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Diffstat (limited to 'finale/project_proposal.tex')
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1 files changed, 20 insertions, 5 deletions
diff --git a/finale/project_proposal.tex b/finale/project_proposal.tex index 912b281..5e0d21c 100644 --- a/finale/project_proposal.tex +++ b/finale/project_proposal.tex @@ -14,7 +14,7 @@ The network inference problem concerns itself with learning the edges and the edge weights of an unknown network. Each edge weight $\theta_e, e\in E$ is a parameter to be estimated. The information at our disposal is the result of a cascade process on the network. Here, we will focus on the Generalized Linear -Cascade (GLC) model introduced in~\cite{} presented below. +Cascade (GLC) model introduced in~\cite{paper} presented below. \paragraph{The GLC model} @@ -56,8 +56,8 @@ linear model. In particular, if $f$ is the sigmoid function, we are performing logistic regression: $$ \begin{cases} -y_i^* = \theta_i \cdot x^t + \epsilon, \text{~where~} \epsilon\sim -Logistic(0, 1) \\ +y_i^* = \theta_i \cdot x_i + \epsilon \text{~where~} \epsilon\sim +Logistic(0, 1) \text{~and~} x = {(x^t)}_{t \in \mathcal{T}_i}\\ y_i = [y_i^* > 0] \text{~where~} y_i^t = x_i^{t+1} \end{cases} $$ @@ -72,10 +72,25 @@ Can you intuitively link certain node-level/graph-level properties with the resulting variance on the estimated parameter? \item Do the previous observations correspond with the theoretical result, given by the Fisher information matrix: $$\hat \beta \sim \mathcal{N}(\beta, -I{(\theta)}^{-1})$$ where $I(\theta) = - \left(\frac{\partial^2\log -\mathcal{L}}{\partial \theta^2} \right)^{-1}$ +I{(\theta)}^{-1})$$ where $I(\theta) = - {\left(\frac{\partial^2\log +\mathcal{L}}{\partial \theta^2} \right)}^{-1}$ \item Are there networks in which the Fisher information matrix is singular? What happens to the estimation of $\beta$ in this case? +\item What if the generative process is generated with a different link +function? Is there a regularization scheme which can mitigate any bias/exploding +variance in the estimated parameters? \end{itemize} +\subsection*{Program plan} + +The project will be a series of simulations to answer each of the above +questions? When possible, we will try to explain the results found in the +simulation with a simplified analysis on toy-networks. Thibaut and I have worked +together in the past, and have kept our contributions balanced. + +\begin{thebibliography}{1} +\bibitem{paper} Pouget-Abadie, J. and Horel, T. \emph{Inferring Graphs from +Cascades: A Sparse Recovery Framework}, ICML 2015 +\end{thebibliography} + \end{document} |