ODIn/paper-BagRelationalPDBsAreHard
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More fixes per Atri's suggestions

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Aaron Huber 2019-08-07 11:57:07 -04:00
parent 179a794592
commit 0af40b80a1
2 changed files with 23 additions and 0 deletions
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analysis.tex
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@ -47,8 +47,31 @@ For a given $\wVec \in \pw$, substituting definitions we have
\expect{\sum\limits_{\substack{\wVecPrime, \wVec \in \pw \st \\
\sketchHashParam{\wVecPrime} = \sketchHashParam{\wVec} \\
\wVecPrime \neq \wVec}}\kMapParam{\wVecPrime}\sketchPolarParam{\wVecPrime}\sketchPolarParam{\wVec}}\label{eq:step-three}\\
=&~\expect{\sum\limits_{\substack{\wVecPrime \in \pw \st \\
\sketchHashParam{\wVecPrime} = \sketchHashParam{\wVec},\\
\wVecPrime = \wVec}}\kMapParam{\wVecPrime}} \cdot \expect{\sum\limits_{\substack{\wVecPrime \in \pw \st \\
\sketchHashParam{\wVecPrime} = \sketchHashParam{\wVec},\\
\wVecPrime = \wVec}}\sketchPolarParam{\wVecPrime}^2} + \nonumber\\
&\qquad\expect{\sum\limits_{\substack{\wVecPrime, \wVec \in \pw \st \\
\sketchHashParam{\wVecPrime} = \sketchHashParam{\wVec} \\
\wVecPrime \neq \wVec}}\kMapParam{\wVecPrime}}\expect{\sum\limits_{\substack{\wVecPrime, \wVec \in \pw \st \\
\sketchHashParam{\wVecPrime} = \sketchHashParam{\wVec} \\
\wVecPrime \neq \wVec}}\sketchPolarParam{\wVecPrime}\sketchPolarParam{\wVec}}\label{eq:step-three-a}\\
=&~{\sum\limits_{\substack{\wVecPrime \in \pw \st \\
\sketchHashParam{\wVecPrime} = \sketchHashParam{\wVec},\\
\wVecPrime = \wVec}}\kMapParam{\wVecPrime}} \cdot \expect{\sum\limits_{\substack{\wVecPrime \in \pw \st \\
\sketchHashParam{\wVecPrime} = \sketchHashParam{\wVec},\\
\wVecPrime = \wVec}}\sketchPolarParam{\wVecPrime}^2} + \nonumber\\
&\qquad\sum\limits_{\substack{\wVecPrime, \wVec \in \pw \st \\
\sketchHashParam{\wVecPrime} = \sketchHashParam{\wVec} \\
\wVecPrime \neq \wVec}}\kMapParam{\wVecPrime}\expect{\sum\limits_{\substack{\wVecPrime, \wVec \in \pw \st \\
\sketchHashParam{\wVecPrime} = \sketchHashParam{\wVec} \\
\wVecPrime \neq \wVec}}\sketchPolarParam{\wVecPrime}\sketchPolarParam{\wVec}}\label{eq:step-three-b}\\
%\] from which the last term evaluates to $0$ and we have
%\[
=&~\sum\limits_{\substack{\wVecPrime \in \pw \st \\
\sketchHashParam{\wVecPrime} = \sketchHashParam{\wVec},\\
\wVecPrime = \wVec}}\kMapParam{\wVecPrime} \label{eq:step-three-c}\\
=&~\expect{\sum\limits_{\substack{\wVecPrime\in \pw \st \\
\sketchHashParam{\wVecPrime} = \sketchHashParam{\wVec},\\
\wVecPrime = \wVec}}\kMapParam{\wVecPrime}\sketchPolarParam{\wVecPrime}^2}\label{eq:step-four}\\

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