213 lines
17 KiB
TeX
213 lines
17 KiB
TeX
% -*- root: main.tex -*-
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\section{Combining Sketches}
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\label{sec:combining}
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\subsection{Adding Sketches}
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When assuming that the variables are independent, as in the TIDB model, it is a known result that
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\[
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\varParam{X + Y} = \varParam{X} + \varParam{Y}.
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\]
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By \eqref{eq:sub-bounds-final} it immediately follows that adding $n$ base (base meaning a sketch that has not previously been added to another sketch) sketches results in the following variance:
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\[
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3n\left(\frac{\sqrt{\norm{\genV}_\infty}\left(|\pw|\right)}{\sqrt{\norm{\genV}_0\norm{\genV}_1} \epsilon^2} + \frac{1}{\epsilon^2}\right).
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\]
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\subsection{Multiplying Sketches}
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There are various ways we might 'consider' the multiplication of sketches. First, estimates might be multiplied, second, the sketches can be multiplied pointwise, taking then the estimate of the resultant sketch (this is the correct way), and finally we could consider an estimate to be the multiplication of corresponding buckets. Stated formally the above variations are
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\begin{align*}
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&\est{1} = \sum_{\wVec \in \pw}\sCom{1}{\hashP{\wVec}}\polP{\wVec} \cdot \sCom{2}{\hashP{\wVec}}\polP{\wVec}\\
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&\est{2} = \sum_{\wVec \in \pw }\left(\sCom{1}{\hashP{\wVec}} \cdot \sCom{2}{\hashP{\wVec}}\right)\polP{\wVec}\\
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&\est{3} = \sum_{j \in \sketchCols}\sCom{1}{j} \cdot \sCom{2}{j}.
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\end{align*}
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Calculating the expectation for $\est{1}$ for a product of two terms evaluates to
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\begin{align*}
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&\expect{\sum_{\wVec \in \pw}\sCom{1}{\hashP{\wVec}}\polP{\wVec} \cdot \sCom{2}{\hashP{\wVec}}\polP{\wVec}}\\
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=& \expect{\sum_{\wVec \in \pw}\polP{\wVec}\polP{\wVec}\sum_{\substack{\wOne \in \pw \st\\ \hashP{\wOne} = \hashP{\wVec}}} \genV_1\paramBox{\wOne}\polP{\wOne} \sum_{\substack{\wTwo \in \pw \st\\ \hashP{\wTwo} = \hashP{\wVec}}}\genV_2\paramBox{\wTwo}\polP{\wTwo}}\\
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=& \mathbb{E}\big[\sum_{\wVec \in \pw}\polP{\wVec}^2\left(\genV_1\paramBox{\wVec}\polP{\wVec} +\sum_{\substack{\wOne \in \pw \st\\
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\wOne \neq \wVec,\\
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\hashP{\wOne} = \hashP{\wVec}}} \genV_1\paramBox{\wOne}\polP{\wOne} \right)\\
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& \qquad \left(\genV_2\paramBox{\wVec}\polP{\wVec} +\sum_{\substack{\wTwo \in \pw \st\\
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\wTwo\neq \wVec,\\
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\hashP{\wTwo} = \hashP{\wVec}}} \genV_2\paramBox{\wTwo}\polP{\wTwo} \right)\big]\\
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=& \mathbb{E}\big[\sum_{\wVec \in \pw}\polP{\wVec}^2\big(\gVP{1}{\wVec}\gVP{2}{\wVec}\polP{\wVec}^2 + \\
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&\qquad \gVP{1}{\wVec}\polP{\wVec}\sum_{\substack{\wTwo \in \pw \st\\
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\hashP{\wTwo} = \hashP{\wVec},\\
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\wTwo \neq \wVec}}\gVP{2}{\wTwo}\polP{\wTwo} + \\
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&\qquad \gVP{2}{\wVec}\polP{\wVec}\sum_{\substack{\wOne \in \pw \st\\
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\hashP{\wOne} = \hashP{\wVec},\\
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\wOne\neq \wVec}}\gVP{1}{\wVecPrime}\polP{\wVecPrime} + \\
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&\qquad \sum_{\substack{\wOne \in \pw \st\\
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\hashP{\wOne} = \hashP{\wVec},\\
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\wOne\neq \wVec}}\gVP{1}{\wOne}\polP{\wOne}\sum_{\substack{\wTwo \in \pw \st\\
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\hashP{\wTwo} = \hashP{\wVec},\\
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\wTwo \neq \wVec}}\gVP{2}{\wTwo}\polP{\wTwo}\big)\big]\\%\polP{\wVec}\genV_1\paramBox{\wVec}\polP{\wVec}\genV_2\paramBox{\wVec}\polP{\wVec}}\\
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=& \sum_{\wVec \in \pw}\left(\genV_1\paramBox{\wVec}\genV_2\paramBox{\wVec} + \sum_{\substack{\wOne \in \pw \st \\
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\wOne \neq \wVec}}\gVP{1}{\wOne}\gVP{2}{\wOne}\right).
