexes preserving the order, i.e., $v \leq w$ implies $f(v) \leq f(w)$. - \subsection*{Directed simplicial complex} \label{directed simplicial complex} + \subsection*{Directed simplicial complex} \label{directed_simplicial_complex} A \textit{directed simplicial complex} is a pair of sets $(V, X)$ with the elements of $X$ being tuples of elements of $V$, i.e., elements in $\bigcup_{n\geq1} V^{\times n}$ such that: \begin{enumerate} @@ -104,9 +113,11 @@ \item if $x$ is in $X$ and $y$ is a subtuple of $x$, then $y$ is in $X$. \end{enumerate} - With appropriate modifications the same terminology and notation introduced for \hyperref[ordered simplicial complex]{ordered simplicial complex} applies to directed simplicial complex. + With appropriate modifications the same terminology and notation introduced for + %\hyperref[ordered_simplical_complex]{ordered simplicial complex} + ordered simplicial complex applies to directed simplicial complex. - \subsection*{Chain complex} \label{chain complex} + \subsection*{Chain complex} \label{chain_complex} A \textit{chain complex} of is a pair $(C_*, \partial)$ where \begin{equation*} @@ -114,52 +125,66 @@ \end{equation*} with $C_n$ a $\Bbbk$-vector space and $\partial_n : C_{n+1} \to C_n$ is a $\Bbbk$-linear map such that $\partial_{n+1} \partial_n = 0$. We refer to $\partial$ as the \textit{boundary map} of the chain complex. - The elements of $C$ are called \textit{chains} and if $c \in C_n$ we say its \textit{degree} is $n$ or simply that it is an $n$-chain. Elements in the kernel of $\partial$ are called \textit{cycles}, and elements in the image of $\partial$ are called \textit{boundaries}. Notice that every boundary is a cycle. This fact is central to the definition of \hyperref[homology]{homology}. + The elements of $C$ are called \textit{chains} and if $c \in C_n$ we say its \textit{degree} is $n$ or simply that it is an $n$-chain. Elements in the kernel of $\partial$ are called \textit{cycles}, and elements in the image of $\partial$ are called \textit{boundaries}. Notice that every boundary is a cycle. This fact is central to the definition of + % \hyperref[homology_and_cohomology]{homology} + homology. A \textit{chain map} is a $\Bbbk$-linear map $f : C \to C'$ between chain complexes such that $f(C_n) \subseteq C'_n$ and $\partial f = f \partial$. Given a chain complex $(C_*, \partial)$, its linear dual $C^*$ is also a chain complex with $C^{-n} = \mathrm{Hom_\Bbbk}(C_n, \Bbbk)$ and boundary map $\delta$ defined by $\delta(\alpha)(c) = \alpha(\partial c)$ for any $\alpha \in C^*$ and $c \in C_*$. - \subsection*{Homology and cohomology} \label{homology and cohomology} + \subsection*{Homology and cohomology} \label{homology_and_cohomology} - Let $(C_*, \partial)$ be a \hyperref[chain complex]{chain complex}. Its \textit{$n$-th homology group} is the quotient of the subspace of $n$-cycles by the subspace of $n$-boundaries, that is, $H_n(C_*) = \mathrm{ker}(\partial_n)/ \mathrm{im}(\partial_{n+1})$. The \textit{homology} of $(C, \partial)$ is defined by $H_*(C) = \bigoplus_{n \in \mathbb Z} H_n(C)$. + Let $(C_*, \partial)$ be a + % \hyperref[chain_complex]{chain complex} + chain complex. Its $n$\textit{-th homology group} is the quotient of the subspace of $n$-cycles by the subspace of $n$-boundaries, that is, $H_n(C_*) = \mathrm{ker}(\partial_n)/ \mathrm{im}(\partial_{n+1})$. The \textit{homology} of $(C, \partial)$ is defined by $H_*(C) = \bigoplus_{n \in \mathbb Z} H_n(C)$. When the chain complex under consideration is the linear dual of a chain complex we sometimes refer to its homology as the \textit{cohomology} of the predual complex and write $H^n$ for $H_{-n}$. A chain map $f : C \to C'$ induces a map between the associated homologies. - \subsection*{Simplicial chains and simplicial homology} \label{simplicial chains and simplicial homology} + \subsection*{Simplicial chains and simplicial homology} \label{simplicial_chains_and_simplicial_homology} - Let $X$ be an ordered or directed simplicial complex. Define its \textit{simplicial chain complex with $\Bbbk$-coefficients} $C_*(X; \Bbbk)$ by + Let $X$ be an ordered or directed simplicial complex. Define its \textit{simplicial chain complex with} $\Bbbk$\textit{-coefficients} $C_*(X; \Bbbk)$ by \begin{equation*} - C_n(X; \Bbbk) = \Bbbk\{X_n\} \qquad \partial_n(x) = \sum_{i=0}^{n} (-1)^i d_ix + C_n(X; \Bbbk) = \Bbbk\{X_n\}, \qquad \partial_n(x) = \sum_{i=0}^{n} (-1)^i d_ix \end{equation*} - and its \textit{homology and cohomology with $\Bbbk$-coefficients} as the \hyperref[homology and cohomology]{homology and cohomology} of this chain complex. We use the notation $H_*(X; \Bbbk)$ and $H^*(X; \Bbbk)$ for these. + and its \textit{homology and cohomology with} $\Bbbk$\textit{-coefficients} as the + % \hyperref[homology_and_cohomology]{homology and cohomology} + homology and cohomology of this chain complex. We use the notation $H_*(X; \Bbbk)$ and $H^*(X; \Bbbk)$ for these. - A \hyperref[abstract simplicial complex]{simplicial map} induces a \hyperref[chain complex]{chain map} between the associated simplicial chain complexes and, therefore, between the associated simplicial (co)homologies. + A + % \hyperref[abstract_simplicial_complex]{simplicial map} + simplicial map induces a + % \hyperref[chain_complex]{chain map} + chain map between the associated simplicial chain complexes and, therefore, between the associated simplicial (co)homologies. - \subsection*{Cubical chains and cubical homology} \label{cubical chains and cubical homology} + \subsection*{Cubical chains and cubical homology} \label{cubical_chains_and_cubical_homology} - Let $X$ be a cubical complex. Define its \textit{cubical chain complex with $\Bbbk$-coefficients} $C_*(X; \Bbbk)$ by + Let $X$ be a cubical complex. Define its \textit{cubical chain complex with} $\Bbbk$\textit{-coefficients} $C_*(X; \Bbbk)$ by \begin{equation*} - C_n(X; \Bbbk) = \Bbbk\{X_n\} \qquad \partial_n x = \sum_{i = 1}^{n} (-1)^{i-1}(d^+_i x - d^-_i x) + C_n(X; \Bbbk) = \Bbbk\{X_n\}, \qquad \partial_n x = \sum_{i = 1}^{n} (-1)^{i-1}(d^+_i x - d^-_i x) \end{equation*} where $x = I_1 \times \cdots \times I_N$ and $s(i)$ is the dimension of $I_1 \times \cdots \times I_i$. - Its \textit{homology and cohomology with $\Bbbk$-coefficients} is the \hyperref[homology and cohomology]{homology and cohomology} of this chain complex. We use the notation $H_*(X; \Bbbk)$ and $H^*(X; \Bbbk)$ for these. + Its \textit{homology and cohomology with} $\Bbbk$\textit{-coefficients} is the + % \hyperref[homology_and_cohomology]{homology and cohomology} + homology and cohomology of this chain complex. We use the notation $H_*(X; \Bbbk)$ and $H^*(X; \Bbbk)$ for these. - \subsection*{Filtered complex} \label{filtered complex} + \subsection*{Filtered complex} \label{filtered_complex} A \textit{filtered