ODIn/paper-BagRelationalPDBsAreHard
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Expectation calculations added.

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Aaron Huber 2019-09-03 10:14:40 -04:00
parent ebe78adb3e
commit 584658c281
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16
combining.tex
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@ -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
\[

2
macros.tex
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@ -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
%%%%%%%%%%%%%%%%