Ajout retours PS

annexe.tex

@@ -76,6 +76,15 @@
     Thus $\ell(\bY;\theta) - \KL{\Ryt}{\log \Prob(\bZ,\bW|\bY;\theta)} = \mathcal{J}(\tau;\theta) \qed$
 \end{frame}
 
+\begin{frame}
+    \frametitle{On the BIC-L}
+    \begin{align*}
+         & \text{ICL}(\hat{\theta})                = \Esp_{\mathbf{Z}, \mathbf{W}|\mathbf{Y}} [\log p(\mathbf{Y}|\mathbf{Z},\mathbf{W};\hat{\theta})] - \frac{1}{2} \text{pen}(\dots)                                                                         \\
+         & \text{BIC}(\hat{\theta})                = \Esp_{\mathbf{Z}, \mathbf{W}|\mathbf{Y}} [\log p(\mathbf{Y}|\mathbf{Z},\mathbf{W};\hat{\theta})] + \mathcal{H}(p(\mathbf{Z},\mathbf{W}|\mathbf{Y})) - \frac{1}{2} \text{pen}(\dots)                      \\
+         & \text{BIC-L}(\hat{\theta}, \hat{\tau})  = \Esp_{\mathcal{R}_{\mathbf{Y}, \hat{\tau}}}[\log \ell_c(\mathbf{Y},\mathbf{Z},\mathbf{W};\hat{\theta}^{\text{var}})] + \mathcal{H}(\mathcal{R}_{\mathbf{Y}, \hat{\tau}}) - \frac{1}{2} \text{pen}(\dots) \\
+    \end{align*}
+\end{frame}
+
 \section{Model selection}
 
 \begin{frame}
@@ -112,6 +121,7 @@
         \end{column}
     \end{columns}
 \end{frame}
+
 \begin{frame}
     \frametitle{Choice of $(Q_1,Q_2)$ - Sliding window}
     \begin{columns}

principal.tex

@@ -1,6 +1,5 @@
 \section{Model Context}
 \label{sec:context-of-the-model}
-
 \begin{frame}
     \frametitle{Why a network?}
     \begin{columns}
@@ -55,6 +54,7 @@
                   \item node level: degree, centrality\dots
                   \item network level: density, nestedness\dots
               \end{itemize}
+              \cite{kolaczykStatisticalAnalysisNetwork2009}
         \item \textbf<2>{Node embedding and/or clustering with latent variable models}
               \\\cite{snijdersEstimationPredictionStochastic1997,hoffLatentSpaceApproaches2002}
         \item Node or network embedding with Graph Convolutional Networks
@@ -64,8 +64,7 @@
 
 \begin{frame}
     \addtocounter{footnote}{1}
-    \frametitle{Latent Block Model (LBM\footnotemark[\thefootnote])}
+    \frametitle{Bipartite Stochastic Block Model (BiSBM\footnotemark[\thefootnote])}\framesubtitle{\cite{govaertEMAlgorithmBlock2005}}
-    \cite{govaertEMAlgorithmBlock2005}.
     \begin{columns}
         \begin{column}{0.40\linewidth}
             \begin{figure}[H]
@@ -108,7 +107,7 @@
             \end{column}}
     \end{columns}
 
-    \footnotetext[\thefootnote]{Which I will henceforth call BiSBM}
+    \footnotetext[\thefootnote]{Commonly Known as \emph{Latent Block Model} (LBM) in the literature.}
 \end{frame}
 
 \begin{frame}
@@ -160,11 +159,11 @@
 \label{sec:extension-of-colsbm-to-bipartite-networks}
 \begin{frame}
     \frametitle{Model 0: sep-BiSBM}
-    \footnotesize
+    \only<1-2>{
-    $
+        \begin{equation*}
-        \forall m \in \{1\dots M\}, Y^m \overset{ind}{\sim} \mathcal{F}\text{-BiSBM}_{n_1^m,n_2^m}(Q_1\alert<2>{^m}, Q_2\alert<2>{^m}, \pi\alert<2>{^m}, \rho\alert<2>{^m}, \alpha\alert<2>{^m})
+            \forall m \in \{1\dots M\}, Y^m \overset{ind}{\sim} \mathcal{F}\text{-BiSBM}_{n_1^m,n_2^m}(Q_1\alert<2->{^m}, Q_2\alert<2->{^m}, \pi\alert<2->{^m}, \rho\alert<2->{^m}, \alpha\alert<2->{^m})
-    $
+        \end{equation*}}
-    \onslide<3>{
+    \only<3>{
         \begin{figure}[ht]
             \centering
             \begin{subfigure}[ht]{0.42\textwidth}
@@ -188,6 +187,7 @@
                 \includegraphics[width=0.5\textwidth]{tikz/applications/baldock/bisbm-mat-Baldock2019_Reading.pdf}
                 \caption{Reading}
             \end{subfigure}
+            \vspace{-\baselineskip}
             \caption{Reordered adjacency matrices, using BiSBM for each network}
             \label{fig:adj-reord}
         \end{figure}
@@ -249,32 +249,32 @@
 %         \end{alertblock}}
 
