Variance Computations for 4-way cases
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analysis.tex
84
analysis.tex
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@ -6,38 +6,88 @@ We begin the analysis by showing that with high probability an estimate is appro
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The first step is to show that the expectation of the estimate of a tuple t's membership across all worlds is $\numWorldsSum$.
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\begin{align}
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&\expect \big[\estimate\big]\\
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=&\expect \big[\estExpOne\big]\\
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=&\expect \big[\sum_{\substack{j \in [B],\\
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&\expect{\estimate}\\
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=&\expect{\estExpOne}\\
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=&\expect{\sum_{\substack{j \in [B],\\
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\wVec \in \pw~|~ \sketchHash{i}[\wVec] = j,\\
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\wVec[w']\in \pw~|~ \sketchHash{i}[\wVec[w']] = j} } v_t[\wVec] \cdot s_i[\wVec] \cdot s_i[\wVec[w']]\big]\\
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=&\expect \big[ \sum_{\substack{j \in [B],\\
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\wVec[w']\in \pw~|~ \sketchHash{i}[\wVec[w']] = j} } v_t[\wVec] \cdot s_i[\wVec] \cdot s_i[\wVec[w']]}\\
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=&\multLineExpect\big[\sum_{\substack{j \in [B],\\
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\wVec~|~\sketchHashParam{\wVec}= j,\\
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\wVecPrime~|~\sketchHashParam{\wVecPrime} = j,\\
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\wVec = \wVecPrime}} \wIndParam{\wVec} \cdot \polarFunc{\wVec} \cdot \polarFunc{\wVecPrime} + \nonumber \\
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&\phantom{{}\wIndParam{\wVec}}\sum_{\substack{j \in [B], \\
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\wVec = \wVecPrime}} \kMapParam{\wVec} \cdot \sketchPolarParam{\wVec} \cdot \sketchPolarParam{\wVecPrime} + \nonumber \\
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&\phantom{{}\kMapParam{\wVec}}\sum_{\substack{j \in [B], \\
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\wVec~|~\sketchHashParam{\wVec} = j,\\
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\wVecPrime ~|~ \sketchHashParam{\wVecPrime} = j,\\ \wVec \neq \wVecPrime}} \wIndParam{\wVec} \cdot \polarFunc{\wVec} \cdot\polarFunc{\wVecPrime}\big]\textit{(by linearity of expectation)}\\
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=&\expect \big[ \sum_{\substack{j \in [B],\\
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\wVecPrime ~|~ \sketchHashParam{\wVecPrime} = j,\\ \wVec \neq \wVecPrime}} \kMapParam{\wVec} \cdot \sketchPolarParam{\wVec} \cdot\sketchPolarParam{\wVecPrime}\big]\textit{(by linearity of expectation)}\\
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=&\expect{\sum_{\substack{j \in [B],\\
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\wVec~|~\sketchHashParam{\wVec}= j,\\
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\wVecPrime~|~\sketchHashParam{\wVecPrime} = j,\\
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\wVec = \wVecPrime}} \wIndParam{\wVec} \cdot \polarFunc{\wVec} \cdot \polarFunc{\wVecPrime}\big] \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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&= \sum_{\substack{j \in [B],\\
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\wVec~|~\sketchHashParam{\wVec}= j,\\}} \wIndParam{\wVec}
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=& \sum_{\substack{j \in [B],\\
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\wVec~|~\sketchHashParam{\wVec}= j,\\}} \kMapParam{\wVec}
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\end{align}
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For the next step, we show that the variance of an estimate is small.$$\var{\estimate}$$
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For the next step, we show that the variance of an estimate is small.$$\varParam{\estimate}$$
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\begin{align}
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&=\var{\estExpOne}\\
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&= \big(\estTwo\big)^2\\
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&=\sum_{\substack{
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&=\varParam{\estExpOne}\\
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&= \expect{\big(\estTwo\big)^2}\\
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&=\expect{\sum_{\substack{
