Changes per Atri's suggestions: Sec. 2-expectation
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analysis.tex
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analysis.tex
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% -*- root: main.tex -*-
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\section{Analysis}
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\label{sec:analysis}
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We begin the analysis by showing that with high probability an estimate is approximately $\numWorldsP$, where $p$ is the probability measure for a given TIPD. Note that
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We begin the analysis by showing that with high probability an estimate is approximately $\numWorldsP$, where $p$ is a tuple's probability measure for a given TIPD. Note that
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\begin{equation}
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\numWorldsP = \numWorldsSum\label{eq:mu}.
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\end{equation}
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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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We begin by making the claim that the expectation of the estimate of a tuple t's membership across all worlds is $\numWorldsSum$, formally
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\begin{equation}
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\expect{\sum_{\wVec \in \pw} \sketchJParam{\sketchHashParam{\wVec}} \cdot \sketchPolarParam{\wVec}} = \sum_{\wVec \in \pw}\kMapParam{\wVec}\label{eq:allWorlds-est}.
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\end{equation}
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To verify this claim, we argue that the expectation of the estimate of a tuple's appearance in single world is it's annotation, i.e.
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\begin{equation}
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\expect{\sketchJParam{\sketchHashParam{\wVec}}\cdot \sketchPolarParam{\wVec}} = \kMapParam{\wVec} \label{eq:single-est}.
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\end{equation}
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\AR{While the analysis below is correct, the way it is stated it seems to `come out of the blue.' I would recommend that you re-structure the argument below as follows. First argue that $\expect{\sketch[i][\sketchHash[\wVec]]\cdot s_i[\wVec]}=v_t[\wVec]$. From this the claim below just follows by linearity of expectation but this result is a good thing for the reader to realize. Also instead of summing over $j\in [B],\wVec|h_i[\wVec]=j,\wVec'|h_i[\wVec']=j$ it would be better to just write it as sum over all $\wVec,\wVec'\in W\text{ s.t. }h_i[\wVec]=h_i[\wVec']$-- the latter is bit more compact and it is easier to comprehend as well.}
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\begin{align}
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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']]}\\
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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}} \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}} \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}} \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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=& \estExp \label{eq:estExpect}
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\end{align}
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\AH{Proof changed as suggested above. I aired on the verbose side for the sake of clarity.}
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For a given $\wVec \in \pw$, substituting definitions we have
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\begin{align*}
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&\expect{\sketchJParam{\sketchHashParam{\wVec}} \cdot \sketchPolarParam{\wVec}} = \nonumber\\
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&\phantom{{}\sketchJParam{\sketchHashParam{\wVec}}}\expect{\big(\sum_{\substack{\wVecPrime \in \pw \st \\
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\sketchHashParam{\wVecPrime} = \sketchHashParam{\wVec}}}\kMapParam{\wVecPrime} \cdot \sketchPolarParam{\wVecPrime}\big) \cdot \sketchPolarParam{\wVec} }.
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\end{align*}
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Since $\wVec \in \pw$, we know that for $j \in [|\pw|], \exists \wVecPrime_j \st \wVecPrime_j = \wVec$. This yields
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\begin{multline*}
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\mathbb{E}\big[\kMapParam{\wVecPrime_0}\cdot \sketchPolarParam{\wVecPrime_0} + \cdots \\
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+\kMapParam{\wVecPrime_j}\cdot \sketchPolarParam{\wVecPrime_j}\cdot \sketchPolarParam{\wVecPrime_j}+ \cdots \\
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+ \kMapParam{\wVecPrime_n}\sketchPolarParam{\wVecPrime_n}\big]
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\end{multline*}
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Due to the uniformity of $\sketchPolar$, we have
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\begin{equation*}
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= \kMapParam{\wVec},
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\end{equation*}
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thus verifying \eqref{eq:single-est}.
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We can now take \eqref{eq:single-est}, substitute it in for \eqref{eq:allWorlds-est} and show by linearity of expectation that \eqref{eq:allWorlds-est} holds.
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\begin{align*}
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&\expect{\sum_{\wVec \in \pw} \sketchJParam{\sketchHashParam{\wVec}} \cdot \sketchPolarParam{\wVec}} \\
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&= \expect{\sum_{\wVecPrime \in \pw}\kMapParam{\wVecPrime} \cdot \sketchPolarParam{\wVecPrime} \cdot \sum_{\substack{\wVec \in \pw \st \\
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\sketchHashParam{\wVecPrime} = \sketchHashParam{\wVec}}}\sketchPolarParam{\wVec}}\\
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&= \sum_{\wVec \in \pw} \expect{\left( \sum_{\substack{\wVecPrime \in \pw \st \\
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\sketchHashParam{\wVecPrime} = \sketchHashParam{\wVec}}}\kMapParam{\wVecPrime}\cdot\sketchPolarParam{\wVecPrime}\right) \cdot \sketchPolarParam{\wVec}}\\
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&= \sum_{\wVec \in \pw}\kMapParam{\wVec}.
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\end{align*}
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%\begin{align}
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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']]}\\
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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}} \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}} \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}} \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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%=& \estExp \label{eq:estExpect}
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%\end{align}
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\AR{A general comment: The last display equation should have a period at the end. The idea is that display equations are considered part of a sentence and every sentence should end with a period.}
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\AH{Thank you for clarifying this, as I have always wondered what the convention was for display equations. Hopefully, I haven't missed any end display equations, and have them all fixed properly.}
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For the next step, we show that the variance of an estimate is small.$$\varParam{\estimate}$$
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@ -7,6 +7,7 @@
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%
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\newcommand{\sketch}{\mathcal{S}}
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\newcommand{\sketchIj}{\sketch[i][j]}
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\newcommand{\sketchJParam}[1]{\sketch\paramBox{i}\paramBox{#1}}
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\newcommand{\sketchCols}{B}
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\newcommand{\sketchRows}{M}
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\newcommand{\sketchHash}[1][i]{h_{#1}}
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