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Verified Commit dbdd45af authored by Björn Ludwig's avatar Björn Ludwig
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Revert "refactor(thesis): introduce commands for the natural and real numbers"

This reverts commit 2ce3ae45 but reintroduces the actually desired changes.
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......@@ -93,9 +93,9 @@
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 \R^N \to \R \colon x \mapsto f
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 \((x_j)_{j=1, \hdots, N} \in \R^N\) the best estimates representing
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
......@@ -123,15 +123,15 @@
As starting point we consider an already trained, fully-connected multi-layer
perceptron (MLP) with input vector \(x = (x_j)_{j=1, \hdots, N}\) comprised of
the input quantities, any number \(l \in \N\) of layers and a possibly
the input quantities, any number \(l \in \mathbb N\) of layers and a possibly
vector-valued output quantity \(Y\) of the last layer.
Each layer consists of \(n^{(i)} \in \N\) neurons, \(i= 0, 1, \hdots, l\)
Each layer consists of \(n^{(i)} \in \mathbb N\) neurons, \(i= 0, 1, \hdots, l\)
where \(n^{(0)} = N\).
We denote for \(i = 1, \hdots, l\) the \(i\)th layer's linear transformations
\(z^{(i)} \colon \R^{n^{(i-1)}} \to \R^{n^{(i)}}\) as
\(z^{(i)} \colon \mathbb R^{n^{(i-1)}} \to \mathbb R^{n^{(i)}}\) as
\[z^{(i)}(X) = W^{(i)} \cdot x + b^{(i)}\]
with weights \(W^{(i)} \in \R^{n^{(i)} \times n^{(i-1)}}\) and biases \({b^{
(i)}} \in \R^{n^{(i)}}\).
with weights \(W^{(i)} \in \mathbb R^{n^{(i)} \times n^{(i-1)}}\) and biases \({b^{
(i)}} \in \mathbb R^{n^{(i)}}\).
As activation functions we consider
\begin{itemize}
\item the so-called \textit{rectifier}
......@@ -144,7 +144,7 @@
\textit{rectifier}:
\begin{gather}
\tag{rectifier}
g(z) = \max\{0, z\}, \quad z \in \R.
g(z) = \max\{0, z\}, \quad z \in \mathbb R\label{eq:rectifier}, \text{ and}
\end{gather}
We define the first derivative of \(g\) at \(0\) to be \(g'(0) \coloneqq 1\).
This results in \(g \in C^\infty([0, \infty))\) and \(g \in C^\infty((-\infty, 0)
......
......@@ -219,8 +219,8 @@ 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
\(f \colon \R^N \to \R \colon x \mapsto f (x) = y\), which takes the
form of an already trained, deep neural network.
\(f \colon \R^N \to \R \colon x \mapsto f (x) = y\), which takes the form of an
already trained, deep neural network.
Let \((x_j)_{j=1, \hdots, N} \in \R^N\) 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}).
......@@ -248,8 +248,8 @@ method's inputs and result.
As starting point we consider an already trained, fully-connected multi-layer
perceptron (MLP) with input vector \(x^{(0)} = \left( x_k^{(0)} \right)_{k=1, \hdots,
n^{(0)}}\) comprised of the estimates of the input quantities, any number \(l \in
\N\) of layers and a possibly vector-valued estimate $y$ of the output
quantity in the last layer.
\N\) of layers and a possibly vector-valued estimate $y$ of the output quantity in
the last layer.
Each layer consists of \(n^{(i)} \in \N\) neurons, \(i= 0, 1, \hdots, l\).
We denote for \(i = 1, \hdots, l\) the \(i\)th layer's linear transformations
\(z^{(i)} \colon \R^{n^{(i-1)}} \to \R^{n^{(i)}}\) as
......@@ -304,7 +304,7 @@ with weights \(W^{(i)} \in \R^{n^{(i)} \times n^{(i-1)}}\) and biases \({b^{(i)}
\begin{gather}
\label{eq:sigmoid}
\tag{sigmoid}
\sigma(x) = \frac{e^x}{1 + e^x} = \frac{1}{1 + e^{-x}}, \quad x \in \R.
\sigma(x) = \frac{e^x}{1 + e^x} = \frac{1}{1 + e^{-x}}.
\end{gather}
\textit{First derivative of sigmoid}:
......@@ -371,8 +371,7 @@ referred to as the
\begin{gather}
\label{eq:parametric_softplus}
\tag{parametric softplus}
\softplus(x) = \frac{\ln \left( 1 + e^{nx} \right)}{n}, \quad n \in
\R^+.
\softplus(x) = \frac{\ln \left( 1 + e^{nx} \right)}{n}, \quad n \in \R^+.
\end{gather}
Its further investigation is justified by the well-known fact, that in the limit
......@@ -393,8 +392,7 @@ transition from $n \to \infty$ one obtains the most wide spread activation funct
\textit{(parametric) softplus first derivative}:
\begin{gather}
\odv{\softplus}{x} (x) = \frac{1}{1 + e^{-nx}} = \sigma(nx), \quad n \in
\R^+.
\odv{\softplus}{x} (x) = \frac{1}{1 + e^{-nx}} = \sigma(nx), \quad n \in \R^+.
\label{eq:parametric-softplus-first-derivative}
\end{gather}
......
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