From ba98a0b27361cb0987fb8c911d256dd1c919f269 Mon Sep 17 00:00:00 2001
From: Thibaut Horel <thibaut.horel@gmail.com>
Date: Tue, 7 Feb 2012 16:11:47 -0800
Subject: Analysis of the submodularity with quadratic form theory.

Add a paper from Richardson on LSE.
---
 notes.tex | 67 +++++++++++++++++++++++++++++++++++++++++++++++++++++++--------
 1 file changed, 59 insertions(+), 8 deletions(-)

(limited to 'notes.tex')

diff --git a/notes.tex b/notes.tex
index 25ca121..c84f989 100644
--- a/notes.tex
+++ b/notes.tex
@@ -1,6 +1,7 @@
 \documentclass{article}
 \usepackage[utf8]{inputenc}
 \usepackage{amsmath,amsthm,amsfonts}
+\usepackage{comment}
 \newtheorem{lemma}{Lemma}
 \newcommand{\var}{\mathop{\mathrm{Var}}}
 \newcommand{\condexp}[2]{\mathop{\mathbb{E}}\left[#1|#2\right]}
@@ -9,6 +10,7 @@
 \newcommand{\tr}[1]{#1^*}
 \newcommand{\ip}[2]{\langle #1, #2 \rangle}
 \newcommand{\mse}{\mathop{\mathrm{MSE}}}
+\newcommand{\trace}{\mathop{\mathrm{tr}}}
 \begin{document}
 
 \section{Understanding the recommender system}
@@ -16,7 +18,7 @@
 \subsection{General problem}
 
 We already have a database $D_n$ of $n$ users. For each user $i$ we
-have a set of explanatory variables (features), this is a vector
+have a set of $k$ explanatory variables (features), this is a vector
 $x_i$.
 
 The problem is the following: we are about to start an experiment where for
@@ -202,7 +204,62 @@ inequality becomes:
 which is trivially true (a more direct proof for the one-dimensional
 case is of course possible).
 
-\subparagraph{Useless attempt}
+In order to understand more precisely under which assumptions the
+above inequality could become true, it is convenient to look at it
+from the quadratic form perspective. Indeed this inequality can be
+rewritten as:
+
+\begin{equation}\label{eq-inequality}
+\tr x B x \geq 0 
+\end{equation}
+
+with:
+\begin{align*}
+ B = &\, \left(1+\norm{x_0}^2\right)\left(1+\norm{z}^2\right)\ip{x_0}{z}
+ (x_0\tr z+z\tr x_0)\\
+& -\ip{x_0}{z}^2\Big( \left(1+\norm{x_0}^2\right)z\tr z +  \left(1+\norm{z}^2\right)x_0\tr z\big)
+\end{align*}
+
+This quadratic form is degenerate, its kernel is $x_0^{\bot}\cap
+z^\bot$ which is of dimension $k-2$.
+
+\paragraph{Case when $\norm{x_0}=\norm{z}=1$} In this case, it suffices to study the quadratic form given by matrix
+$B'$ with:
+\begin{displaymath}
+  B' = 2\ip{x_0}{z}(x_0\tr z+z\tr x_0) -\ip{x_0}{z}^2(z\tr z + x_0\tr x_0)
+\end{displaymath}
+
+Writing $a = \ip{x_0}{z}$, the two non-zero eigenvalues are:
+\begin{align*}
+  \lambda_1 & = -2a^3 + 2a^2 + 4a = -2a(a+1)(a-2)\\
+  \lambda_2 & = 2a^3 + 2a^2 - 4a = 2a(a-1)(a+2)
+\end{align*}
+
+which are respectively associated with the eigenvectors:
+\begin{align*}
+  x_1 & = x_0+z\\
+  x_2 & = x_0 - z
+\end{align*}
+
+By the Cauchy-Schwarz inequality, $a\in]-1,1[$, and the two
+eigenvalues are of opposite sign on this interval. Thus inequality
+\eqref{eq-inequality} does not hold for all $x$.
+
+\paragraph{In expectation?} If we assume a prior knowledge on the
+distribution of $x$, writing $\Sigma$ the variance-covariance matrix
+of $x$ and $\mu$ its mean vector, then taking the expectation of
+\eqref{eq-inequality} we get:
+\begin{displaymath}
+  \expt{\tr x B' x} = \trace(B'\Sigma) + \tr\mu B'\mu
+\end{displaymath}
+
+\nocite{shapley,inverse,recommendation,cook,shapleyor,subsetselection11,lse}
+\bibliographystyle{plain}
+\bibliography{notes.bib}
+
+\section*{Appendix}
+
+\paragraph{Previous attempt at taming the submodularity}
 
 The inequality only depends on the projection of $x$ on the plane
 spanned by $x_0$ and $z$. Writing
@@ -237,11 +294,5 @@ This inequality will be true for all $\lambda$ and $\mu$ if and only
 if the quadratic form is positive semidefinite. As its trace is
 positive, this is equivalent to the positiveness of its determinant.
 
-\subparagraph{General derivation}
-
-
-\nocite{shapley,inverse,recommendation,cook,shapleyor,subsetselection11}
-\bibliographystyle{plain}
-\bibliography{notes.bib}
 
 \end{document}
\ No newline at end of file
-- 
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