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| -rw-r--r-- | paper/sections/results.tex | 27 |
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diff --git a/paper/sections/results.tex b/paper/sections/results.tex index 0c1cc3b..6c8a35a 100644 --- a/paper/sections/results.tex +++ b/paper/sections/results.tex @@ -310,14 +310,27 @@ case to the assumption made in the Lasso analysis of \cite{bickel:2009}. \paragraph{(RE) with high probability} The Generalized Linear Cascade model yields a probability distribution over the -set observed nodes $x^t$. It is then natural to ask whether the restricted -eigenvalue condition is likely to occur under this probabilistic model. Several -recent papers show that large classes of correlated designs obey the restricted -eigenvalue property with high probability \cite{raskutti:10} -\cite{rudelson:13}. +observed sets of infeceted nodes $(x^t)_{t\in\mathcal{T}}$. It is then natural +to ask whether the restricted eigenvalue condition is likely to occur under +this probabilistic model. Several recent papers show that large classes of +correlated designs obey the restricted eigenvalue property with high +probability \cite{raskutti:10, rudelson:13}. -Expressing the minimum restricted eigenvalue $\gamma$ as a function of the -cascade model parameters is highly non-trivial. Yet, the restricted eigenvalue +In our case, we can show that if (RE)-condition holds for the expected Hessian +matrix $\E[\nabla^2\mathcal{L}(\theta^*)]$, then it holds for the finite sample +Hessian matrix $\nabla^2\mathcal{L}(\theta)$ with high probability. Note that +the expected Hessian matrix is exactly the Fisher Information matrix of the +generalized linear cascade model which captures the amount of information about +$\theta$ conveyed by the random observations. Therefore, under an assumption +which only involves the probabilistic model and not the actual observations, we +can reformulate Theorem~\ref{thm:main}. + +We will need the following additional assumptions on the inverse link +function $f$: + + + +Yet, the restricted eigenvalue property is however well behaved in the following sense: under reasonable assumptions, if the population matrix of the hessian $\mathbb{E} \left[\nabla^2 {\cal L}(\theta) \right]$, corresponding to the \emph{Fisher Information |