complex} is a collection of simplicial or cubical complexes $\{X_s\}_{s \in \mathbb R}$ such that $X_s$ is a subcomplex of $X_t$ for each $s \leq t$. - \subsection*{Cellwise filtration} \label{cellwise filtration} + \subsection*{Cellwise filtration} \label{cellwise_filtration} - A \textit{cellwise filtration} is a simplicial or cubical complex $X$ together with a total order $\leq$ on its simplices or elementary cubes such that for each $y \in X$ the set $\{x \in X\ :\ x \leq y\}$ is a subcomplex of $X$. A cellwise filtration can be naturally thought of as a \hyperref[filtered complex]{filtered complex}. + A \textit{cellwise filtration} is a simplicial or cubical complex $X$ together with a total order $\leq$ on its simplices or elementary cubes such that for each $y \in X$ the set $\{x \in X\ :\ x \leq y\}$ is a subcomplex of $X$. A cellwise filtration can be naturally thought of as a + % \hyperref[filtered_complex]{filtered complex} + filtered complex. - \subsection*{Clique and flag complexes} \label{clique and flag complexes} + \subsection*{Clique and flag complexes} \label{clique_and_flag_complexes} Let $G$ be a $1$-dimensional abstract (resp. directed) simplicial complex. The abstract (resp. directed) simplicial complex $\langle G \rangle$ has the same set of vertices as $G$ and $\{v_0, \dots, v_n\}$ \big(resp. $(v_0, \dots, v_n)$\big) is a simplex in $\langle G \rangle$ if an only if $\{v_i, v_j\}$ \big(resp. $(v_i, v_j)$\big) is in $G$ for each pair of vertices $v_i, v_j$. - An abstract (resp. directed) simplicial complex $X$ is a \textit{clique (resp. flag) complex} if $X = \langle G \rangle$ for some $G$. + An abstract (resp. directed) simplicial complex $X$ is a \textit{clique (resp.\ flag) complex} if $X = \langle G \rangle$ for some $G$. Given a function \begin{equation*} @@ -174,53 +199,72 @@ 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{align*} - and define the \hyperref[filtered complex]{filtered complex} $\{\langle G \rangle_{s}\}_{s \in \mathbb R}$ by + and define the % \hyperref[filtered_complex]{filtered complex} + filtered complex $\{\langle G \rangle_{s}\}_{s \in \mathbb R}$ by \begin{equation*} \langle G \rangle_s = \{\sigma \in \langle G \rangle\ |\ w(\sigma) \leq s\}. \end{equation*} - A filtered complex $\{X_s\}_{s \in \mathbb R}$ is a \textit{filtered clique (resp. flag) complex} if $X_s = \langle G \rangle_s$ for some $(G,w)$. + A filtered complex $\{X_s\}_{s \in \mathbb R}$ is a \textit{filtered clique (resp.\ flag) complex} if $X_s = \langle G \rangle_s$ for some $(G,w)$. - \subsection*{Persistence module} \label{persistence module} + \subsection*{Persistence module} \label{persistence_module} A \textit{persistence module} is a collection containing a $\Bbbk$-vector spaces $V(s)$ for each real number $s$ together with $\Bbbk$-linear maps $f_{st} : V(s) \to V(t)$, referred to as \textit{structure maps}, for each pair $s \leq t$, satisfying naturality, i.e., if $r \leq s \leq t$, then $f_{rt} = f_{st} \circ f_{rs}$ and tameness, i.e., all but finitely many structure maps are isomorphisms. A \textit{morphism of persistence modules} $F : V \to W$ is a collection of linear maps $F(s) : V(s) \to W(s)$ such that $F(t) \circ f_{st} = f_{st} \circ F(s)$ for each par of reals $s \leq t$. We say that $F$ is an \textit{isomorphisms} if each $F(s)$ is. - \subsection*{Persistent simplicial (co)homology} \label{persistent simplicial (co)homology} + \subsection*{Persistent simplicial (co)homology} \label{persistent_simplicial_(co)homology} Let $\{X(s)\}_{s \in \mathbb R} $ be a set of ordered or directed simplicial complexes together with simplicial maps $f_{st} : X(s) \to X(t)$ for each pair $s \leq t$, such that \begin{equation*} r \leq s \leq t\ \quad\text{implies} \quad f_{rt} = f_{st} \circ f_{rs} \end{equation*} - for example, a \hyperref[filtered complex]{filtered complex}. Its \textit{persistent simplicial homology with $\Bbbk$-coefficients} is the persistence module + for example, a + % \hyperref[filtered_complex]{filtered complex} + filtered complex. Its \textit{persistent simplicial homology with} $\Bbbk$\textit{-coefficients} is the persistence module \begin{equation*} H_*(X(s); \Bbbk) \end{equation*} - with structure maps $H_*(f_{st}) : H_*(X(s); \Bbbk) \to H_*(X(t); \Bbbk)$ induced form the maps $f_{st.}$ In general, the collection constructed this way needs not satisfy the tameness condition of a \hyperref[persistence module]{persistence module}, but we restrict attention to the cases where it does. Its \textit{persistence simplicial cohomology with $\Bbbk$-coefficients} is defined analogously. + with structure maps $H_*(f_{st}) : H_*(X(s); \Bbbk) \to H_*(X(t); \Bbbk)$ induced form the maps $f_{st.}$ In general, the collection constructed this way needs not satisfy the tameness condition of a + % \hyperref[persistence_module]{persistence module} + persistence module, but we restrict attention to the cases where it does. Its \textit{persistence simplicial cohomology with} $\Bbbk$\textit{-coefficients} is defined analogously. - \subsection*{Vietoris-Rips complex and Vietoris-Rips persistence} \label{vietoris-rips complex and vietoris-rips persistence} + \subsection*{Vietoris-Rips complex and Vietoris-Rips persistence} \label{vietoris-rips_complex_and_vietoris-rips_persistence} - Let $(X, d)$ be a \hyperref[finite metric spaces and point clouds]{finite metric space}. Define the Vietoris-Rips complex of $X$ as the \hyperref[filtered complex]{filtered complex} $VR_s(X)$ that contains a subset of $X$ as a simplex if all pairwise distances in the subset are less than or equal to $s$, explicitly + Let $(X, d)$ be a + % \hyperref[finite_metric_spaces_and_point_clouds]{finite metric space} + finite metric space. Define the Vietoris-Rips complex of $X$ as the + % \hyperref[filtered_complex]{filtered complex} + filtered complex $VR_s(X)$ that contains a subset of $X$ as a simplex if all pairwise distances in the subset are less than or equal to $s$, explicitly \begin{equation*} VR_s(X) = \Big\{ [v_0,\dots,v_n]\ \Big|\ \forall i,j\ \,d(v_i, v_j) \leq s \Big\}. \end{equation*} - The \textit{Vietoris-Rips persistence} of $(X, d)$ is the \hyperref[persistent simplicial (co)homology]{persistent simplicial (co)homology} of $VR_s(X)$. + The \textit{Vietoris-Rips persistence} of $(X, d)$ is the + % \hyperref[persistent_simplicial_(co)homology]{persistent simplicial (co)homology} + persistent simplicial (co)homology of $VR_s(X)$. A more general version is obtained by replacing the distance function with an arbitrary function \begin{equation*} w : X \times X \to \mathbb R \cup \{\infty\} \end{equation*} - and defining $VR_s(X)$ as the \hyperref[clique and flag complexes]{filtered clique complex} associated to $(X \times X ,w)$. + and defining $VR_s(X)$ as the + % \hyperref[clique_and_flag_complexes]{filtered clique complex} + filtered clique complex associated to $(X \times X ,w)$. - \subsection*{\v{C}ech complex and \v{C}ech persistence} \label{cech complex and cech persistence} + \subsection*{\v{C}ech complex and \v{C}ech persistence} \label{cech_complex_and_cech_persistence} - Let $(X, d)$ be a \hyperref[finite metric spaces and point clouds]{point cloud}. Define the \v{C}ech complex of $X$ as the \hyperref[filtered complex]{filtered complex} $\check{C}_s(X)$ that is empty if $s<0$ and, if $s \geq 0$, contains a subset of $X$ as a simplex if the balls of radius $s$ with centers in the subset have a non-empty intersection, explicitly + Let $(X, d)$ be a + % \hyperref[finite_metric_spaces_and_point_clouds]{point cloud} + point cloud. Define the \v{C}ech complex of $X$ as the + % \hyperref[filtered_complex]{filtered complex} + filtered complex $\check{C}_s(X)$ that is empty if $s<0$ and, if $s \geq 0$, contains a subset of $X$ as a simplex if the balls of radius $s$ with centers in the subset have a non-empty intersection, explicitly \begin{equation*} \check{C}_s(X) = \Big\{ [v_0,\dots,v_n]\ \Big|\ \bigcap_{i=0}^n B_s(x_i) \neq \emptyset \Big\}. \end{equation*} - The \textit{\v Cech persistence (co)homology} of $(X,d)$ is the \hyperref[persistent simplicial (co)homology]{persistent simplicial (co)homo-logy} of $\check{C}_s(X)$. + The \textit{\v Cech persistence (co)homology} of $(X,d)$ is the + % \hyperref[persistent_simplicial_(co)homology]{persistent simplicial (co)homology} + persistent simplicial (co)homology of $\check{C}_s(X)$. \subsection*{Multiset} \label{multiset} @@ -234,19 +278,23 @@ \end{cases} \end{equation*} - \subsection*{Persistence diagram} \label{persistence diagram} + \subsection*{Persistence diagram} \label{persistence_diagram} - A \textit{persistence diagram} is a \hyperref[multiset]{multiset} of points in + A \textit{persistence diagram} is a + %\hyperref[multiset]{multiset} + multiset of points in \begin{equation*} \mathbb R \times \big( \mathbb{R} \cup \{+\infty\} \big). \end{equation*} - Given a \hyperref[persistence module]{persistence module} its associated persistence diagram is determined by the following condition: for each pair $s,t$ the number counted with multiplicity of points $(b,d)$ in the multiset, satisfying $b \leq s \leq t < d$ is equal to the rank of $f_{st.}$ + Given a + % \hyperref[persistence_module]{persistence module} + persistence module, its associated persistence diagram is determined by the following condition: for each pair $s,t$ the number counted with multiplicity of points $(b,d)$ in the multiset, satisfying $b \leq s \leq t < d$ is equal to the rank of $f_{st.}$ A well known result establishes that there exists an isomorphism between two persistence module if and only if their persistence diagrams are equal. - \subsection*{Wasserstein and bottleneck distance} \label{wasserstein and bottleneck distance} + \subsection*{Wasserstein and bottleneck distance} \label{wasserstein_and_bottleneck_distance} - The \textit{$p$-Wasserstein distance} between two persistence diagrams $D_1$ and $D_2$ is the infimum over all bijections $\gamma: D_1 \cup \Delta \to D_2 \cup \Delta$ of + The $p$\textit{-Wasserstein distance} between two persistence diagrams $D_1$ and $D_2$ is the infimum over all bijections $\gamma: D_1 \cup \Delta \to D_2 \cup \Delta$ of \begin{equation*} \Big(\sum_{x \in D_1 \cup \Delta} ||x - \gamma(x)||_\infty^p \Big)^{1/p} \end{equation*} @@ -257,64 +305,69 @@ \sup_{x \in D_1 \cup \Delta} ||x - \gamma(x)||_{\infty.} \end{equation*} - \paragraph{\\ Reference:} \cite{kerber2017geometry} - - \subsection*{Persistence landscape} \label{persistence landscape} - - A \textit{persistence landscape} isa set $\{\lambda_k\}_{k \in \mathbb N}$ of functions - \begin{equation*} - \lambda : \mathbb R \to \overline{\mathbb R} - \end{equation*} - where $\lambda_k$ is referred to as the \textit{$k$-layer of the persistence landscape}. - - Let $\{(b_i, d_i)\}_{i \in I}$ be a \hyperref[persistence diagram] {persistence diagram}. Its \textit{associated persistence landscape} $\lambda$ is defined by letting $\lambda_k$ be the $k$-th largest value of the set $\{\Lambda_i(t)\}_ {i \in I}$ where - \begin{equation*} - \Lambda_i(t) = \left[ \min \{t-b_i, d_i-t\}\right]_+ - \end{equation*} - and $c_+ := \max(c,0)$. - - Intuitively, we can describe the set of graphs of a persistence landscape by first joining each of the points in the multiset to the diagonal via a horizontal as well as a vertical line, then clockwise rotating the figure 45 degrees and rescaling it by $1/\sqrt{2}$. - - \paragraph{\\ Reference:} \cite{bubenik2015statistical} + The set of persistence diagrams together with any of the distances above is a + %\hyperref[metric space]{metric space}. + metric space. - \subsection*{Persistence landscape norm} \label{persistence landscape norm} + \paragraph{\\ Reference:} \cite{kerber2017geometry} - Given a function $f : \mathbb R \to \overline{\mathbb R}$ define - \begin{equation*} - ||f||_p = \left( \int_{\mathbb R} f^p(x)\, dx \right)^{1/p} - \end{equation*} - whenever the right hand side exists and is finite. + \subsection*{Persistence landscape} \label{persistence_landscape} - The \textit{$p$-norm} of a \hyperref[persistence landscape]{persistence landscape} $\lambda = \{\lambda_k\}_{k \in \mathbb N}$ is defined to be + Let $\{(b_i, d_i)\}_{i \in I}$ be a + %\hyperref[persistence diagram] {persistence diagram} + persistence diagram. Its \textit{persistence landscape} is the set $\{\lambda_k\}_{k \in \mathbb N}$ of functions + \begin{equation*} + \lambda_k : \mathbb R \to \overline{\mathbb R} + \end{equation*} + defined by letting $\lambda_k$ be the $k$-th largest value of the set $\{\Lambda_i(t)\}_ {i \in I}$ where + \begin{equation*} + \Lambda_i(t) = \left[ \min \{t-b_i, d_i-t\}\right]_+ + \end{equation*} + and $c_+ := \max(c,0)$. The function $\lambda_k$ is referred to as the \textit{$k$-layer of the persistence landscape}. + + Intuitively, we can describe the set of graphs of a persistence landscape by first joining each of the points in the multiset to the diagonal via a horizontal as well as a vertical line, then clockwise rotating the figure 45 degrees and rescaling it by $1/\sqrt{2}$. + + The persistence landscape construction defines a + %\hyperref[vectorization, kernel and amplitude]{vectorization} + vectorization of the set of persistence diagrams with target the vector space of real-valued function on $\mathbb N \times \mathbb R$. For any $p = 1,\dots,\infty$ we can restrict attention to persistence diagrams $D$ whose associated persistence landscape $\lambda$ is + %\hyperref[l^p norm]{$p$-integrable} + $p$-integrable, that is to say, + \begin{equation} \label{equation: persistence landscape norm} + ||\lambda||_p = \left( \sum_{i \in \mathbb N} ||\lambda_i||^p_p \right)^{1/p} + \end{equation} + where + \begin{equation*} + ||\lambda_i||_p = \left( \int_{\mathbb R} \lambda_i^p(x)\, dx \right)^{1/p} + \end{equation*} + is finite. In this case we refer to (\ref{equation: persistence landscape norm}) as the + %\hyperref[vectorization, kernel and amplitude]{amplitude} + \textit{landscape} $p$-\textit{amplitude} of $D$. - \begin{equation*} - ||\lambda||_p = \left( \sum_{i \in \mathbb N} ||\lambda_i||^p_p \right)^{1/p} - \end{equation*} - whenever the right hand side exists and is finite. - - \paragraph{\\ References:} \cite{stein2011functional, bubenik2015statistical} + \paragraph{\\ References:} \cite{bubenik2015statistical} - \subsection*{Weighted silhouette} \label{weighted silhouettes} + \subsection*{Weighted silhouette} \label{weighted_silhouette} - Let $D = {(b_i, d_i)}_{i \in I}$ be a \hyperref[persistence diagram] {persistence diagram}. A \textit{weighted silhouette} associated to $D$ is a continuous function $\phi : \mathbb R \to \mathbb R$ of the form - \begin{equation*} - \phi(t) = \frac{\sum_{i \in I}w_i \Lambda_i(t)}{\sum_{i \in I}w_i}, - \end{equation*} - where $\{w_i\}_{i \in I}$ is a set of positive real numbers and - \begin{equation*} \label{equation: lambda for persistence landscapes} - \Lambda_i(t) = \left[ \min \{t-b_i, d_i-t\}\right]_+ - \end{equation*} - with $c_+ := \max(c,0)$. The particular choice $w_i = \vert d_i - b_i \vert^p$ for $0 < p \leq \infty$ is referred to as \textit{power-weighted silhouettes}. + Let $D = \{(b_i, d_i)\}_{i \in I}$ be a + %\hyperref[persistence diagram] {persistence diagram} + persistence diagram and $w = \{w_i\}_{i \in I}$ a set of positive real numbers. The \textit{silhouette of $D$ weighted by $w$} is the function $\phi : \mathbb R \to \mathbb R$ defined by + \begin{equation*} + \phi(t) = \frac{\sum_{i \in I}w_i \Lambda_i(t)}{\sum_{i \in I}w_i}, + \end{equation*} + where + \begin{equation*} \label{equation: lambda for persistence landscapes} + \Lambda_i(t) = \left[ \min \{t-b_i, d_i-t\}\right]_+ + \end{equation*} + and $c_+ := \max(c,0)$. When $w_i = \vert d_i - b_i \vert^p$ for $0 < p \leq \infty$ we refer to $\phi$ as the \textit{$p$-power-weighted silhouette} of $D$. The silhouette construction defines a + %\hyperref[vectorization_kernel_and_amplitude]{vectorization} + vectorization of the set of persistence diagrams with target the vector space of continuous real-valued functions on $\mathbb R$. \paragraph{\\ References:} \cite{chazal2014stochastic} - \subsection*{Amplitude} \label{amplitude} + \subsection*{Persistence entropy} \label{persistence_entropy} - Given a function assigning a real number to a pair of persistence diagrams, we define the \textit{amplitude} of a persistence diagram $D$ to be the value assigned to the pair $(D \cup \Delta, \Delta)$. Important examples of such functions are: \hyperref[wasserstein and bottleneck distance]{Wasserstein and bottleneck distances} and \hyperref[persistence landscape norm]{landscape distance}. - - \subsection*{Persistence entropy} \label{persistence entropy} - - Intuitively, this is a measure of the entropy of the points in a \hyperref[persistence diagram]{persistence diagram}. Precisely, let $D = \{(b_i, d_i)\}_{i \in I}$ be a persistence diagram with each $d_i < +\infty$. The \textit{persistence entropy} of $D$ is defined by + Intuitively, this is a measure of the entropy of the points in a + % \hyperref[persistence_diagram]{persistence diagram} + persistence diagram. Precisely, let $D = \{(b_i, d_i)\}_{i \in I}$ be a persistence diagram with each $d_i < +\infty$. The \textit{persistence entropy} of $D$ is defined by \begin{equation*} E(D) = - \sum_{i \in I} p_i \log(p_i) \end{equation*} @@ -325,79 +378,132 @@ \paragraph{\\ References:} \cite{rucco2016characterisation} - \subsection*{Betti curve} \label{betti curve} + \subsection*{Betti curve} \label{betti_curve} - Let $D$ be a \hyperref[persistence diagram]{persistence diagram}. Its \textit{Betti curve} is the function $\beta_D : \mathbb R \to \mathbb N$ whose value on $s \in \mathbb R$ is the number, counted with multiplicity, of points $(b_i,d_i)$ in $D$ such that $b_i \leq s 0 - \end{equation*} - \begin{equation*} - \langle u, v\rangle = \langle v, u\rangle - \end{equation*} - \begin{equation*} - \langle au+v, w \rangle = a\langle u, w \rangle + \langle v, w \rangle. + \begin{equation*} + \langle -, - \rangle : V \times V \to \mathbb R \end{equation*} - In this case the function $\langle -, - \rangle$ is referred to as the \textit{inner product} and the function given by - \begin{equation*} - ||u|| = \sqrt{\langle u, u \rangle} + is said to be an \textit{inner product space} if for all $u,v,w \in V$ + \begin{equation*} + u \neq 0\ \Rightarrow\ \langle u, u \rangle > 0 + \end{equation*} + \begin{equation*} + \langle u, v\rangle = \langle v, u\rangle \end{equation*} - as its associated \textit{norm}. An inner product space is naturally a metric space with distance function \begin{equation*} - d(u,v) = ||u-v||. - \end{equation*} + \langle au+v, w \rangle = a\langle u, w \rangle + \langle v, w \rangle. + \end{equation*} + The function $\langle -, - \rangle$ is referred to as the \textit{inner product}. - A \textit{kernel} on a set $X$ is a function - \begin{equation*} - k : X \times X - \end{equation*} - for which there exists a function $\phi : X \to V$ to an inner product space such that - \begin{equation*} - k(x, y) = \langle \phi(x), \phi(y) \rangle. + A vector space $V$ together with a function + \begin{equation*} + ||-|| : V \to \mathbb R + \end{equation*} + is said to be an \textit{normed space} if the values of $||-||$ are all non-negative and for all $u,v \in V$ and $a \in \mathbb R$ + \begin{equation*} + ||v|| = 0\ \Leftrightarrow\ u = 0 + \end{equation*} + \begin{equation*} + ||a u || = |a|\, ||u|| + \end{equation*} + \begin{equation*} + ||u+v|| = ||u|| + ||v||. + \end{equation*} + The function $||-||$ is referred to as the \textit{norm}. + + An inner product space is naturally a norm space with + \begin{equation*} + ||u|| = \sqrt{\langle u, u \rangle} + \end{equation*} + and a norm space is naturally a + %\hyperref[metric_space]{metric space} + metric space with distance function + \begin{equation*} + d(u,v) = ||u-v||. \end{equation*} - \subsection*{Euclidean distance and norm} \label{euclidean distance and norm} + \subsection*{Euclidean distance and norm} \label{euclidean_distance_and_norm} - The vector space $\mathbb R^n$ is an \hyperref[metric, inner product and kernel]{inner product space} with inner product + The vector space $\mathbb R^n$ is an + % \hyperref[metric_inner_product_and_kernel]{inner product space} + inner product space with inner product \begin{equation*} \langle x, y \rangle = (x_1-y_1)^2 + \cdots + (x_n-y_n)^2. \end{equation*} - The