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\end{align*}
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This result for three sketches in the product is
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\begin{align*}
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&\sum_{\wVec \in \pw}\gVP{1}{\wVec}\gVP{2}{\wVec}\gVP{3}{\wVec} + \gVP{1}{\wVec}\sum_{\substack{\wTwo \in \pw \st \\
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\hashP{\wTwo} = \hashP{\wVec}\\
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\wTwo \neq \wVec}} \gVP{2}{\wTwo}\gVP{3}{\wTwo}\\
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&\qquad +\gVP{2}{\wVec}\sum_{\substack{\wOne\in \pw \st \\ \hashP{\wOne} = \hashP{\wVec}\\
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\wOne \neq \wVec}} \gVP{1}{\wOne}\gVP{3}{\wOne}\\
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&\qquad + \gVP{3}{\wVec}\sum_{\substack{\wOne'\in \pw \st \\ \hashP{\wOne} = \hashP{\wVec}\\
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\wOne \neq \wVec}} \gVP{1}{\wOne}\gVP{2}{\wOne}.
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\end{align*}
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In expectation $\est{2}$ results in
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\begin{align*}
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&\expect{\sum_{\wVec \in \pw }\left(\sCom{1}{\hashP{\wVec}} \cdot \sCom{2}{\hashP{\wVec}}\right)\polP{\wVec}}\\
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= &\expect{\sum_{\wVec \in \pw}\left(\sum_{\substack{\wVecPrime \in \pw \st\\
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\hashP{\wVecPrime} = \hashP{\wVec}}} \genV_1\paramBox{\wVecPrime}\polP{\wVecPrime}\sum_{\substack{\wVecPrime \in \pw \st\\
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\hashP{\wVecPrime} = \hashP{\wVec}}}\genV_2\paramBox{\wVecPrime}\polP{\wVecPrime}\right)\polP{\wVec}}\\
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= &\mathbb{E}\big[\sum_{\wVec \in \pw}\polP{\wVec}\left(\sum_{\substack{\wVecPrime \in \pw \st\\ \hashP{\wVecPrime} = \hashP{\wVec}\\\wVecPrime \neq \wVec}}\genV_1\paramBox{\wVecPrime}\polP{\wVecPrime} + \genV_1\paramBox{\wVec}\polP{\wVec}\right)\\
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&\qquad\left(\sum_{\substack{\wVecPrime \in \pw \st\\ \hashP{\wVecPrime} = \hashP{\wVec}\\\wVecPrime \neq \wVec}}\genV_2\paramBox{\wVecPrime}\polP{\wVecPrime} + \genV_2\paramBox{\wVec}\polP{\wVec}\right)\big]\\
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= &\expect{\sum_{\wVec \in \pw}\polP{\wVec}\genV_1\paramBox{\wVec}\polP{\wVec}\genV_2\paramBox{\wVec}\polP{\wVec}}\\
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= & 0.