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\wVec_1, \wVec_2,\\
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\wVecPrime_1, \wVecPrime_2 \in \pw,\\
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\sketchHashParam{\wVec_1} = \sketchHashParam{\wVecPrime_1},\\
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\sketchHashParam{\wVec_2} = \sketchHashParam{\wVecPrime_2}
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}}\wIndParam{\wVec_1} \cdot \wIndParam{\wVec_2}\cdot\polarFunc{\wVec_1}\cdot\polarFunc{\wVec_2}\cdot\polarFunc{\wVecPrime_1}\cdot\polarFunc{\wVecPrime_2}
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}}\kMapParam{\wVec_1} \cdot \kMapParam{\wVec_2}\cdot\sketchPolarParam{\wVec_1}\cdot\sketchPolarParam{\wVec_2}\cdot\sketchPolarParam{\wVecPrime_1}\cdot\sketchPolarParam{\wVecPrime_2} }\label{eq:var-sum-w}
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\end{align}
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Note that four-wise independence is assumed across all four random variables of \eqref{eq:var-sum-w}. Zooming in on the inner products of the $\sketchPolar$ functions,
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\begin{equation}
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\polarProdEq \label{eq:polar-product}
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\end{equation}
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note that all four random variables in \eqref{eq:polar-product} take their values from the same set of possible worlds $\pw$. Thus, there are four possible patterns of distribution between the $\wVec$ variables, namely:
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\begin{align*}
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&\distPattern{1}:&\cOne\\
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&\distPattern{2}:&\cTwo \textit{*} \\
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&\distPattern{3}:&\cThree \textit{*} \\
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&\distPattern{4}:&\cFour \textit{*}\\
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&\distPattern{5}:&\cFive
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\end{align*}
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$$\text{ }^*\textit{(and all variants of the respective pattern)}$$
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We are interested in those particular cases whose expecation does not equal zero, since an expectation of zero will not add to the summation of \eqref{eq:var-sum-w}. In expectation we have that
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\begin{align}
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&\expect{\sum_{\substack{\elems \\
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\st \cOne}} \polarProdEq} = 1 \label{eq:polar-prod-all}\\
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&\expect{\sum_{\substack{\elems \\
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\st \cTwo}} \polarProdEq} = 1 \label{eq:polar-prod-two-and-two}\\
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&\expect{\sum_{\substack{\elems \\
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\st \cThree}} \polarProdEq} = 0 \nonumber \\
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&\expect{\sum_{\substack{\elems \\
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\st \cFour}} \polarProdEq} = 0 \nonumber \\
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&\expect{\sum_{\substack{\elems \\
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\st \cFive}} \polarProdEq} = 0 \nonumber
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\end{align}
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Only equation \eqref{eq:polar-prod-all} (which maps to $\cOne$) and \eqref{eq:polar-prod-two-and-two} (mapping to $\cTwo$) affect the $\var$ computation.
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Thus, when considering $\distPattern{1}$ the variance results in
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\begin{equation}
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\sum_{\wVec \in \pw} \kMapParam{\wVec}^2
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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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\begin{align*}
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&\vCase{1}:&\cTwo \\
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&\vCase{2}:&\cTwoV{\wOne}{\wTwo}{\wOneP}{\wTwoP}\\
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&\vCase{3}:&\cTwoV{\wOne}{\wTwoP}{\wOneP}{\wTwo}
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\end{align*}
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When considered separately, the variants have the following $\var$.
