epigraph gls and sym

Author: alexjungaalto

ADictML_English.pdf

@@ -45,6 +45,8 @@
 \usepackage{caption}
 \captionsetup[figure]{name=Fig., labelsep=period} % Customize figure caption format
 
+\usepgfplotslibrary{fillbetween}
+\usetikzlibrary{intersections}
 
 
 \usepackage{xspace}

ADictML_English.tex

@@ -1,10 +1,64 @@
 % !TeX spellcheck = en_GB
 
 \newglossaryentry{minimum}
-{
-	name=minimum,
-	description={Given a set of real numbers, the minimum\index{minimum} is the smallest of those numbers.},
-	first={minimum},text={minimum}
+{name=minimum,
+ description={Given a set of real numbers, the minimum\index{minimum} is the smallest of those numbers.},
+ first={minimum},text={minimum}
+}
+
+\newglossaryentry{epigraph}
+{name={epigraph},
+  description={The epigraph\index{epigraph} of a real-valued function $f : \mathbb{R}^n \to \mathbb{R} \cup \{+\infty\}$ 
+  	is the set of points lying on or above its graph:
+		\[
+		\operatorname{epi}(f) = \left\{ (\mathbf{x}, t) \in \mathbb{R}^n \times \mathbb{R} \,\middle|\, f(\mathbf{x}) \leq t \right\}.
+		\]
+		A function is \gls{convex} if and only if its epigraph is a \gls{convex} set \cite{BoydConvexBook,BertCvxAnalOpt}.
+		\begin{figure}[H]
+			\centering
+			\begin{tikzpicture}[scale=1.0]
+				\begin{axis}[
+					axis lines = middle,
+					xlabel = $x$,
+					ylabel = {$$},
+					xmin=-2, xmax=2,
+					ymin=0, ymax=4.5,
+					samples=100,
+					domain=-1.5:1.5,
+					thick,
+					width=8cm,
+					height=6cm,
+					grid=none,
+					axis on top,
+					]
+					% Function
+					\addplot [blue, thick, domain=-1.5:1.5] {x^2} node [pos=0.85, anchor=south west, xshift=5pt] {$f(x)$};
+					% Epigraph shading
+					\addplot [
+					name path=f,
+					draw=none,
+					ytick=\empty,
+					domain=-1.5:1.5,
+					] {x^2};
+					\path[name path=top] (axis cs:-1.5,4) -- (axis cs:1.5,4);
+					\addplot [
+					blue!20,
+					opacity=0.6,
+					draw=none,
+					] fill between [
+					of=f and top,
+					soft clip={domain=-1.5:1.5},
+					];
+					    \node[font=\small] at (axis cs:-1.0,2.3) {$\operatorname{epi} f$};
+				%	\node[align=center, fill=white, draw=black, rounded corners, font=\small] at (axis cs:0.5,3.5) {Epigraph\\$\{(x,t) \mid f(x) \le t\}$};
+				\end{axis}
+			\end{tikzpicture}
+			\caption{Epigraph of the function $f(x) = x^2$ (shaded area).}
+		\end{figure}
+	},
+	first={epigraph},
+	text={epigraph},
+	plural={epigraphs}
 }
 
 

ADictML_Glossary_English.tex

@@ -27,14 +27,16 @@ \section*{Sets and Functions}
 	 &\log a &\quad &\mbox{The logarithm of the positive number $a \in \mathbb{R}_{++}$.  } \nonumber \\[2mm] \hline \nonumber\\[-5mm]
 	 &\hypothesis(\cdot)\!:\!\mathcal{A}\!\rightarrow\!\mathcal{B} :  a \!\mapsto\!h(a) &\quad &\parbox{.75\textwidth}{
 	 	A function (i.e., a map) that accepts any element $a \in \mathcal{A}$ from a set $\mathcal{A}$ 
-	 	as input and delivers a well-defined element $h(a) \in \mathcal{B}$ of a set $\mathcal{B}$. 
-	 	The set $\mathcal{A}$ is the domain of the function $h$ and the set $\mathcal{B}$ is the 
+	 	as input and delivers a well-defined element $\hypothesis(a) \in \mathcal{B}$ of a set $\mathcal{B}$. 
+	 	The set $\mathcal{A}$ is the domain of the function $\hypothesis$ and the set $\mathcal{B}$ is the 
 	 	codomain of $\hypothesis$. \Gls{ml} aims at finding (or learning) a function $\hypothesis$ (i.e., a \gls{hypothesis}) 
 	 	that reads in the \gls{feature}s $\featurevec$ of a \gls{datapoint} and delivers a \gls{prediction} $\hypothesis(\featurevec)$
 	 	for its \gls{label} $\truelabel$.} \nonumber \\[2mm] \hline \nonumber\\[-5mm]
+	 	&\operatorname{epi}(f)  & \quad & \parbox{.75\textwidth}{The \gls{epigraph} of real-valued function 
+	 	$f: \mathbb{R}^{\featuredim}\rightarrow \mathbb{R}$.} \nonumber \\[2mm]  \hline \nonumber\\[-5mm]
 	 	&\nabla f(\weights) & \quad & \parbox{.75\textwidth}{The \gls{gradient} of a \gls{differentiable} real-valued function 
 	 	$f: \mathbb{R}^{\featuredim}\rightarrow \mathbb{R}$ is the vector 
-	 	$\nabla f(\weights) = \big( \frac{\partial f}{\partial \weight_{1}},\ldots,\frac{\partial f}{\partial \weight_{\featuredim}}  \big)^{T} \in \mathbb{R}^{\featuredim}$ \cite[Ch. 9]{RudinBookPrinciplesMatheAnalysis}.}   \nonumber
+	 	$\nabla f(\weights) = \big( \frac{\partial f}{\partial \weight_{1}},\ldots,\frac{\partial f}{\partial \weight_{\featuredim}}  \big)^{T} \in \mathbb{R}^{\featuredim}$ \cite[Ch. 9]{RudinBookPrinciplesMatheAnalysis}.}    \nonumber
 \end{align} 
 \section*{Matrices and Vectors} 
 

ListSymbols_English.tex

@@ -0,0 +1,6 @@
+#!/bin/bash  
+echo "$1"
+bash -c 'rm -f *.aux *.log *.out *.toc *.bbl *.dvi *.ist *.blg *.fls *.fdb_latexmk *.synctex.gz *.glo *.gls *.glg *.*-glg'
+git add . 
+git commit -m "$1"
+git push origin main