associated norm and distance function are referred to as \textit{euclidean norm} and \textit{euclidean distance}. + This inner product is referred to as \textit{dot product} and the associated norm and distance function are respectively named \textit{euclidean norm} and \textit{euclidean distance}. + + \subsection*{$L^p$-norm:} \label{l^p_norm} - \subsection*{Finite metric spaces and point clouds} \label{finite metric spaces and point clouds} + ... - A \textit{finite metric space} is a finite set together with a \hyperref[metric space]{metric}. A \textit{distance matrix} associated to a finite metric space is obtained by choosing a total order on the finite set and setting the $(i,j)$-entry to be equal to the distance between the $i$-th and $j$-th elements. + \subsection*{Vectorization, kernel and amplitude} \label{vectorization_kernel_and_amplitude} + Let $X$ be a set, for example, the set of all + %\hyperref[persistence_diagram]{persistence diagrams} + persistence diagrams. A \textit{vectorization} for $X$ is a function + \begin{equation*} + \phi : X \to V + \end{equation*} + where $V$ is a vector space. A \textit{kernel} on the set $X$ is a function + \begin{equation*} + k : X \times X \to \mathbb R + \end{equation*} + for which there exists a vectorization $\phi : X \to V$ with $V$ an + %\hyperref[inner_product_and_norm]{inner product space} + inner product space such that + \begin{equation*} + k(x,y) = \langle \phi(x), \phi(y) \rangle + \end{equation*} + for each $x,y \in X$. Similarly, an \textit{amplitude} on $X$ is a function + \begin{equation*} + A : X \to \mathbb R + \end{equation*} + for which there exists a vectorization $\phi : X \to V$ with $V$ a + %\hyperref[inner_product_and_norm]{normed space} + normed space such that + \begin{equation*} + A(x) = ||\phi(x)|| + \end{equation*} + for all $x \in X$. - A \textit{point cloud} is a finite subset of $\mathbb{R}^n$ (for some $n$) together with the metric induced from the \hyperref[euclidean distance and norm]{eucliden distance}. + \subsection*{Finite metric spaces and point clouds} \label{finite_metric_spaces_and_point_clouds} + + A \textit{finite metric space} is a finite set together with a + % \hyperref[metric_inner_product_and_kernel]{metric} + metric. A \textit{distance matrix} associated to a finite metric space is obtained by choosing a total order on the finite set and setting the $(i,j)$-entry to be equal to the distance between the $i$-th and $j$-th elements. + + A \textit{point cloud} is a finite subset of $\mathbb{R}^n$ (for some $n$) together with the metric induced from the + % \hyperref[euclidean_distance_and_norm]{eucliden distance} + euclidean distance. \section{Time series} - \subsection*{Time series} \label{time series} + \subsection*{Time series} \label{time_series} A \textit{time series} is a sequence $\{y_i\}_{i = 0}^n$ of real numbers. @@ -417,13 +523,15 @@ \end{equation*} using a function $f : M \to \mathbb R$ as follows: let $\{t_i\}_{i=0}^n$ be a time series taking values in $U$, then \begin{equation*} - \{f(\varphi(t_i, m))\}_{i=0}^n. + \{f(\varphi(t_i, m))\}_{i=0}^n \end{equation*} for an arbitrarily chosen $m \in M$. - \subsection*{Takens embedding} \label{takens embedding} + \subsection*{Takens embedding} \label{takens_embedding} - Let $M \subset \mathbb R^d$ be a \hyperref[manifold]{compact manifold} of dimension $n$. Let + Let $M \subset \mathbb R^d$ be a + %\hyperref[manifold]{compact manifold} + compact manifold of dimension $n$. Let \begin{equation*} \varphi : \mathbb R \times M \to M \end{equation*} @@ -456,12 +564,13 @@ \paragraph{\\ References:} \cite{milnor1997topology,guillemin2010differential} - \subsection*{Compact subset} \label{compact subset} - + \subsection*{Compact subset} \label{compact_subset} + A subset $K$ of a metric space $(X,d)$ is said to be \textit{bounded} if there exist a real number $D$ such that for each pair of elements in $K$ the distance between them is less than $D$. It is said to be \textit{complete} if for any $x \in X$ it is the case that $x \in K$ if for any $\epsilon > 0$ the intersection between $K$ and $\{y \,;\ d(x,y) < \epsilon \}$ is not empty. It is said to be \textit{compact} if it is both bounded and complete. - + \section{Bibliography} \bibliography{bibliography}{} \bibliographystyle{alpha} - + + \end{document} From daf033a05cf1582ff232bbe37d31923803b69129 Mon Sep 17 00:00:00 2001 From: ammedmar Date: Mon, 23 Mar 2020 18:15:00 +0100 Subject: [PATCH 04/11] Change [] for \lbrack \rbrack Signed-off-by: ammedmar --- doc/theory/glossary.tex | 12 ++++++------ 1 file changed, 6 insertions(+), 6 deletions(-) diff --git a/doc/theory/glossary.tex b/doc/theory/glossary.tex index d9d00b2b5..caf10fc47 100644 --- a/doc/theory/glossary.tex +++ b/doc/theory/glossary.tex @@ -39,7 +39,7 @@ An \textit{elementary interval} $I_a$ is a subset of $\mathbb{R}$ of the form $[a, a+1]$ or $[a,a] = \{a\}$ for some $a \in \mathbb{R}$. These two types are called respectively \textit{non-degenerate} and \textit{degenerate}. To a non-degenerate elementary interval we assign two degenerate elementary intervals \begin{equation*} - d^+ I_a = [a+1, a+1] \qquad \text{and} \qquad d^- I_a = [a, a]. + d^+ I_a = \lbrack a+1, a+1 \rbrack \qquad \text{and} \qquad d^- I_a = \lbrack a, a \rbrack. \end{equation*} An \textit{elementary cube} is a subset of the form \begin{equation*} @@ -62,9 +62,9 @@ A set $\{v_0, \dots, v_n\} \subset \mathbb{R}^N$ is said to be \textit{geometrically independent} if the vectors $\{v_0-v_1, \dots, v_0-v_n\}$ are linearly independent. In this case, we refer to their convex closure as a \textit{simplex}, explicitly \begin{equation*} - [v_0, \dots , v_n] = \left\{ \sum c_i (v_0 - v_i)\ \big|\ c_1+\dots+c_n = 1,\ c_i \geq 0 \right\} + \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\} \end{equation*} - and to $n$ as its \textit{dimension}. The $i$\textit{-th face} of $[v_0, \dots, v_n]$ is defined by + and to $n$ as its \textit{dimension}. The $i$\textit{-th face} of $\lbrack v_0, \dots, v_n \rbrack$ is defined by \begin{equation*} d_i[v_0, \ldots, v_n] = [v_0, \dots, \widehat{v}_i, \dots, v_n] \end{equation*} @@ -101,7 +101,7 @@ An \textit{ordered simplicial complex} is an % \hyperref[abstract_simplicial_complex]{abstract simplicial complex} - 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 $n$-simplex using its ordered vertices by $[v_0, \dots, v_n]$. + 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 $n$-simplex using its ordered vertices by $\lbrack v_0, \dots, v_n \rbrack$. A \textit{simplicial map} between ordered simplicial complexes is a simplicial map $f$ between their underlying simplicial complexes preserving the order, i.e., $v \leq w$ implies $f(v) \leq f(w)$. @@ -237,7 +237,7 @@ % \hyperref[filtered_complex]{filtered complex} filtered complex $VR_s(X)$ that contains a subset of $X$ as a simplex if all pairwise distances in the subset are less than or equal to $s$, explicitly \begin{equation*} - VR_s(X) = \Big\{ [v_0,\dots,v_n]\ \Big|\ \forall i,j\ \,d(v_i, v_j) \leq s \Big\}. + VR_s(X) = \Big\{ \lbrack v_0,\dots,v_n \rbrack \ \Big|\ \forall i,j\ \,d(v_i, v_j) \leq s \Big\}. \end{equation*} The \textit{Vietoris-Rips persistence} of $(X, d)$ is the % \hyperref[persistent_simplicial_(co)homology]{persistent simplicial (co)homology} @@ -260,7 +260,7 @@ % \hyperref[filtered_complex]{filtered complex} filtered complex $\check{C}_s(X)$ that is empty if $s<0$ and, if $s \geq 0$, contains a subset of $X$ as a simplex if the balls of radius $s$ with centers in the subset have a non-empty intersection, explicitly \begin{equation*} - \check{C}_s(X) = \Big\{ [v_0,\dots,v_n]\ \Big|\ \bigcap_{i=0}^n B_s(x_i) \neq \emptyset \Big\}. + \check{C}_s(X) = \Big\{ \lbrack v_0,\dots,v_n \rbrack \ \Big|\ \bigcap_{i=0}^n B_s(x_i) \neq \emptyset \Big\}. \end{equation*} The \textit{\v Cech persistence (co)homology} of $(X,d)$ is the % \hyperref[persistent_simplicial_(co)homology]{persistent simplicial (co)homology} From 0c93a9bc2f91131d903d2ea711f9ecc36b40aadd Mon Sep 17 00:00:00 2001 From: ammedmar Date: Wed, 1 Apr 2020 22:32:21 +0200 Subject: [PATCH 05/11] Update after W's comments Signed-off-by: ammedmar --- doc/theory/glossary.tex | 18 +++++++++--------- 1 file changed, 9 insertions(+), 9 deletions(-) diff --git a/doc/theory/glossary.tex b/doc/theory/glossary.tex index caf10fc47..ed9229185 100644 --- a/doc/theory/glossary.tex +++ b/doc/theory/glossary.tex @@ -319,27 +319,27 @@ \begin{equation*} \lambda_k : \mathbb R \to \overline{\mathbb R} \end{equation*} - defined by letting $\lambda_k$ be the $k$-th largest value of the set $\{\Lambda_i(t)\}_ {i \in I}$ where + defined by letting $\lambda_k(t)$ be the $k$-th largest value of the set $\{\Lambda_i(t)\}_ {i \in I}$ where \begin{equation*} \Lambda_i(t) = \left[ \min \{t-b_i, d_i-t\}\right]_+ \end{equation*} - and $c_+ := \max(c,0)$. The function $\lambda_k$ is referred to as the \textit{$k$-layer of the persistence landscape}. + and $c_+ := \max(c,0)$. The function $\lambda_k$ is referred to as the \textit{$k$-layer of the persistence landscape}. - Intuitively, we can describe the set of graphs of a persistence landscape by first joining each of the points in the multiset to the diagonal via a horizontal as well as a vertical line, then clockwise rotating the figure 45 degrees and rescaling it by $1/\sqrt{2}$. + We describe the graph of each $\lambda_k$ intuitively. For each $i \in I$, draw an isosceles triangle with base the interval $(b_i, d_i)$ on the horizontal $t$-axis, and sides with slope 1 and $-1$. This subdivides the plane into a number of polygonal regions. Label each of these regions by the number of triangles containing it. If $P_k$ is the union of the polygonal regions with values at least $k$, then the graph of $\lambda_k$ is the upper contour of $P_k$, with $\lambda_k(a) = 0$ if the vertical line $t=a$ does not intersect $P_k$. The persistence landscape construction defines a %\hyperref[vectorization, kernel and amplitude]{vectorization} vectorization of the set of persistence diagrams with target the vector space of real-valued function on $\mathbb N \times \mathbb R$. For any $p = 1,\dots,\infty$ we can restrict attention to persistence diagrams $D$ whose associated persistence landscape $\lambda$ is %\hyperref[l^p norm]{$p$-integrable} $p$-integrable, that is to say, - \begin{equation} \label{equation: persistence landscape norm} + \begin{equation} \label{equation:persistence_landscape_norm} ||\lambda||_p = \left( \sum_{i \in \mathbb N} ||\lambda_i||^p_p \right)^{1/p} \end{equation} where \begin{equation*} ||\lambda_i||_p = \left( \int_{\mathbb R} \lambda_i^p(x)\, dx \right)^{1/p} \end{equation*} - is finite. In this case we refer to (\ref{equation: persistence landscape norm}) as the + is finite. In this case we refer to (\eqref{equation:persistence_landscape norm}) as the %\hyperref[vectorization, kernel and amplitude]{amplitude} \textit{landscape} $p$-\textit{amplitude} of $D$. @@ -355,7 +355,7 @@ \phi(t) = \frac{\sum_{i \in I}w_i \Lambda_i(t)}{\sum_{i \in I}w_i}, \end{equation*} where - \begin{equation*} \label{equation: lambda for persistence landscapes} + \begin{equation*} \Lambda_i(t) = \left[ \min \{t-b_i, d_i-t\}\right]_+ \end{equation*} and $c_+ := \max(c,0)$. When $w_i = \vert d_i - b_i \vert^p$ for $0 < p \leq \infty$ we refer to $\phi$ as the \textit{$p$-power-weighted silhouette} of $D$. The silhouette construction defines a @@ -387,12 +387,12 @@ The name is inspired from the case when the persistence diagram comes from persistent homology. - \subsection*{Metric space} \label{metric space} + \subsection*{Metric space} \label{metric_space} A set $X$ with a function \begin{equation*} d : X \times X \to \mathbb R \end{equation*} - is said to be a \textit{metric space} if the values of $d$ are all non-negative and for all $x,y,z \in X$ ans $a \in \mathbb R$ + is said to be a \textit{metric space} if the values of $d$ are all non-negative and for all $x,y,z \in X$ \begin{equation*} d(x,y) = 0\ \Leftrightarrow\ x = y \end{equation*} @@ -410,7 +410,7 @@ \begin{equation*} \langle -, - \rangle : V \times V \to \mathbb R \end{equation*} - is said to be an \textit{inner product space} if for all $u,v,w \in V$ + is said to be an \textit{inner product space} if for all $u,v,w \in V$ and $a \in \mathbb R$ \begin{equation*} u \neq 0\ \Rightarrow\ \langle u, u \rangle > 0 \end{equation*} From ba81385b45d4193ea5c503814449fdcac262f63d Mon Sep 17 00:00:00 2001 From: ammedmar Date: Wed, 1 Apr 2020 22:32:21 +0200 Subject: [PATCH 06/11] Update after W's comments Signed-off-by: ammedmar --- doc/theory/glossary.tex | 18 +++++++++--------- 1 file changed, 9 insertions(+), 9 deletions(-) diff --git a/doc/theory/glossary.tex b/doc/theory/glossary.tex index caf10fc47..ed9229185 100644 --- a/doc/theory/glossary.tex +++ b/doc/theory/glossary.tex @@ -319,27 +319,27 @@ \begin{equation*} \lambda_k : \mathbb R \to \overline{\mathbb R} \end{equation*} - defined by letting $\lambda_k$ be the $k$-th largest value of the set $\{\Lambda_i(t)\}_ {i \in I}$ where + defined by letting $\lambda_k(t)$ be the $k$-th largest value of the set $\{\Lambda_i(t)\}_ {i \in I}$ where \begin{equation*} \Lambda_i(t) = \left[ \min \{t-b_i, d_i-t\}\right]_+ \end{equation*} - and $c_+ := \max(c,0)$. The function $\lambda_k$ is referred to as the \textit{$k$-layer of the persistence landscape}. + and $c_+ := \max(c,0)$. The function $\lambda_k$ is referred to as the \textit{$k$-layer of the persistence landscape}. - Intuitively, we can describe the set of graphs of a persistence landscape by first joining each of the points in the multiset to