 % \end{frame}
-
+\section{Inference and model selection}
+\label{sec:inference-and-model-selection}
 \begin{frame}{Parameter estimation}
     By \emph{Variational EM}, as proposed
     by~\cite{daudinMixtureModelRandom2008,
         chabert-liddellLearningCommonStructures2024}.
     \begin{block}{Variational approximation of $\bm{Z,W|Y},\theta^{(t-1)}$}
-        $\mathcal{R}_{Y^m,\tau}(\mathbf{Z}^m, \mathbf{W}^m) =
+        $\mathcal{R}_{Y^m,\tau}(Z^m, W^m) =
-            \mathcal{R}^1_{Y^m,\tau}(\mathbf{Z}^m)
+            \mathcal{R}^1_{Y^m,\tau}(Z^m)
             {\color{red}\times}
-            \mathcal{R}^2_{Y^m,\tau}(\mathbf{W}^m) \Rightarrow$ independence rows, columns.
+            \mathcal{R}^2_{Y^m,\tau}(W^m) \Rightarrow$ independence between rows and columns.
     \end{block}
     \begin{multline*}
         \ell (\bm{Y};\theta) \geq \color{red}\sum_{m=1}^{M} \bigg(
         \color{black} \mathcal{Q}^m(\theta\mid\theta^{(t)}) +
         \mathcal{H}(\mathcal{R}_{Y^m,\theta^{(t)}}
-        (\mathbf{Z}^m, \mathbf{W}^m))
+        (Z^m, W^m))
         \color{red}\bigg) \color{black}
         \eqcolon \mathcal{J}(\tau;\theta)
     \end{multline*}
     where $\mathcal{Q}^m(\theta\mid\theta^{(t)}) =
-        \mathbb{E}_{\mathbf{Z}^m,\mathbf{W}^m
+        \mathbb{E}_{Z^m,W^m
         \sim \mathcal{R}_{Y^m,\tau}(.)}
-        \left[ \ell_c(Y^m,\mathbf{Z}^m,\mathbf{W}^m | \theta) \right] \,$
+        \left[ \ell_c(Y^m,Z^m,W^m | \theta) \right] \,$
 \end{frame}
 
-\section{Model selection}
 \begin{frame}
     \frametitle{Problem of choosing $(Q_1, Q_2)$}
     Need to select $Q_1$ and $Q_2$. BIC-Like criterion\footnote{ICL + entropy - penalty}
@@ -289,7 +289,7 @@
             \item Exploration of a 2D grid is costly. \uncover<2->{$\rightarrow$ \textbf{Greedy
                           approach} and \textbf{sliding window}}
             \item Sensitivity to initializations. \uncover<3->{$\rightarrow$ \textbf{Spectral
-                          clustering} and \textbf{reuse of previous inits}}
+                          clustering} and \textbf{split \& merge} approach}
         \end{itemize}
     \end{alertblock}
 \end{frame}
@@ -328,25 +328,26 @@
 
 \begin{frame}
     \frametitle{Results~\cite{baldockSystemsApproachReveals2019} focus on Leeds}
+    \captionsetup{font=normalsize}
     \begin{figure}[ht]
         \centering
         \begin{subfigure}[t]{0.5\textwidth}
             \centering
             \includegraphics[width=1\textwidth]{tikz/applications/baldock/bisbm-mat-Baldock2019_Leeds.pdf}
-            \caption{Separate model}
+            \caption{Leeds with sep-BiSBM}
         \end{subfigure}\hfill
         \begin{subfigure}[t]{0.5\textwidth}
             \centering
             \includegraphics[width=1\textwidth]{tikz/applications/baldock/colbisbm-mat-Baldock2019_Leeds.pdf}
-            \caption{Joint model}
+            \caption{Leeds with \emph{iid}-colBiSBM}
         \end{subfigure}
-        \caption{Reordered adjacency matrix by sep-BiSBM (left) and by \emph{iid}-colBiSBM (right),~\cite{baldockSystemsApproachReveals2019}}
     \end{figure}
 \end{frame}
 
 \begin{frame}{\emph{Bombus}}
     \only<1>{
         \begin{figure}
+            \captionsetup{font=normalsize}
             \begin{subfigure}[t]{0.5\textwidth}
                 \centering
                 \includegraphics[width=0.5\textwidth]{img/baldock/bombus-hortorum.jpeg}
@@ -420,66 +421,21 @@
     }
 
