%There have been a series of papers arguing that the standard Lasso is an inappropriate exact variable selection method \cite{Zou:2006}, \cite{vandegeer:2011}, since it relies on the essentially necessary irrepresentability condition, introduced in \cite{Zhao:2006}. This is the condition on which the analysis of \cite{Daneshmand:2014} relies. It has been noted this condition is rather stringent and rarely holds in practical situations where correlation between variables occurs. Several alternatives have been suggested, including the adaptive and thresholded lasso, which relax this assumption. We defer an extended analysis of the irrepresentability assumption to Section~\ref{sec:assumptions}.

In this section, we compare the main assumption of \cite{Daneshmand:2014}, commonly known as the {\it irrepresentability condition}, to the restricted eigenvalue condition. We argue that the restricted eigenvalue is more likely to hold in practical situations.

\subsection{The Irrepresentability Condition}

\cite{Daneshmand:2014} rely on an `incoherence' condition on the hessian of the likelihood function. It is in fact easy to see that their condition is equivalent to the more commonly called {\it (S,s)-irrepresentability} condition:

\begin{definition}
Following similar notation, let $Q^* \defeq \nabla^2 f(\theta^*)$. Let $S$ and $S^c$ be the set of indices of all the parents and non-parents respectively and $Q_{S,S}$, $Q_{S^c,S}$, $Q_{S, S^c}$, and $Q_{S^c, S^c}$ the induced sub-matrices.


The {\it (S,s)-irrepresentability} condition is defined as:
\begin{equation}
\nu_{\text{irrepresentable}}(S) \defeq \max_{\tau \in \mathbb{R}^p \ :\ \| \tau \|_{\infty} \leq 1} \|Q_{S^c, S} Q_{S, S}^{-1} \tau\|_{\infty}
\end{equation}
\end{definition}

If our objective is to recover the support of the graph exactly, the irrepresentability condition has been shown to be essentially necessary \cite{Zhao:2006}. However, several recent papers \cite{vandegeer:2011}, \cite{Zou:2006}, argue this condition is unrealistic in situations where there is a correlation between variables. Consider the following simplified example from \cite{vandegeer:2011}:
\begin{equation}
\left( 
\begin{array}{cccc}
I_s & \rho J \\
\rho J & I_s \\
\end{array}
\right)
\end{equation}
where $I_s$ is the $s \times s$ identity matrix, $J$ is the all-ones matrix and $\rho \in \mathbb{R}^+$. It is easy to see that $\nu_{\text{irrepresentable}}(S) = \rho s$ and $\lambda_{\min}(Q) \geq 1 - \rho$, such that for any $\rho > \frac{1}{s}$ and $\rho < 1$, the restricted eigenvalue holds trivially but the (S,s)-irrepresentability does not hold.

As mentioned previously, it is intuitive that the irrepresentability condition is stronger than our suggested {\bf(RE)} assumption. In fact, by adapting slightly a result from \cite{vandegeer:2009}, we can show that a `strong' irrepresentability condition directly {\it implies} the {\bf(RE)} condition for $\ell2$-recovery:

\begin{proposition}
\label{prop:irrepresentability}
If the irrepresentability condition holds with $\epsilon > \frac{2}{3}$, then the restricted eigenvalue condition holds with constant  $\gamma_n \geq \frac{ (1 - 3(1 -\epsilon))^2 \lambda_{\min}^2}{4s}n$, where $\lambda_{\min} > 0$ is the smallest eigenvalue of $Q^*_{S,S}$, on which the results of \cite{Daneshmand:2014} also depend.
\end{proposition}


\subsection{The Restricted Eigenvalue Condition}

Yet, assuming we only wish to recover all edges above a certain threshold, bounding the $\ell2$-error allows us to recover all edges with weights above a certain minimum threshold under an intuitively weaker {\bf(RE)} condition. In practical scenarios, such as in social networks, where one seeks to recover significant edges, this is a reasonable assumption.

Expressing the restricted eigenvalue assumption for correlated measurements as parameters of the graph and the cascade diffusion process is non-trivial. The restricted eigenvalue property is however well behaved in the following sense:

\begin{lemma}
\label{lem:expected_hessian}
Expected hessian analysis!
\end{lemma}

This result is easily proved by adapting slightly a result from \cite{vandegeer:2009} XXX. Similarly to the analysis conducted in \cite{Daneshmand:2014}, we can show that if the eigenvalue can be showed to hold for the `expected' hessian, it can be showed to hold for the hessian itself. It is easy to see that:

\begin{proposition}
\label{prop:expected_hessian}
If result holds for the expected hessian, then it holds for the hessian!
\end{proposition}

It is most likely possible to remove this extra s factor. See sub-gaussian paper by ... but the calculations are more involved.
