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Expecations for Variant 1 corrected; Other variants need correction still

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Aaron Huber 2019-08-29 11:42:00 -04:00
parent ec80c510dd
commit ac1e0048e6
1 changed files with 29 additions and 8 deletions
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combining.tex
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@ -20,18 +20,39 @@ There are various ways we might 'consider' the multiplication of sketches. Firs
&\est{3} = \sum_{j \in \sketchCols}\sCom{1}{j} \cdot \sCom{2}{j}.
\end{align*}
Calculating the expectation for $\est{1}$ evaluates to
Calculating the expectation for $\est{1}$ for a product of two terms evaluates to
\begin{align*}
&\expect{\sum_{\wVec \in \pw}\sCom{1}{\hashP{\wVec}}\polP{\wVec} \cdot \sCom{2}{\hashP{\wVec}}\polP{\wVec}}\\
=& \expect{\sum_{\wVec \in \pw}\polP{\wVec}\polP{\wVec}\sum_{\substack{\wVecPrime \in \pw \st\\ \hashP{\wVecPrime} = \hashP{\wVec}}} \genV_1\paramBox{\wVecPrime}\polP{\wVecPrime} \sum_{\substack{\wVecPrime \in \pw \st\\ \hashP{\wVecPrime} = \hashP{\wVec}}}\genV_2\paramBox{\wVecPrime}\polP{\wVecPrime}}\\
=& \mathbb{E}\big[\sum_{\wVec \in \pw}\polP{\wVec}\polP{\wVec}\left(\sum_{\substack{\wVecPrime \in \pw \st\\
\wVecPrime \neq \wVec}} \genV_1\paramBox{\wVecPrime}\polP{\wVecPrime} + \genV_1\paramBox{\wVec}\polP{\wVec}\right)\\
& \qquad \left(\sum_{\substack{\wVecPrime \in \pw \st\\
\wVecPrime \neq \wVec}} \genV_2\paramBox{\wVecPrime}\polP{\wVecPrime} + \genV_2\paramBox{\wVec}\polP{\wVec}\right)\big]\\
=& \expect{\sum_{\wVec \in \pw}\polP{\wVec}\polP{\wVec}\genV_1\paramBox{\wVec}\polP{\wVec}\genV_2\paramBox{\wVec}\polP{\wVec}}\\
=& \genV_1\paramBox{\wVec}\genV_2\paramBox{\wVec}.
=& \mathbb{E}\big[\sum_{\wVec \in \pw}\polP{\wVec}^2\left(\genV_1\paramBox{\wVec}\polP{\wVec} +\sum_{\substack{\wVecPrime \in \pw \st\\
\wVecPrime \neq \wVec}} \genV_1\paramBox{\wVecPrime}\polP{\wVecPrime} \right)\\
& \qquad \left(\genV_2\paramBox{\wVec}\polP{\wVec} +\sum_{\substack{\wVecPrime \in \pw \st\\
\wVecPrime \neq \wVec}} \genV_2\paramBox{\wVecPrime}\polP{\wVecPrime} \right)\big]\\
=& \mathbb{E}\big[\sum_{\wVec \in \pw}\polP{\wVec}^2\big(\gVP{1}{\wVec}\gVP{2}{\wVec}\polP{\wVec}^2 + \\
&\qquad \gVP{1}{\wVec}\polP{\wVec}\sum_{\substack{\wVecPrime \in \pw \st\\
\hashP{\wVecPrime = \wVec}\\
\wVec \neq \wVecPrime}}\gVP{2}{\wVecPrime}\polP{\wVecPrime} + \\
&\qquad \gVP{2}{\wVec}\polP{\wVec}\sum_{\substack{\wVecPrime \in \pw \st\\
\hashP{\wVecPrime = \wVec}\\
\wVec \neq \wVecPrime}}\gVP{1}{\wVecPrime}\polP{\wVecPrime} + \\
&\qquad \sum_{\substack{\wVecPrime \in \pw \st\\
\hashP{\wVecPrime = \wVec}\\
\wVec \neq \wVecPrime}}\gVP{1}{\wVecPrime}\polP{\wVecPrime}\sum_{\substack{\wVecPrime \in \pw \st\\
\hashP{\wVecPrime = \wVec}\\
\wVec \neq \wVecPrime}}\gVP{2}{\wVecPrime}\polP{\wVecPrime}\big)\big]\\%\polP{\wVec}\genV_1\paramBox{\wVec}\polP{\wVec}\genV_2\paramBox{\wVec}\polP{\wVec}}\\
=& \genV_1\paramBox{\wVec}\genV_2\paramBox{\wVec} + \sum_{\substack{\wVecPrime \in \pw \st \\
\wVec \neq \wVecPrime}}\gVP{1}{\wVecPrime}\gVP{2}{\wVecPrime}.
\end{align*}
This result for three sketches in the product is
\begin{align*}
&\sum_{\wVec \in \pw}\gVP{1}{\wVec}\gVP{2}{\wVec}\gVP{3}{\wVec} + \gVP{1}{\wVec}\sum_{\substack{\wVec'' \in \pw \st \\
\hashP{\wVec''} = \hashP{\wVec}\\
\wVec'' \neq \wVec}} \gVP{2}{\wVec''}\gVP{3}{\wVec''}\\
&\qquad +\gVP{2}{\wVec}\sum_{\substack{\wVec'\in \pw \st \\ \hashP{\wVec'} = \hashP{\wVec}\\
\wVec' \neq \wVec}} \gVP{1}{\wVec'}\gVP{3}{\wVec'}\\
&\qquad + \gVP{3}{\wVec}\sum_{\substack{\wVec'\in \pw \st \\ \hashP{\wVec'} = \hashP{\wVec}\\
\wVec' \neq \wVec}} \gVP{1}{\wVec'}\gVP{2}{\wVec'}\\
\end{align*}
This result is consistent for an arbitrary number of sketches in the product.
In expectation $\est{2}$ results in
\begin{align*}
&\expect{\sum_{\wVec \in \pw }\left(\sCom{1}{\hashP{\wVec}} \cdot \sCom{2}{\hashP{\wVec}}\right)\polP{\wVec}}\\