Up to Var[estimate] completed
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analysis.tex
37
analysis.tex
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@ -23,8 +23,8 @@ The first step is to show that the expectation of the estimate of a tuple t's me
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\wVecPrime~|~\sketchHashParam{\wVecPrime} = j,\\
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\wVecPrime~|~\sketchHashParam{\wVecPrime} = j,\\
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\wVec = \wVecPrime}} \kMapParam{\wVec} \cdot \sketchPolarParam{\wVec} \cdot \sketchPolarParam{\wVecPrime}} \nonumber \\
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\wVec = \wVecPrime}} \kMapParam{\wVec} \cdot \sketchPolarParam{\wVec} \cdot \sketchPolarParam{\wVecPrime}} \nonumber \\
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&\phantom{{}\big[}\textit{(by uniform distribution in the second summation)}\\
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&\phantom{{}\big[}\textit{(by uniform distribution in the second summation)}\\
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=& \sum_{\substack{j \in [B],\\
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=& \estExp \sum_{\substack{j \in [B],\\
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\wVec~|~\sketchHashParam{\wVec}= j,\\}} \kMapParam{\wVec}
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\wVec~|~\sketchHashParam{\wVec}= j,\\}} \kMapParam{\wVec} \label{eq:estExpect}
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\end{align}
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\end{align}
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For the next step, we show that the variance of an estimate is small.$$\varParam{\estimate}$$
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For the next step, we show that the variance of an estimate is small.$$\varParam{\estimate}$$
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@ -72,7 +72,7 @@ Only equation \eqref{eq:polar-prod-all} (which maps to $\cOne$) and \eqref{eq:po
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Thus, when considering $\distPattern{1}$ the variance results in
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Thus, when considering $\distPattern{1}$ the variance results in
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\begin{equation}
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\begin{equation}
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\sum_{\wVec \in \pw} \kMapParam{\wVec}^2
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\distPatOne\label{eq:distPatOne}
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\end{equation}
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\end{equation}
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For the distribution pattern $\cTwo$, we have three variants to consider.
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For the distribution pattern $\cTwo$, we have three variants to consider.
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@ -83,12 +83,33 @@ For the distribution pattern $\cTwo$, we have three variants to consider.
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\end{align*}
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\end{align*}
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When considered separately, the variants have the following $\var$.
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When considered separately, the variants have the following $\var$.
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\begin{align}
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\begin{align}
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\cTwo&=\sum_{\wOne \neq \wTwo}\kMapParam{\wOne} \cdot \kMapParam{\wTwo}\\
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\cTwo&= \variantOne \label{eq:variantOne}\\
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\cTwoV{\wOne}{\wTwo}{\wOneP}{\wTwoP}&=\sum_{\substack{\wOne \neq \wOneP,\\
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\cTwoV{\wOne}{\wTwo}{\wOneP}{\wTwoP}&=\variantTwo \label{eq:variantTwo}\\
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\wOne = \wTwo,\\
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\cTwoV{\wOne}{\wTwoP}{\wOneP}{\wTwo}&=\variantThree\label{eq:variantThree}
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\sketchHashParam{\wOne} = \sketchHashParam{\wOneP}}} \big| \sketchHashParam{\wOne}\neq \sketchHashParam{\wOneP} \big|\cdot \kMapParam{\wOne}\cdot \kMapParam{\wTwo}\\
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\cTwoV{\wOne}{\wTwoP}{\wOneP}{\wTwo}&=\sum_{\wOne \neq \wTwo} \kMapParam{\wOne} \cdot \kMapParam{\wTwo}
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\end{align}
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\end{align}
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Note that at the start of the analysis of $\var$, the second term (expectation \eqref{eq:estExpect} squared) of the $\var$ calculation was not considered. This is because it is cancelled out by \eqref{eq:distPatOne} and \eqref{eq:variantOne}.