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\begin{align}
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\cTwo&=\sum_{\wOne \neq \wTwo}\kMapParam{\wOne} \cdot \kMapParam{\wTwo}\\
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\cTwoV{\wOne}{\wTwo}{\wOneP}{\wTwoP}&=\sum_{\substack{\wOne \neq \wOneP,\\
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\wOne = \wTwo,\\
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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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46
macros.tex
46
macros.tex
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@ -12,30 +12,58 @@
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\newcommand{\sketchHash}[1][i]{h_{#1}}
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\newcommand{\sketchHashParam}[1]{\sketchHash\paramBox{#1}}
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\newcommand{\sketchPolar}[1][i]{s_{#1}}
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\newcommand{\polarFunc}[1]{\sketchPolar\paramBox{#1}}
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\newcommand{\sketchPolarParam}[1]{\sketchPolar\paramBox{#1}}
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%
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%TIDB
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%
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\newcommand{\paramBox}[1]{\left[{#1}\right]}
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\newcommand{\bigParamBox}[1]{\big[{#1}\big]}
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\newcommand{\st}{~|~}
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\newcommand{\pw}{W}
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\newcommand{\numWorlds}{2^N}
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\newcommand{\numWorldsP}{\numWorlds \cdot p}
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\newcommand{\numWorldsSum}{\sum_{\wVec \in \pw}\wIndicator{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{\wIndicator}{v_t}
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\newcommand{\wIndicator}[1]{v_{#1}}
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\newcommand{\wIndParam}[1]{\wIndicator{t}\paramBox{#1}}
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%\newcommand{\kMap}{v_t}
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\newcommand{\kMap}[1]{v_{#1}}
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\newcommand{\kMapParam}[1]{\kMap{t}\paramBox{#1}}
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\newcommand{\wVec}[1][w]{\textbf{#1}}
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\newcommand{\wVecPrime}{\wVec[w']}
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%%%%%%%%%%%%%%%%
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%maybe easier this way:
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%WVector Notation
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%%%%%%%%%%%%%%%%
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\newcommand{\w}{\wVec}
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\newcommand{\wOneP}{\wVecPrime_1}
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\newcommand{\wOne}{\wVec_1}
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\newcommand{\wTwoP}{\wVecPrime_2}
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\newcommand{\wTwo}{\wVec_2}
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%%%%%%%%%%%%%%%%
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%4-way cases
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%%%%%%%%%%%%%%%%
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\newcommand{\polarProdEq}{\sketchPolarParam{\wVec_1}\cdot\sketchPolarParam{\wVec_2}\cdot\sketchPolarParam{\wVecPrime_1}\cdot\sketchPolarParam{\wVecPrime_2}}
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\newcommand{\elems}{\wOne, \wOneP, \wTwo, \wTwoP \in \pw}
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\newcommand{\lab}[1]{\textit{#1}}
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\newcommand{\distPattern}[1]{\lab{Pattern}{\textit{ {#1}}}}
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\newcommand{\vCase}[1]{\lab{Variant }{#1}}
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\newcommand{\cOne}{\wOne = \wOneP = \wTwo = \wTwoP}
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\newcommand{\cTwo}{\wOne = \wOneP \neq \wTwo = \wTwoP}
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\newcommand{\cThree}{\wOne = \wOneP = \wTwo \neq \wTwoP}
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\newcommand{\cFour}{\wOne = \wOneP \neq \wTwo \neq \wTwoP}
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\newcommand{\cFive}{\wOne \neq \wOneP \neq \wTwo \neq \wTwoP}
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\newcommand{\cTwoV}[4]{{#1} = {#2} \neq {#3} = {#4}}
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\newcommand{\relation}{R}
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\newcommand{\expect}{\mathop{\mathbb{E}}}
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\newcommand{\var}[1]{Var\big[{#1}\big]}
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\newcommand{\expect}[1]{\mathop{\mathbb{E}}\bigParamBox{#1}}
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\newcommand{\multLineExpect}{\mathop{\mathbb{E}}}
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\newcommand{\var}{Var}
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\newcommand{\varParam}[1]{Var\bigParamBox{#1}}
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\newcommand{\polarFuncSum}[1][]{\sum_{\substack{\wVecPrime ~|~ \\
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\sketchHash\left[\wVecPrime\right] = j\\
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{#1}}}\polarFunc{\wVecPrime}}
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{#1}}}\sketchPolarParam{\wVecPrime}}
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\newcommand{\estimate}{\sum_{j \in \sketchCols} \sketchIj \cdot \polarFuncSum }
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\newcommand{\estExpOne}{\sum_{\substack{j \in \sketchCols, \\
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\wVec \in \pw ~|~\sketchHash\left[\wVec\right] = j}} \wIndicator{t} \cdot\polarFunc{\wVec} \cdot \polarFuncSum}
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\wVec \in \pw ~|~\sketchHash\left[\wVec\right] = j}} \kMap{t} \cdot\sketchPolarParam{\wVec} \cdot \polarFuncSum}
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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[w']\in \pw~|~ \sketchHash{[\wVec[w']]} = j} } v_t[\wVec] \cdot s_i[\wVec] \cdot s_i[\wVec[w']]}
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1
main.tex
1
main.tex
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@ -9,6 +9,7 @@
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\usepackage{amsthm}
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\usepackage{mathtools}
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\usepackage{etoolbox}
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\usepackage{xstring} %for conditionals in \newcommand
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\usepackage{stmaryrd}
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\usepackage[normalem]{ulem}
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@ -4,7 +4,7 @@
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The following notation is used to reason about the sketching of world membership for a given tuple. We denote the set of all possible worlds as $\pw$. A given sketch $\sketch$ can be viewed as an $\sketchRows \times \sketchCols$ matrix, i.e. a matrix with $\sketchRows$ rows and $\sketchCols$ columns. Each row of $\sketch$ is an estimation of the of $\kDom$ frequency for the given tuple represented by $\sketch$ across all possible worlds.