the diagonal via a horizontal as well as a vertical line, then clockwise rotating the figure 45 degrees and rescaling it by $1/\sqrt{2}$. + We describe the graph of each $\lambda_k$ intuitively. For each $i \in I$, draw an isosceles triangle with base the interval $(b_i, d_i)$ on the horizontal $t$-axis, and sides with slope 1 and $-1$. This subdivides the plane into a number of polygonal regions. Label each of these regions by the number of triangles containing it. If $P_k$ is the union of the polygonal regions with values at least $k$, then the graph of $\lambda_k$ is the upper contour of $P_k$, with $\lambda_k(a) = 0$ if the vertical line $t=a$ does not intersect $P_k$. The persistence landscape construction defines a %\hyperref[vectorization, kernel and amplitude]{vectorization} vectorization of the set of persistence diagrams with target the vector space of real-valued function on $\mathbb N \times \mathbb R$. For any $p = 1,\dots,\infty$ we can restrict attention to persistence diagrams $D$ whose associated persistence landscape $\lambda$ is %\hyperref[l^p norm]{$p$-integrable} $p$-integrable, that is to say, - \begin{equation} \label{equation: persistence landscape norm} + \begin{equation} \label{equation:persistence_landscape_norm} ||\lambda||_p = \left( \sum_{i \in \mathbb N} ||\lambda_i||^p_p \right)^{1/p} \end{equation} where \begin{equation*} ||\lambda_i||_p = \left( \int_{\mathbb R} \lambda_i^p(x)\, dx \right)^{1/p} \end{equation*} - is finite. In this case we refer to (\ref{equation: persistence landscape norm}) as the + is finite. In this case we refer to (\eqref{equation:persistence_landscape norm}) as the %\hyperref[vectorization, kernel and amplitude]{amplitude} \textit{landscape} $p$-\textit{amplitude} of $D$. @@ -355,7 +355,7 @@ \phi(t) = \frac{\sum_{i \in I}w_i \Lambda_i(t)}{\sum_{i \in I}w_i}, \end{equation*} where - \begin{equation*} \label{equation: lambda for persistence landscapes} + \begin{equation*} \Lambda_i(t) = \left[ \min \{t-b_i, d_i-t\}\right]_+ \end{equation*} and $c_+ := \max(c,0)$. When $w_i = \vert d_i - b_i \vert^p$ for $0 < p \leq \infty$ we refer to $\phi$ as the \textit{$p$-power-weighted silhouette} of $D$. The silhouette construction defines a @@ -387,12 +387,12 @@ The name is inspired from the case when the persistence diagram comes from persistent homology. - \subsection*{Metric space} \label{metric space} + \subsection*{Metric space} \label{metric_space} A set $X$ with a function \begin{equation*} d : X \times X \to \mathbb R \end{equation*} - is said to be a \textit{metric space} if the values of $d$ are all non-negative and for all $x,y,z \in X$ ans $a \in \mathbb R$ + is said to be a \textit{metric space} if the values of $d$ are all non-negative and for all $x,y,z \in X$ \begin{equation*} d(x,y) = 0\ \Leftrightarrow\ x = y \end{equation*} @@ -410,7 +410,7 @@ \begin{equation*} \langle -, - \rangle : V \times V \to \mathbb R \end{equation*} - is said to be an \textit{inner product space} if for all $u,v,w \in V$ + is said to be an \textit{inner product space} if for all $u,v,w \in V$ and $a \in \mathbb R$ \begin{equation*} u \neq 0\ \Rightarrow\ \langle u, u \rangle > 0 \end{equation*} From 7d10f5b86640edb3369759432063b90b5ecf6875 Mon Sep 17 00:00:00 2001 From: ammedmar Date: Fri, 3 Apr 2020 12:54:32 +0200 Subject: [PATCH 07/11] Update afte W's second comments Signed-off-by: ammedmar --- doc/theory/glossary.tex | 14 +++++--------- 1 file changed, 5 insertions(+), 9 deletions(-) diff --git a/doc/theory/glossary.tex b/doc/theory/glossary.tex index 599757796..e43727870 100644 --- a/doc/theory/glossary.tex +++ b/doc/theory/glossary.tex @@ -30,7 +30,7 @@ $\mathbb R^d$ & The vector space of $d$-tuples of real numbers. \\ $\Delta$ & The %\hyperref[multiset]{multiset} - multiset $ \lbrace (s, s) \mid s \in \mathbb{R} \rbrace $ with multiplicity $ ( s,s ) \mapsto +\infty$. + multiset $ \lbrace (s, s) \mid s \in \mathbb{R} \rbrace $ with multiplicity $ ( s,s ) \mapsto +\infty$. \end{tabular} \section{Homology} @@ -325,7 +325,7 @@ \end{equation*} and $c_+ := \max(c,0)$. The function $\lambda_k$ is referred to as the \textit{$k$-layer of the persistence landscape}. - We describe the graph of each $\lambda_k$ intuitively. For each $i \in I$, draw an isosceles triangle with base the interval $(b_i, d_i)$ on the horizontal $t$-axis, and sides with slope 1 and $-1$. This subdivides the plane into a number of polygonal regions. Label each of these regions by the number of triangles containing it. If $P_k$ is the union of the polygonal regions with values at least $k$, then the graph of $\lambda_k$ is the upper contour of $P_k$, with $\lambda_k(a) = 0$ if the vertical line $t=a$ does not intersect $P_k$. + We describe the graph of each $\lambda_k$ intuitively. For each $i \in I$, draw an isosceles triangle with base the interval $(b_i, d_i)$ on the horizontal $t$-axis, and sides with slope 1 and $-1$. This subdivides the plane into a number of polygonal regions. Label each of these regions by the number of triangles containing it. If $P_k$ is the union of the polygonal regions with values at least $k$, then the graph of $\lambda_k$ is the upper contour of $P_k$, with $\lambda_k(a) = 0$ if the vertical line $t=a$ does not intersect $P_k$. The persistence landscape construction defines a %\hyperref[vectorization, kernel and amplitude]{vectorization} @@ -339,11 +339,10 @@ \begin{equation*} ||\lambda_i||_p = \left( \int_{\mathbb R} \lambda_i^p(x)\, dx \right)^{1/p} \end{equation*} - is finite. In this case we refer to (\eqref{equation:persistence_landscape_norm}) as the + is finite. In this case we refer to \eqref{equation:persistence_landscape_norm} as the %\hyperref[vectorization, kernel and amplitude]{amplitude} \textit{landscape} $p$-\textit{amplitude} of $D$. - \paragraph{\\ References:} \cite{bubenik2015statistical} \subsection*{Weighted silhouette} \label{weighted_silhouette} @@ -367,7 +366,7 @@ \subsection*{Persistence entropy} \label{persistence_entropy} Intuitively, this is a measure of the entropy of the points in a - % \hyperref[persistence_diagram]{persistence diagram} + % \hyperref[persistence_diagram]{persistence diagram} persistence diagram. Precisely, let $D = \{(b_i, d_i)\}_{i \in I}$ be a persistence diagram with each $d_i < +\infty$. The \textit{persistence entropy} of $D$ is defined by \begin{equation*} E(D) = - \sum_{i \in I} p_i \log(p_i) @@ -459,11 +458,8 @@ \end{equation*} This inner product is referred to as \textit{dot product} and the associated norm and distance function are respectively named \textit{euclidean norm} and \textit{euclidean distance}. - \subsection*{$L^p$-norm:} \label{l^p_norm} - - ... - \subsection*{Vectorization, kernel and amplitude} \label{vectorization_kernel_and_amplitude} + Let $X$ be a set, for example, the set of all %\hyperref[persistence_diagram]{persistence diagrams} persistence diagrams. A \textit{vectorization} for $X$ is a function From 441c215dd02df3a06838ab328aed4ee877a13cd8 Mon Sep 17 00:00:00 2001 From: ammedmar Date: Sun, 5 Apr 2020 21:51:58 +0200 Subject: [PATCH 08/11] Update after Umbe's comments Signed-off-by: ammedmar --- doc/theory/glossary.tex | 12 ++++++------ 1 file changed, 6 insertions(+), 6 deletions(-) diff --git a/doc/theory/glossary.tex b/doc/theory/glossary.tex index e43727870..005c1735f 100644 --- a/doc/theory/glossary.tex +++ b/doc/theory/glossary.tex @@ -232,7 +232,7 @@ \subsection*{Vietoris-Rips complex and Vietoris-Rips persistence} \label{vietoris-rips_complex_and_vietoris-rips_persistence} Let $(X, d)$ be a - % \hyperref[finite_metric_spaces_and_point_clouds]{finite metric space} + % \hyperref[finite_metric_spaces_and_point_clouds]{finite metric_space} finite metric space. Define the Vietoris-Rips complex of $X$ as the % \hyperref[filtered_complex]{filtered complex} filtered complex $VR_s(X)$ that contains a subset of $X$ as a simplex if all pairwise distances in the subset are less than or equal to $s$, explicitly @@ -314,7 +314,7 @@ \subsection*{Persistence landscape} \label{persistence_landscape} Let $\{(b_i, d_i)\}_{i \in I}$ be a - %\hyperref[persistence diagram] {persistence diagram} + %\hyperref[persistence diagram] {persistence_diagram} persistence diagram. Its \textit{persistence landscape} is the set $\{\lambda_k\}_{k \in \mathbb N}$ of functions \begin{equation*} \lambda_k : \mathbb R \to \overline{\mathbb R} @@ -328,9 +328,9 @@ We describe the graph of each $\lambda_k$ intuitively. For each $i \in I$, draw an isosceles triangle with base the interval $(b_i, d_i)$ on the horizontal $t$-axis, and sides with slope 1 and $-1$. This subdivides the plane into a number of polygonal regions. Label each of these regions by the number of triangles containing it. If $P_k$ is the union of the polygonal regions with values at least $k$, then the graph of $\lambda_k$ is the upper contour of $P_k$, with $\lambda_k(a) = 0$ if the vertical line $t=a$ does not intersect $P_k$. The persistence landscape construction defines a - %\hyperref[vectorization, kernel and amplitude]{vectorization} + %\hyperref[vectorization_kernel_and_amplitude]{vectorization} vectorization of the set of persistence diagrams with target the vector space of real-valued function on $\mathbb N \times \mathbb R$. For any $p = 1,\dots,\infty$ we can restrict attention to persistence diagrams $D$ whose associated persistence landscape $\lambda$ is - %\hyperref[l^p norm]{$p$-integrable} + %\hyperref[lp_norm]{$p$-integrable} $p$-integrable, that is to say, \begin{equation} \label{equation:persistence_landscape_norm} ||\lambda||_p = \left( \sum_{i \in \mathbb N} ||\lambda_i||^p_p \right)^{1/p} @@ -340,7 +340,7 @@ ||\lambda_i||_p = \left( \int_{\mathbb R} \lambda_i^p(x)\, dx \right)^{1/p} \end{equation*} is finite. In this case we refer to \eqref{equation:persistence_landscape_norm} as the - %\hyperref[vectorization, kernel and amplitude]{amplitude} + %\hyperref[vectorization_kernel_and_amplitude]{amplitude} \textit{landscape} $p$-\textit{amplitude} of $D$. \paragraph{\\ References:} \cite{bubenik2015statistical} @@ -348,7 +348,7 @@ \subsection*{Weighted silhouette} \label{weighted_silhouette} Let $D = \{(b_i, d_i)\}_{i \in I}$ be a - %\hyperref[persistence diagram] {persistence diagram} + %\hyperref[persistence_diagram]{persistence diagram} persistence diagram and $w = \{w_i\}_{i \in I}$ a set of positive real numbers. The \textit{silhouette of $D$ weighted by $w$} is the function $\phi : \mathbb R \to \mathbb R$ defined by \begin{equation*} \phi(t) = \frac{\sum_{i \in I}w_i \Lambda_i(t)}{\sum_{i \in I}w_i}, From 34d56ffbcc1fa40f40dd268fceb4cb8cb845ec2a Mon Sep 17 00:00:00 2001 From: ammedmar Date: Mon, 6 Apr 2020 09:14:06 +0200 Subject: [PATCH 09/11] Update after Umbe's second comments Signed-off-by: ammedmar --- doc/theory/glossary.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/doc/theory/glossary.tex b/doc/theory/glossary.tex index 005c1735f..3df3d0e08 100644 --- a/doc/theory/glossary.tex +++ b/doc/theory/glossary.tex @@ -442,7 +442,7 @@ ||u|| = \sqrt{\langle u, u \rangle} \end{equation*} and a norm space is naturally a - %\hyperref[metric_space]{metric space} + %\hyperref[metric_space]{metric_space} metric space with distance function \begin{equation*} d(u,v) = ||u-v||. From c8ec04f30357a3abf73942bc5f3f84c8a456fed9 Mon Sep 17 00:00:00 2001 From: ammedmar Date: Mon, 6 Apr 2020 12:12:53 +0200 Subject: [PATCH 10/11] Update after Umbe's third comments Signed-off-by: ammedmar --- doc/theory/glossary.tex | 6 +++--- 1 file changed, 3 insertions(+), 3 deletions(-) diff --git a/doc/theory/glossary.tex b/doc/theory/glossary.tex index 3df3d0e08..f06a15105 100644 --- a/doc/theory/glossary.tex +++ b/doc/theory/glossary.tex @@ -306,7 +306,7 @@ \end{equation*} The set of persistence diagrams together with any of the distances above is a - %\hyperref[metric space]{metric space}. + %\hyperref[metric_space]{metric space}. metric space. \paragraph{\\ Reference:} \cite{kerber2017geometry} @@ -314,7 +314,7 @@ \subsection*{Persistence landscape} \label{persistence_landscape} Let $\{(b_i, d_i)\}_{i \in I}$ be a - %\hyperref[persistence diagram] {persistence_diagram} + %\hyperref[persistence_diagram]{persistence diagram} persistence diagram. Its \textit{persistence landscape} is the set $\{\lambda_k\}_{k \in \mathbb N}$ of functions \begin{equation*} \lambda_k : \mathbb R \to \overline{\mathbb R} @@ -442,7 +442,7 @@ ||u|| = \sqrt{\langle u, u \rangle} \end{equation*} and a norm space is naturally a - %\hyperref[metric_space]{metric_space} + %\hyperref[metric_space]{metric space} metric space with distance function \begin{equation*} d(u,v) = ||u-v||. From c45f10e68afbc2baa958d0e6b4521aaefa941a9f Mon Sep 17 00:00:00 2001 From: ammedmar Date: Mon, 6 Apr 2020 12:39:18 +0200 Subject: [PATCH 11/11] Update after Umbe's 4th comments Signed-off-by: ammedmar --- doc/theory/glossary.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/doc/theory/glossary.tex b/doc/theory/glossary.tex index f06a15105..0a9c8f518 100644 --- a/doc/theory/glossary.tex +++ b/doc/theory/glossary.tex @@ -232,7 +232,7 @@ \subsection*{Vietoris-Rips complex and Vietoris-Rips persistence} \label{vietoris-rips_complex_and_vietoris-rips_persistence} Let $(X, d)$ be a - % \hyperref[finite_metric_spaces_and_point_clouds]{finite metric_space} + % \hyperref[finite_metric_spaces_and_point_clouds]{finite metric space} finite metric space. Define the Vietoris-Rips complex of $X$ as the % \hyperref[filtered_complex]{filtered complex} filtered complex $VR_s(X)$ that contains a subset of $X$ as a simplex if all pairwise distances in the subset are less than or equal to $s$, explicitly