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\begin{equation*}
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\big(\estExp\big)^2 = \distPatOne + \variantOne
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\end{equation*}
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With only \eqref{eq:variantTwo} and \eqref{eq:variantThree} remaining, we have
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\begin{multline*}
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\varParam{\estimate} = \\
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\variantTwo ~+ \\
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\variantThree
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\end{multline*}
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Converting terms into their space requirements yields
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\begin{align}
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&\variantTwo \Rightarrow\numWorldsP \cdot \frac{\numWorlds - 1}{\sketchCols}\label{eq:spaceOne}\\
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&\variantThree \Rightarrow \numWorldsP \cdot \frac{\numWorldsP - 1}{\sketchCols}\label{eq:spaceTwo}
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\end{align}
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\eqref{eq:spaceOne} and \eqref{eq:spaceTwo} further reduce to
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\begin{equation}
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\frac{2^{2N}(\prob + \prob^2)}{\sketchCols} - \numWorlds(\frac{\prob}{\sketchCols} + \prob)
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\end{equation}
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16
macros.tex
16
macros.tex
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@ -21,7 +21,8 @@
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\newcommand{\st}{~|~}
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\newcommand{\st}{~|~}
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\newcommand{\pw}{W}
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\newcommand{\pw}{W}
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\newcommand{\numWorlds}{2^N}
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\newcommand{\numWorlds}{2^N}
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\newcommand{\numWorldsP}{\numWorlds \cdot p}
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\newcommand{\prob}{p}
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\newcommand{\numWorldsP}{\numWorlds\prob}
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\newcommand{\numWorldsSum}{\sum_{\wVec \in \pw}\kMap{t}[\wVec]}
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\newcommand{\numWorldsSum}{\sum_{\wVec \in \pw}\kMap{t}[\wVec]}
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\newcommand{\numTup}{N}
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\newcommand{\numTup}{N}
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%\newcommand{\kMap}{v_t}
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%\newcommand{\kMap}{v_t}
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@ -58,6 +59,9 @@
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\newcommand{\multLineExpect}{\mathop{\mathbb{E}}}
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\newcommand{\multLineExpect}{\mathop{\mathbb{E}}}
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\newcommand{\var}{Var}
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\newcommand{\var}{Var}
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\newcommand{\varParam}[1]{Var\bigParamBox{#1}}
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\newcommand{\varParam}[1]{Var\bigParamBox{#1}}
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%%%%%%%%%%%%%%%%%
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% Equations
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%%%%%%%%%%%%%%%%%
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\newcommand{\polarFuncSum}[1][]{\sum_{\substack{\wVecPrime ~|~ \\
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\newcommand{\polarFuncSum}[1][]{\sum_{\substack{\wVecPrime ~|~ \\
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\sketchHash\left[\wVecPrime\right] = j\\
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\sketchHash\left[\wVecPrime\right] = j\\
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{#1}}}\sketchPolarParam{\wVecPrime}}
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{#1}}}\sketchPolarParam{\wVecPrime}}
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@ -67,6 +71,14 @@
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\newcommand{\estTwo}{\sum_{\substack{j \in [B],\\
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\newcommand{\estTwo}{\sum_{\substack{j \in [B],\\
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\wVec \in \pw~|~ \sketchHash{[\wVec]} = j,\\
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\wVec \in \pw~|~ \sketchHash{[\wVec]} = j,\\
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\wVec[w']\in \pw~|~ \sketchHash{[\wVec[w']]} = j} } v_t[\wVec] \cdot s_i[\wVec] \cdot s_i[\wVec[w']]}
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\wVec[w']\in \pw~|~ \sketchHash{[\wVec[w']]} = j} } v_t[\wVec] \cdot s_i[\wVec] \cdot s_i[\wVec[w']]}
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\newcommand{\estExp}{ \sum_{\substack{j \in [B],\\
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\wVec~|~\sketchHashParam{\wVec}= j,\\}} \kMapParam{\wVec}}
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\newcommand{\distPatOne}{\sum_{\wVec \in \pw} \kMapParam{\wVec}^2}
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\newcommand{\variantOne}{\sum_{\wOne \neq \wTwo}\kMapParam{\wOne} \cdot \kMapParam{\wTwo}}
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\newcommand{\variantTwo}{\sum_{\substack{\wVec \neq \wVecPrime,\\
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\sketchHashParam{\wVec} = \sketchHashParam{\wVecPrime}}} \big| \sketchHashParam{\wVec} = \sketchHashParam{\wVecPrime} \big|\cdot \kMapParam{\wVec}^2}
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\newcommand{\variantThree}{\sum_{\substack{\wOne \neq \wTwo,\\
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\sketchHashParam{\wOne} = \sketchHashParam{\wTwo}}} \kMapParam{\wOne} \cdot \kMapParam{\wTwo}}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% COMMENTS
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% COMMENTS
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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@ -209,7 +221,7 @@
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\newcommand{\bestG}[1]{BestGuess(#1)}
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\newcommand{\bestG}[1]{BestGuess(#1)}
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\newcommand{\uadb}{\db_{UA}}
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\newcommand{\uadb}{\db_{UA}}
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\newcommand{\bgdb}{D_{bg}}
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\newcommand{\bgdb}{D_{bg}}
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\newcommand{\prob}{P}
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%\newcommand{\prob}{P}
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\newcommand{\probOf}[1]{\prob(#1)}
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\newcommand{\probOf}[1]{\prob(#1)}
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\newcommand{\pTupMult}[2]{\prob({#1}, \geq {#2})}
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\newcommand{\pTupMult}[2]{\prob({#1}, \geq {#2})}
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\newcommand{\rowsmr}{\mathcal{K}_{uncert}}
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\newcommand{\rowsmr}{\mathcal{K}_{uncert}}
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