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To facilitate binning the $\kDom$ values for a given world $\wVec$, each row has two pairwise independent hash functions $\sketchHash{i}:\pw \to [B]$ and $\sketchPolar{i}:\pw \to \{-1,1\}$, where all functions are independent of one another. Finally, the function $\wIndicator{t}$ defined as $\wIndicator{t} : \{0, 1\}^\numTup \rightarrow \kDom$ is used to determine the tuple's $\kDom$ annotation for a given world.
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To facilitate binning the $\kDom$ values for a given world $\wVec$, each row has two pairwise independent hash functions $\sketchHash{i}:\pw \to [B]$ and $\sketchPolar{i}:\pw \to \{-1,1\}$, where all functions are independent of one another. Finally, the function $\kMap{t}$ defined as $\kMap{t} : \{0, 1\}^\numTup \rightarrow \kDom$ is used to determine the tuple's $\kDom$ annotation for a given world.
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\AR{I do not like this notation. I prefer vectors being typeset in bold, i.e. $\mathbf{w}$. $\wVec$ is good for writing on the board but it is more standard to bold vectors in linear algebra. Also the $\kDom$ values are not binned by $\sketchHash{i}$ but the actual $\wVec$s are.}
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\AH{Done.}
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@ -14,11 +14,11 @@ To facilitate binning the $\kDom$ values for a given world $\wVec$, each row has
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\AR{While in general I'm a fan of using English to define things, one of the exceptions if when you are defining a function. It would be better to explicit state that $\sketchHash{i}:W\to [B]$ and $\sketchPolar{i}:W\to \{-1,1\}$. Of course for these definitions you need to define $W$ upfront.}
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\AH{Done}
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When a world value $\wVec$'s $\kDom$ value is updated, it's $\kDom$ value is first retrieved via $\wIndicator{t}$ and then multiplied by the output of the $i^{th}$ row's polarity function $\sketchPolar{i}$. The resulting computation is then added to the current value contained in the bin mapping. Formally:
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$$\sketch[\sketchHash{i}(\wVec)] ~+=~ \sketchPolar{i}(\wVec) \times \wIndicator{t}(\wVec)$$
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When a world value $\wVec$'s $\kDom$ value is updated, it's $\kDom$ value is first retrieved via $\kMap{t}$ and then multiplied by the output of the $i^{th}$ row's polarity function $\sketchPolar{i}$. The resulting computation is then added to the current value contained in the bin mapping. Formally:
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$$\sketch[\sketchHash{i}(\wVec)] ~+=~ \sketchPolar{i}(\wVec) \times \kMap{t}(\wVec)$$
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When referring to Tuple Independent Databases (TIDB), a database $\relation$ contains $\numTup$ tuples, with $\numWorlds$ possible worlds $\pw$. $\pw$ is denoted as $\{0, 1\}^\numTup$, where a specific world $\wVec$ is defined as $\wVec \in \{0, 1\}^\numTup$.
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\AR{I'm fine $\wIndicator{t}$ defined as a function instead of a vector in $\kDom^W$ but I'm not sure if one would be easier than the other to write arguments. I guess we can re-consider this later as it is defined as a macro.}
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\AR{I'm fine $\kMap{t}$ defined as a function instead of a vector in $\kDom^W$ but I'm not sure if one would be easier than the other to write arguments. I guess we can re-consider this later as it is defined as a macro.}
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\AH{I too am unsure of which way would be best to go on this. I think originally we had proposed to define $\wVec$ as a mapping to the tuple's $\kDom$ annotation.}
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