From 584658c281ef0823e107d1109d770c8828cc5c19 Mon Sep 17 00:00:00 2001 From: Aaron Huber Date: Tue, 3 Sep 2019 10:14:40 -0400 Subject: [PATCH] Expectation calculations added. --- combining.tex | 16 ++++++++++++++++ macros.tex | 2 ++ 2 files changed, 18 insertions(+) diff --git a/combining.tex b/combining.tex index 96c3032..a6c9b5d 100644 --- a/combining.tex +++ b/combining.tex @@ -96,6 +96,22 @@ Following the reversal of the pattern of $\est{2}$, an odd number of sketches wo = & 0. \end{align*} +The case for an odd number of sketches can be reduced to the even case by including the one's vector as an operand in the product, whose sketch is simply $\gIJ = \sum_{\wVec \in \pw}\polP{\wVec}$. The expectation, albeit, does not yield the ground truth, +\begin{align*} +&\expect{\sum_{j \in \sketchCols}\sCom{1}{j}\sCom{2}{j}\sCom{3}{j}\gIJ}\\ +&= \sum_{j \in \sketchCols}\sum_{\substack{\wOne \in \pw \st\\ + \hashP{\wOne} = j}}\gVP{1}{\wOne}\gVP{2}{\wOne}\gVP{3}{\wOne} \\ +&\qquad + \gVP{1}{\wOne}\gVP{3}{\wOne}\sum_{\substack{\wTwo \in \pw \st \\ + \hashP{\wTwo} = j,\\ + \wTwo \neq \wOne}}\gVP{2}{\wTwo}\\ +&\qquad + \gVP{1}{\wOne}\gVP{2}{\wOne}\sum_{\substack{\wThree \in \pw \st\\ + \hashP{\wThree} = j,\\ + \wThree \neq \wOne}}\gVP{3}{\wThree} \\ +&\qquad + \gVP{1}{\wOne}\sum_{\substack{\wTwo \in \pw \st\\ + \hashP{\wTwo} = j,\\ + \wTwo \neq \wOne}}\gVP{2}{\wTwo}\gVP{3}{\wTwo}. +\end{align*} + For the case of multiplication, when assumming independent variables, it is a known result that \[ diff --git a/macros.tex b/macros.tex index 5908bf5..8c20e84 100644 --- a/macros.tex +++ b/macros.tex @@ -69,6 +69,8 @@ \newcommand{\wOne}{\wVec_1} \newcommand{\wTwoP}{\wVecPrime_2} \newcommand{\wTwo}{\wVec_2} +\newcommand{\wThree}{\wVec_3} +\newcommand{\wFour}{\wVec_4} %%%%%%%%%%%%%%%% %4-way cases %%%%%%%%%%%%%%%%