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Diffstat (limited to 'paper/sections/results.tex')
| -rw-r--r-- | paper/sections/results.tex | 5 |
1 files changed, 2 insertions, 3 deletions
diff --git a/paper/sections/results.tex b/paper/sections/results.tex index 6fba6b2..cd0a5f8 100644 --- a/paper/sections/results.tex +++ b/paper/sections/results.tex @@ -20,8 +20,7 @@ a set $\mathcal{T}$ of observations. We will write $n\defeq |\mathcal{T}|$. \label{sec:main_theorem} In this section, we analyze the case where $\theta^*$ is exactly sparse. We -write $S\defeq\text{supp}(\theta^*)$ and $s=|S|$. Our main theorem will rely -on the now standard \emph{restricted eigenvalue condition}. +write $S\defeq\text{supp}(\theta^*)$ and $s=|S|$. In our context, $S$ is the set of all nodes susceptible to influence our node $i$. In other words, if $\theta^*$ is exactly $s$-sparse, then node $i$ has at most $s$ parents. Our main theorem will rely on the now standard \emph{restricted eigenvalue condition}. \begin{definition} Let $\Sigma\in\mathcal{S}_m(\reals)$ be a real symmetric matrix and $S$ be @@ -55,7 +54,7 @@ In the voter model, $\frac{f'}{f}(z) = \frac{1}{z}$ and $\frac{f'}{f}(1-z) 1-\alpha$ for all $(i,j)\in E$. Similarly, in the Independent Cascade Model, $\frac{f'}{f}(z) = \frac{1}{1-e^z}$ and $\frac{f'}{1-f}(z) = -1$ and (LF) holds if $p_{i, j}\leq 1-\alpha$ for all $(i, j)\in E$. Remember that in this case we -have $\Theta_{i,j} = \log(1-p_{i,j})$. +have $\Theta_{i,j} = \log(1-p_{i,j})$. These assumptions are reasonable: if an edge has a weight very close to 0, then the ``infection'' will never happen along that edge for our set of observations and we can never hope to recover it. \begin{theorem} \label{thm:main} |