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Diffstat (limited to 'paper/sections/results.tex')
| -rw-r--r-- | paper/sections/results.tex | 12 |
1 files changed, 6 insertions, 6 deletions
diff --git a/paper/sections/results.tex b/paper/sections/results.tex index e91cad4..6b9fd7a 100644 --- a/paper/sections/results.tex +++ b/paper/sections/results.tex @@ -121,10 +121,10 @@ Assume {\bf(LF)} holds for some $\alpha>0$. For any $\delta\in(0,1)$: \end{lemma} The proof of Lemma~\ref{lem:ub} relies crucially on Azuma-Hoeffding's -inequality, which allows us to handle correlated observations. This departs from -the usual assumptions made in sparse recovery settings, where the sequence of -measurements are assumed to be independent from one another. We now show how -to use Theorem~\ref{thm:main} to recover the support of $\theta^*$, that is, to +inequality, which allows us to handle correlated observations. This departs +from the usual assumptions made in sparse recovery settings, that the +measurements are independent from one another. We now show how to +use Theorem~\ref{thm:main} to recover the support of $\theta^*$, that is, to solve the Network Inference problem. \begin{corollary} @@ -225,7 +225,7 @@ Observe that the Hessian of $\mathcal{L}$ can be seen as a re-weighted \bigg[x_i^{t+1}\frac{f''f-f'^2}{f^2}(\inprod{\theta^*}{x^t})\\ -(1-x_i^{t+1})\frac{f''(1-f) + f'^2}{(1-f)^2}(\inprod{\theta^*}{x^t})\bigg] \end{multline*} -If $f$ and $1-f$ are $c$-strictly log-convex~\cite{bagnoli2005log} for $c>0$, +If $f$ and $(1-f)$ are $c$-strictly log-convex for $c>0$, then $ \min\left((\log f)'', (\log (1-f))'' \right) \geq c $. This implies that the $(S, \gamma)$-({\bf RE}) condition in Theorem~\ref{thm:main} and Theorem~\ref{thm:approx_sparse} reduces to a condition on the \emph{Gram @@ -267,7 +267,7 @@ cascade, which are independent, we can apply Theorem 1.8 from \cite{rudelson:13}, lowering the number of required cascades to $s\log m \log^3( s\log m)$. -If $f$ and $1-f$ are strictly log-convex, then the previous observations show +If $f$ and $(1-f)$ are strictly log-convex, then the previous observations show that the quantity $\E[\nabla2\mathcal{L}(\theta^*)]$ in Proposition~\ref{prop:fi} can be replaced by the expected \emph{Gram matrix}: $A \equiv \mathbb{E}[X^T X]$. This matrix $A$ has a natural interpretation: the |