From 95cb5ebd324391bd8c6bb88fa0056f3439ce8425 Mon Sep 17 00:00:00 2001 From: robertchristensen Date: Mon, 15 Sep 2014 15:14:19 -0600 Subject: [PATCH] made suggested changes in lecture 2 notes. --- 2-rectangles/L2.tex | 81 ++++++++++++++++++++---------- 2-rectangles/figs/ge_matrix_2.ipe | 6 +-- 2-rectangles/figs/ge_matrix_2.pdf | Bin 42807 -> 18964 bytes 3 files changed, 56 insertions(+), 31 deletions(-) diff --git a/2-rectangles/L2.tex b/2-rectangles/L2.tex index 9085e70..4ffe186 100644 --- a/2-rectangles/L2.tex +++ b/2-rectangles/L2.tex @@ -133,10 +133,11 @@ We will define the function $\gt(x,y)$ to be \begin{marginfigure} \centering \includegraphics{figs/ge_matrix_2} +\caption{Example matrix for $\gt(x,y)$ when $|x| = |y| = 5$ with fooling set elements of size 2 drawn.} \end{marginfigure} -\sunote{Better to have just the figure as a PDF and use the caption for the - text. This allows you to use the \gt command for the function name, to get a - consistent look.} +%\sunote{Better to have just the figure as a PDF and use the caption for the +% text. This allows you to use the \gt command for the function name, to get a +% consistent look.} According to the definition of $\gt(x,y)$ the lower triangle @@ -159,26 +160,28 @@ follows directly. % recording ~20:20 \paragraph{Example: \disjoint} -\sunote{Didn't we define \disjoint in the last lecture notes?} - -We define $x$ and $y$ to be a vector of bits -which define set membership. - -\begin{align*} - & x,y \in [0,1]^n \\ -\text{and } & x_i \Leftrightarrow i \in S_x \\ - & y_i \Leftrightarrow i \in S_y -\end{align*} - -the function $\disjoint(x,y)$ is defined as follows - -\begin{equation*} -\disjoint(x,y) = \left\{ - \begin{array}{l l} - 1 & \quad \text{if $(x,y)$ are disjoint: } \left\{ \forall i, x_i \land y_i = 0 \right\} \\ - 0 & \quad \text{otherwise} - \end{array} \right. -\end{equation*} +%\sunote{Didn't we define \disjoint in the last lecture notes?} +%\ronote{In response, we did define \disjoint, but in in lecture 1 +% we did not reason about the definition with as much detail.} +% +%We define $x$ and $y$ to be a vector of bits +%which define set membership. +% +%\begin{align*} +% & x,y \in [0,1]^n \\ +%\text{and } & x_i \Leftrightarrow i \in S_x \\ +% & y_i \Leftrightarrow i \in S_y +%\end{align*} +% +%the function $\disjoint(x,y)$ is defined as follows +% +%\begin{equation*} +%\disjoint(x,y) = \left\{ +% \begin{array}{l l} +% 1 & \quad \text{if $(x,y)$ are disjoint: } \left\{ \forall i, x_i \land y_i = 0 \right\} \\ +% 0 & \quad \text{otherwise} +% \end{array} \right. +%\end{equation*} \begin{marginfigure} @@ -191,9 +194,9 @@ $1$ & $0$ & $1$ & $1$ & $0$ & $0$ \\ $1$ & $1$ & $1$ & $0$ & $0$ & $0$ \\ \end{tabular} \\ \vspace{1em} -Matrix for $DISJ(x,y)$ when $n=4$ showing which +\caption{Matrix for $DISJ(x,y)$ when $n=4$ showing which combinations will result in a $1$ or -a $0$. +a $0$.