refactor(thesis): evaluate uncertainty propagation runtimes

src/thesis/Thesis_408230.tex

@@ -2856,6 +2856,39 @@ and possible delays due to other operations on the used machine.
   \label{tab:computational_time_quadlu}
 \end{table}
 
+It should be noted, however, that this is not the total time of execution of the
+required examples, which are also included in the code
+repository~\citep{ludwig_pytorch_gum_uncertainty_propagation_2023}.
+Some pre- and post-processing code was needed around the main code snippets measured,
+but its runtime is not relevant for the desired comparison, as it is always the same.
+For interested readers, please refer to the column \enquote*{others} in
+table~\ref{tab:computational_time_robustness} in
+chapter~\ref{ch:robustness_verification} regarding the runtime of these pre- and
+post-processing steps.
+This contains almost identical operations in otherwise the same execution environment.
+
+The following observations apply to all three activation functions.
+In general, one can notice that for tiny networks with 10 neurons, the nonlinear
+parts require more effort than the linear parts.
+Already at 2.7e+01 or for QuadLU at 1.0e+01, this ratio changes with increasing
+degree.
+For the largest instances, the execution times for the non-linear components are
+already three orders of magnitude smaller than those of the linear components.
+Despite the apparently more complex structure in dependence on a parameter for
+QuadLU, all three activation functions are in the same order of magnitude with regard
+to the computing times, with a few exceptions, and otherwise only differ from each
+other by one order of magnitude.
+However, these differences can only be observed up to a neuron count of 2,500.
+The differences in the smaller networks can most likely be explained by overhead in
+calling functions and size comparisons, which is compensated for by the more
+efficient calculation of the function values of QuadLU for larger vectors and matrices.
+Interestingly, no systematic difference can be found between the calculations with
+and without uncertainties, although the two cases are handled separately in the
+implementations.
+In the case of unoccupied uncertainties, care was taken to really only perform
+calculations for the values.
+This is surprising insofar as two multiplications of fully populated matrices are
+carried out less in the case of unpopulated covariance matrices, which should also lead to corresponding runtime savings.
 
 \chapter{Robustness Verification}\label{ch:robustness_verification}