wip(thesis): introduce section outline and robustness verification task

src/thesis/Thesis_408230.tex

@@ -192,23 +192,29 @@
 
 \section{Objective}\label{sec:objective}
 
-  We first want to provide a mathematical foundation for propagating the uncertainty
-  in neural network inputs through the networks in a GUM-compliant way.
-  The task we eventually aim to solve is:
-  \begin{task}
-    Let \(X = (X_j)_{j=1, \hdots, N}\) the input quantities of a
-    measurement function \(f \colon \mathbb R^N \to \mathbb R \colon x \mapsto f
-    (x) = y\), which takes the form of an already trained, deep neural network.
-    Let \((x_j)_{j=1, \hdots, N} \in \mathbb R^N\) the best estimates representing
-    the input quantities and \(u(x_j), j=1, \ldots, N\) the associated standard
-    uncertainties (\cite[paragraph 3.18]{jcgm_evaluation_2008}).
-    Propagate the \(x_j\) and \(u(x_j), j=1, \ldots, N\) through
-    \(f\) to form an estimate \(y\) of the output quantity \(Y\) and the
-    associated standard uncertainty \(u(y)\) as suggested in ~\cite[paragraph
-    7.2]{jcgm_evaluation_2009}.
-  \end{task}
-  Afterwards we will apply an existing robustness verification method to our network
-  and measure its performance.
+We first want to provide a mathematical foundation for propagating the uncertainty
+in neural network inputs through the networks in a GUM-compliant way.
+The task we eventually aim to solve is:
+\begin{task}[Uncertainty propagation]
+  Let \(X = (X_j)_{j=1, \hdots, N}\) the input quantities of a measurement function
+  \(f \colon \mathbb R^N \to \mathbb R \colon x \mapsto f (x) = y\), which takes the
+  form of an already trained, deep neural network.
+  Let \((x_j)_{j=1, \hdots, N} \in \mathbb R^N\) the best estimates representing the
+  input quantities and \(u(x_j), j=1, \ldots, N\) the associated standard
+  uncertainties (\cite[paragraph 3 .18]{jcgm_evaluation_2008}).
+  Propagate the \(x_j\) and \(u(x_j), j=1, \ldots, N\) through \(f\) to form an
+  estimate \(y\) of the output quantity \(Y\) and the associated standard uncertainty
+  \(u(y)\) as suggested in ~\cite[paragraph 7 .2]{jcgm_evaluation_2009}.
+\end{task}
+Afterwards we will apply an existing robustness verification method to our network
+and measure its performance.
+\begin{task}[Robustness verification]
+  Given a classification deep neural network (CDNN) $f$ with an input region
+  $\Theta$ comprised of a set of uncertain inputs, does the robustness property hold?
+\end{task}
+
+
+\section{Outline}\label{sec:outline}
 
 \include{preliminaries}