 \end{frame}
-\begin{frame}
+\section{Extension and conclusion}
-    \frametitle{Network clustering}
-    \begin{figure}[ht]
-        \includegraphics[width=0.45\textwidth]{tikz/applications/baldock/mat-Baldock2011_TB+Baldock2011_JN.pdf}
-        \caption{Adjacency matrix,~\cite{baldockDailyTemporalStructure2011}}
-    \end{figure}
-\end{frame}
 
 \begin{frame}[allowframebreaks]
     \frametitle{Application to~\cite{baldockDailyTemporalStructure2011,
             baldockSystemsApproachReveals2019}}
-    TODO pivot or remove slide
+    TODO Put $\alpha$ plots and tree structure of partition
     \begin{figure}[t]
         \centering
-        \begin{subfigure}{0.5\textwidth}
+        \includegraphics[scale=0.1]{backup-app-iid-struct1.png}
-            \centering
+        \includegraphics[scale=0.2]{backup-app-iid-struct2.png}
-            \includegraphics[scale=0.1]{backup-app-iid-struct1.png}
+        \caption{Model $iid$, separate African (left) and English (right) networks}
-            \includegraphics[scale=0.2]{backup-app-iid-struct2.png}
-            \caption{Model $iid$,\\
-                separate African (left) and English (right) networks}
-        \end{subfigure}%
-        ~
-        \begin{subfigure}{0.5\textwidth}
-            \centering
-            \includegraphics[scale=0.2]{backup-app-pirho-struct.png}
-            \caption{Model $\pi\rho$,\\
-                merge African and English networks}
-        \end{subfigure}%
-        \caption{Structures detected for networks
-            of~\cite{baldockDailyTemporalStructure2011,
-                baldockSystemsApproachReveals2019}}
     \end{figure}
 \end{frame}
-
-\begin{frame}{Results}
-    \begin{figure}[ht]
-        \centering
-        \begin{subfigure}{0.5\textwidth}
-            \centering
-            \includegraphics[width=1\textwidth]{tikz/applications/baldock/bisbm-mat-Baldock2011_TB+Baldock2011_JN.pdf}
-            \caption{Reordered by LBM}
-        \end{subfigure}\hfil
-        \begin{subfigure}{0.5\textwidth}
-            \centering
-            \includegraphics[width=1\textwidth]{tikz/applications/baldock/pirho-colbisbm-mat-Baldock2011_TB+Baldock2011_JN.pdf}
-            \caption{Reordered by $\pi\rho$-colBiSBM}
-        \end{subfigure}
-
-        \caption{Reordered adjacency matrix by $\pi\rho$-colBiSBM,~\cite{baldockDailyTemporalStructure2011}}
-    \end{figure}
-\end{frame}
-
-\section{Conclusion}
-\label{sec:conclusion}
 \begin{frame}
     \frametitle{Conclusion and perspectives}
-    % DONE Add a conclusion perspective slide
-    % Recall models with clustering
-    % Mention analysis of corrected networks for sampling
-    % Link to the package
     \begin{block}{Capabilities}
         \begin{itemize}
             \item 4 models including 3 with flexibility on at least one of
@@ -488,21 +444,10 @@
             \item Partition a set of networks according to their structures.
         \end{itemize}
     \end{block}
-
-\end{frame}
-
-\begin{frame}{Perspectives}
-
-    \begin{block}{Future work}
-        \begin{itemize}
-            \item Multi-layer networks (account for sampling bias, presence/absence)
-            \item Graph Convolutional Network to allow for scalability
-        \end{itemize}
-    \end{block}
-
     \begin{block}{Package and applications}
         \begin{itemize}
-            \item CRAN submission
+            \item \texttt{ArXiv} preprint in redaction
+            \item \texttt{CRAN} submission
             \item Integrate the possibility of an additional criterion for clustering (e.g.
                   urbanization gradient~\cite{fisogniSeasonalTrajectoriesPlantpollinator2022})
             \item Apply clustering to data from

references.bib

@@ -708,6 +708,25 @@ Read\_Status\_Date: 2025-05-09T11:54:37.094Z},
   file = {/home/louis/snap/zotero-snap/common/Zotero/storage/5THEWLW6/Kipf et Welling - 2016 - Variational Graph Auto-Encoders.pdf;/home/louis/snap/zotero-snap/common/Zotero/storage/BBTHQNRZ/1611.html}
 }
 
+@book{kolaczykStatisticalAnalysisNetwork2009,
+  title = {Statistical {{Analysis}} of {{Network Data}}: {{Methods}} and {{Models}}},
+  shorttitle = {Statistical {{Analysis}} of {{Network Data}}},
+  author = {Kolaczyk, Eric D.},
+  date = {2009},
+  series = {Springer {{Series}} in {{Statistics}}},
+  publisher = {Springer New York},
+  location = {New York, NY},
+  doi = {10.1007/978-0-387-88146-1},
+  url = {https://link.springer.com/10.1007/978-0-387-88146-1},
+  urldate = {2025-05-26},
+  isbn = {978-0-387-88145-4 978-0-387-88146-1},
+  langid = {english},
+  keywords = {/unread},
+  annotation = {Read\_Status: New\\
+Read\_Status\_Date: 2025-05-26T11:42:27.939Z},
+  file = {/home/louis/snap/zotero-snap/common/Zotero/storage/RQPMHFGB/Kolaczyk - 2009 - Statistical Analysis of Network Data Methods and Models.pdf}
+}
+
 @online{kumpulainenYourBlockOur2024,
   title = {From Your {{Block}} to Our {{Block}}: {{How}} to {{Find Shared Structure}} between {{Stochastic Block Models}} over {{Multiple Graphs}}},
   shorttitle = {From Your {{Block}} to Our {{Block}}},