From fc56a8f69d6eaeaecacdd4fdfa40df7e3d16599d Mon Sep 17 00:00:00 2001
From: Stratis Ioannidis <stratis@Stratiss-MacBook-Air.local>
Date: Sun, 22 Sep 2013 22:41:24 +0200
Subject: space saving

---
 approximation.tex | 4 ++--
 1 file changed, 2 insertions(+), 2 deletions(-)

(limited to 'approximation.tex')

diff --git a/approximation.tex b/approximation.tex
index cc278e1..1b94615 100755
--- a/approximation.tex
+++ b/approximation.tex
@@ -47,7 +47,7 @@ expectation of $V$ under the  distribution $P_\mathcal{N}^\lambda$:
     F(\lambda) 
     \defeq \mathbb{E}_{S\sim P_\mathcal{N}^\lambda}\big[V(S)\big]
   %  = \sum_{S\subseteq\mathcal{N}} P_\mathcal{N}^\lambda(S) V(S)
-= \mathbb{E}_{S\sim P_\mathcal{N}^\lambda}\left[ \log\det\left( I_d + \sum_{i\in S} x_i\T{x_i}\right) \right],\quad \lambda\in[0,1]^n.
+= \mathbb{E}_{S\sim P_\mathcal{N}^\lambda}\big[ \log\det\big( I_d + \textstyle\sum_{i\in S} x_i\T{x_i}\big) \big],\quad \lambda\in[0,1]^n.
 \end{equation}
 Function $F$ is an extension of $V$ to the domain  $[0,1]^n$, as it equals $V$ on integer inputs: $F(\id_S) = V(S)$ for all
 $S\subseteq\mathcal{N}$, where $\id_S$ denotes the indicator vector of $S$. %\citeN{pipage} have shown how to use this extension to obtain approximation guarantees for an interesting class of optimization problems through the \emph{pipage rounding} framework, which has been successfully applied in \citeN{chen, singer-influence}. 
@@ -56,7 +56,7 @@ multi-linear extension \eqref{eq:multi-linear} cannot be optimized in
 polynomial time for  the value function $V$  we study here, given by \eqref{modified}. Hence, we introduce an extension $L:[0,1]^n\to\reals$ s.t.~
 \begin{equation}\label{eq:our-relaxation}
         L(\lambda) \defeq
-\log\det\left(I_d + \sum_{i\in\mathcal{N}} \lambda_i x_i\T{x_i}\right),
+\log\det\big(I_d + \textstyle\sum_{i\in\mathcal{N}} \lambda_i x_i\T{x_i}\big),
 %= \log\det\left(\mathbb{E}_{S\sim P_\mathcal{N}^\lambda}\bigg[I_d + \sum_{i\in S} x_i\T{x_i} \bigg]\right),
 \quad \lambda\in[0,1]^n.
 \end{equation}
-- 
cgit v1.2.3-70-g09d2

