\paragraph{MURGS experiments}

In this section we report results of a separate simulation study to assess the feature selection capability of MURGS. Figure~\ref{fig:murgs_sim} demonstrates in terms of F1 score (see Section~\ref{sec:metrics}) how MURGS is able to select active groups across with increasing sample size. Encouragingly, group size does not seem to play a significant role in the performance of MURGS. The first row of Figure~\ref{fig:murgs_sim} is based on data generated according to the synthetic data generation strategy described in Sections~\ref{sec:experiments} and~\ref{sec:synthetic_data}. We select sink nodes in a graph making sure that the number of parents is the same across group size. Then we generate data with increasing sample size and track the F1 score. Besides some numerical instabilities MURGS shows convergence behavior already on small sample sizes.

To test out the boundaries, we reiterate the experiments with a different data generation strategy (second row). Nonlinear functions are now generated via randomly initialized deep neural networks with ReLU activation function such that additive models will have a hard time approximating the regression functions. Indeed, MURGS struggles much more to recover the active nodes when the number of nodes in the graph is larger. Despite the challenging nonlinearity F1 score depicts high values throughout again without requiring extraordinarily high sample size.

\paragraph{Additional real data and synthetic data results}

Figures~\ref{fig:real_data_grandag},~\ref{fig:real_data_gpc}, and~\ref{fig:real_data_glingam} present supplementary results for the remaining algorithms in the real-data experiment (Section~\ref{sec:real_data}) that were omitted from the main text.

Additionally, the ensuing boxplots provide additional results regarding the synthetic experiment described in Sections~\ref{sec:experiments} and~\ref{sec:synthetic_data}. The results plots are ordered by node size and group size.

\clearpage\newpage
\onecolumn

\begin{figure}
  \centering
  \scalebox{0.6}{
    \input{./Figures/real_data_grandag.tex}
  }
  \caption{Learned causal edges from the real-world dataset using \textit{GroupGraN-DAG}.}
  \label{fig:real_data_grandag}
\end{figure}

\begin{figure}
  \centering
  \scalebox{0.6}{
    \input{./Figures/real_data_gpc.tex}
  }
  \caption{Learned causal edges from the real-world dataset using \textit{GroupPC}.}
  \label{fig:real_data_gpc}
\end{figure}

\begin{figure}
  \centering
  \scalebox{0.6}{
    \input{./Figures/real_data_lingam.tex}
  }
  \caption{Learned causal edges from the real-world dataset using \textit{GroupDirectLiNGAM}.}
  \label{fig:real_data_glingam}
\end{figure}
\newpage\clearpage
\begin{centering}
  \includegraphics[width=.82\textwidth]{Figures/result_boxplot_10_2_1000.pdf}
  \includegraphics[width=.82\textwidth]{Figures/result_boxplot_10_5_1000.pdf}
  \includegraphics[width=.82\textwidth]{Figures/result_boxplot_10_10_1000.pdf}

  \includegraphics[width=.82\textwidth]{Figures/result_boxplot_10_5_2000.pdf}
  \includegraphics[width=.82\textwidth]{Figures/result_boxplot_10_10_2000.pdf}

  \includegraphics[width=.82\textwidth]{Figures/result_boxplot_15_2_1000.pdf}
  \includegraphics[width=.82\textwidth]{Figures/result_boxplot_15_5_1000.pdf}
  \includegraphics[width=.82\textwidth]{Figures/result_boxplot_15_10_1000.pdf}

  \includegraphics[width=.82\textwidth]{Figures/result_boxplot_15_2_2000.pdf}

  \includegraphics[width=.82\textwidth]{Figures/result_boxplot_20_2_1000.pdf}
  \includegraphics[width=.82\textwidth]{Figures/result_boxplot_20_5_1000.pdf}

  \includegraphics[width=.82\textwidth]{Figures/result_boxplot_20_2_2000.pdf}

\end{centering}
