From ac1e0048e61ec958437757c819fa334b65d454b5 Mon Sep 17 00:00:00 2001 From: Aaron Huber Date: Thu, 29 Aug 2019 11:42:00 -0400 Subject: [PATCH] Expecations for Variant 1 corrected; Other variants need correction still --- combining.tex | 37 +++++++++++++++++++++++++++++-------- 1 file changed, 29 insertions(+), 8 deletions(-) diff --git a/combining.tex b/combining.tex index 60c6e7b..44d7815 100644 --- a/combining.tex +++ b/combining.tex @@ -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}}\\