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diff --git a/finale/sections/related.tex b/finale/sections/related.tex new file mode 100644 index 0000000..59fb759 --- /dev/null +++ b/finale/sections/related.tex @@ -0,0 +1,41 @@ +\paragraph{Related works.} +The study of edge prediction in graphs has been an active field of research for +over a decade~\cite{Nowell08, Leskovec07, AdarA05}.~\cite{GomezRodriguez:2010} +introduced the {\scshape Netinf} algorithm, which approximates the likelihood of +cascades represented as a continuous process. The algorithm, relying on +Maximum-Likelihood and convex relaxations was improved in later +work~\cite{gomezbalduzzi:2011}, but is not known to have any theoretical +guarantees beside empirical validation on synthetic networks. + +Let us denote by $m$ the number of nodes in the graph and for a given node, we +denote by $s$ its in-degree. In what follows, we call a \emph{recovery +guarantee} an upper bound on the number of observations required (expressed as +a function of $s$ and $m$) to learn the incoming edges of a node to a fixed +arbitrary accuracy level. \cite{Netrapalli:2012} studied the discrete-time +version of the independent cascade model and obtained the first ${\cal O}(s^2 +\log m)$ recovery guarantee on general networks, by using unregularized MLE and +making a {\it correlation decay\/} assumption, which limits the number of new +infections at every step. They also suggest a {\scshape Greedy} algorithm, +with provably good performance in tree graphs, which maintains a counter of +each node's antecedents, i.e. the set of nodes infected at a prior time step to +its infection time. The work of~\cite{Abrahao:13} studies the same +continuous-model framework as \cite{GomezRodriguez:2010} and obtains an ${\cal +O}(s^9 \log^2 s \log m)$ support recovery algorithm, without the +\emph{correlation decay} assumption. \cite{du2013uncover,Daneshmand:2014} +propose Lasso regularization of MLE for recovering the weights of the graph +under a continuous-time independent cascade model. The work +of~\cite{du2014influence} is slightly orthogonal to ours since they suggest +learning the \emph{influence} function, rather than the parameters of the +network directly. + +More recently, the continuous process studied in previous work has been +reformulated as a Hawkes process, with recent papers +\cite{linderman2014discovering, linderman2015scalable, simma2012modeling}, +focusing on Expectation-Maximization, and Variational Inference methods to learn +these processes. Because of the discrete nature of the GLC model, we hope to +bring a better understanding to the link between the properties of a graph and +its \emph{learnability}. We wish to further explore the idea of non-product +priors raised in \cite{linderman2014discovering, linderman2015scalable}, since +the experimental validation of their work focused on simple graph priors. +Finally, the \emph{Active Learning} formulation is, to the best of the +authors' knowledge, novel in this context. |