} \end{marginfigure} Let us observe one of the items in the @@ -201,7 +204,7 @@ fooling set using our example. Using the pair $(x_1, y_1) = (01, 10)$ and $(x_2, y_2) = (10, 01)$ we can see this is part of the fooling set. This set is interesting because $(x_1,y_1) = (\overline{x_2}, \overline{y_2})$. -In fact, the fooling st for $DISJ(x,y)$ are the bit patterns with the bits flipped; +In fact, the fooling st for $\disjoint(x,y)$ are the bit patterns with the bits flipped; it is the set of all $A$ and $\overline{A}$. % recording ~33:20 @@ -348,7 +351,31 @@ rest we do not care about. We are choosing some subset of the $r$ bits to be fixed to $1$, the rest we can pick to be anything else. Doing this results in $2^n - 2^{n-r}$ -\sunote{See the note that I provided in the discussion group and insert it here} +% \sunote{See the note that I provided in the discussion group and insert it here} +The statement that the rank of the IP matrix over GF($2$)$=2^n$ is +not true. A proof of this can be found in Sherstov's +notes\footnote{http://www.cs.ucla.edu/~sherstov/teaching2012-winter/docs/lecture03.pdf}, +which we will briefly discuss here. + +The first observation is that the rank argument works over any field. So +we can choose the field we wish to evaluate matrix rank over. In particular, +the rank over the reals is larger than the rank over any other field. + +Recall that we would like to show that the inner product has deterministic +communication complexity $n$ via the rank method. So we need to show that +the rank of the induced matrix (where $M_{ij}=<\!\!x_i,y_j\!\!>$) is large: +remember that IP is a function from $n$ bit string to $\{0, 1\}$. + +Firstly, consider the auxiliary matrix $M'=2M-1$ (where $1$ is the all ons matrix). +All this does is change $0$ to $-1$ and $1$ to $1$, which is more convenient +to work with. It is a nonsingular transformation, so $\text{rank}(M) \ge \text{rank}(M')-1$. + +Now it is easy to see that $M'_{\text{IP}}$ on one-bit strings is the Hadamard +matrix\footnote{$\begin{bmatrix} 1 & 1 \\ 1 & -1 \end{bmatrix}$} and the IP matrix for $n$ bit strings is merely the $n$-th order +version of this (or the $n$th tensor power of this matrix). But this matrix +has full rank: in particular its rank is $2^n$. Therefor the rank of the IP matrix +is at least $2^n-1$. + %recording ~1:00:00 \section{Randomized Algorithms} diff --git a/2-rectangles/figs/ge_matrix_2.ipe b/2-rectangles/figs/ge_matrix_2.ipe index f81a747..1d0f5e0 100644 --- a/2-rectangles/figs/ge_matrix_2.ipe +++ b/2-rectangles/figs/ge_matrix_2.ipe @@ -1,7 +1,7 @@ - + \usepackage{amsmath} @@ -233,9 +233,7 @@ h -Example matrix for $gt(x,y)$ when \\$|x| = |y| = 5$ with fooling set elements of size 2 drawn. -\\ \\ -$+$ \begin{bmatrix} 1 & 0 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 \\ diff --git a/2-rectangles/figs/ge_matrix_2.pdf b/2-rectangles/figs/ge_matrix_2.pdf index 13407eeaaaba4a359f439f8ab4a76ce592b46cd6..0dfd92a15eab58e8118a790385cbbd4141ab87bd 100644 GIT binary patch delta 9527 zcmZvB1yCH!vULa=++7xTUsxotxVyW%yTbwjg6rb0!Gi?|?(R--hoC_ML0W6*Zu3g z%PdRrl%c>FJrOIaAdwQ}3Zrn7Tgz1|SCznO(us$=bt)3&6|C!S>C8^G~z5dh-g zNdaV0L*hITfjQ?(yjYGB66_zyixbU=GMFo2cZE&>+sotl-F5D^t zOx~g56To~L6j@)lLSn&wl}6ib4mR(=vVpEcAbjPf+Z-%LU(ow^Uv6g^M1~PRhz!6) 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