From 3cd73c735e26805d449ac73cefac5b5fa43e2590 Mon Sep 17 00:00:00 2001
From: Thibaut Horel <thibaut.horel@gmail.com>
Date: Mon, 1 Oct 2012 18:15:51 -0700
Subject: Fix a typo, add a remark about the value function

---
 notes.tex | 18 ++++++++++++++----
 1 file changed, 14 insertions(+), 4 deletions(-)

(limited to 'notes.tex')

diff --git a/notes.tex b/notes.tex
index ec0435c..1ce4cc4 100644
--- a/notes.tex
+++ b/notes.tex
@@ -1,7 +1,7 @@
-\documentclass[twocolumn]{article}
+\documentclass{IEEEtran}
+%\usepackage{mathptmx}
 \usepackage[utf8]{inputenc}
 \usepackage{amsmath,amsthm,amsfonts}
-\usepackage{comment}
 \newtheorem{lemma}{Lemma}
 \newtheorem{fact}{Fact}
 \newtheorem{example}{Example}
@@ -16,6 +16,7 @@
 \DeclareMathOperator{\trace}{tr}
 \DeclareMathOperator*{\argmax}{arg\,max}
 \title{Value of data}
+\author{Stratis Ionnadis \and Thibaut Horel}
 \begin{document}
 \maketitle
 
@@ -160,12 +161,12 @@ The payment received by user $i$ will be denoted by $p_i$.
 The mechanism should satisfy the following conditions:
 \begin{itemize}
     \item \textbf{Normalized} if $s_i = 0$ then $p_i = 0$.
-    \item \textbf{Indiviually rational} $p_i \geq s_ic_i$.
+    \item \textbf{Individually rational} $p_i \geq s_ic_i$.
     \item \textbf{Truthful} $p_i - s_ic_i \geq p_i' - s_i'c_i$, where $p_i'$
         and $s_i'$ are the payment and allocation of user $i$ had he reported
         a cost $c_i'$ different from his true cost $c_i$ (keeping the costs
         reported by the other users the same).
-    \item \textbf{Budget feasible} the payments should be withing budget:
+    \item \textbf{Budget feasible} the payments should be within budget:
         \begin{displaymath}
             \sum_{i\in \mathcal{I}} s_ip_i \leq B
         \end{displaymath}
@@ -360,6 +361,15 @@ Finally, we can use the Sylvester's formula to get the result.
         it is clear that our value function is non-decreasing and submodular.
         The positivity follows from a direct application of the spectral
         theorem.
+    \item the matrix which appears in the  value function:
+        \begin{displaymath}
+            I_d + \frac{\Sigma}{\sigma^2}X_S^*X_S
+        \end{displaymath}
+    is also the inverse of the covariance matrix of the ridge regression
+    estimator. In optimal experiment design, it is common to use the
+    determinant of the inverse of the estiamator's covariance matrix as
+    a mesure of the quality of the predicion. Indeed, this directly relates to
+    the inverse of the volume of the confidence ellipsoid.
     \item This value function can be computed up to a fixed decimal precision in
         polynomial time.
 \end{enumerate}
-- 
cgit v1.2.3-70-g09d2