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\end{align*}
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Note that with an odd number of sketches being multiplied, such as 3, we get the same as for three sketches in $\est{1}$,
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\begin{align*}
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&\sum_{\wVec \in \pw}\gVP{1}{\wVec}\gVP{2}{\wVec}\gVP{3}{\wVec} + \gVP{1}{\wVec}\sum_{\substack{\wTwo \in \pw \st \\
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\hashP{\wTwo} = \hashP{\wVec}\\
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\wTwo \neq \wVec}} \gVP{2}{\wTwo}\gVP{3}{\wTwo}\\
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&\qquad +\gVP{2}{\wVec}\sum_{\substack{\wOne\in \pw \st \\ \hashP{\wOne} = \hashP{\wVec}\\
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\wOne \neq \wVec}} \gVP{1}{\wOne}\gVP{3}{\wOne}\\
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&\qquad + \gVP{3}{\wVec}\sum_{\substack{\wOne'\in \pw \st \\ \hashP{\wOne} = \hashP{\wVec}\\
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\wOne \neq \wVec}} \gVP{1}{\wOne}\gVP{2}{\wOne}.
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\end{align*}
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The even case can be reduced to the odd case by including the one's vector as an operand in the product, whose sketch is simply $\gIJ = \sum_{\wVec \in \pw}\polP{\wVec}$. The expectation then works out to
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\begin{align*}
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&\sum_{\wVec \in \pw}\gVP{1}{\wVec}\gVP{2}{\wVec} + \gVP{1}{\wVec}\sum_{\substack{\wTwo \in \pw \st \\ \wTwo \neq \wVec}}\gVP{2}{\wTwo} + \\
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&\qquad\gVP{2}{\wVec}\sum_{\substack{\wOne \in \pw \st\\\wOne \neq \wVec}}\gVP{1}{\wOne} + \sum_{\substack{\wOne \in \pw \st\\\wOne \neq \wVec}}\gVP{1}{\wOne}\gVP{2}{\wOne}.
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\end{align*}
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For $\est{3}$, multiplying two sketches yields
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\begin{align}
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&\expect{\sum_{j \in \sketchCols}\sCom{1}{j} \cdot \sCom{2}{j}} \nonumber\\
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=&\expect{\sum_{j \in \sketchCols}\left(\sum_{\substack{\wOne \in \pw \st\\\hashP{\wOne} = j}}\gVP{1}{\wOne}\polP{\wOne}\cdot \sum_{\substack{\wTwo \in \pw \st\\\hashP{\wTwo} = j}}\gVP{2}{\wTwo}\polP{\wTwo}\right)} \nonumber\\
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=&\mathbb{E}\big[\sum_{j \in \sketchCols}\sum_{\substack{\wOne, \wTwo \in \pw \st\\\hashP{\wOne} = j\\\wOne = \wTwo}}\gVP{1}{\wOne}\gVP{2}{\wOne}\polP{\wOne}\polP{\wOne} + \nonumber\\
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&\qquad \gVP{1}{\wOne}\polP{\wOne}\sum_{\substack{\wOne, \wTwo \in \pw \st\\\hashP{\wOne} = j\\\wOne \neq \wTwo}}\gVP{2}{\wTwo}\polP{\wTwo}\big] \nonumber\\
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=&\expect{\sum_{\wOne \in \pw}\gVP{1}{\wOne}\gVP{2}{\wOne}} \nonumber \\
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=&\sum_{\wOne \in \pw}\gVP{1}{\wOne}\gVP{2}{\wOne}\label{eq:two-sk-prod}
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\end{align}
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Following the reversal of the pattern of $\est{2}$, an odd number of sketches would produce an expectation of $0$, since each product in the sum has an operand whose expectation evaluates to $0$, as seen in the following,
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\begin{align*}
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&\expect{\sum_{\wVec \in \pw}\gVP{1}{\wVec}\polP{\wVec} \cdot \sum_{\wVecPrime \in \pw}\gVP{2}{\wVecPrime}\polP{\wVecPrime}\cdot\sum_{\wVec'' \in \pw}\gVP{3}{\wVec''}\polP{\wVec''}}\\
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= &\mathbb{E}\big[\sum_{\wVec \in \pw}\gVP{1}{\wVec}\polP{\wVec} \left(\gVP{2}{\wVec}\polP{\wVec} + \sum_{\substack{\wVecPrime \in \pw \st\\\wVecPrime \neq \wVec}}\gVP{2}{\wVecPrime}\polP{\wVecPrime}\right)\\
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&\qquad \left(\gVP{3}{\wVec}\polP{\wVec} + \sum_{\substack{\wVecPrime' \in \pw \st\\\wVecPrime' \neq \wVec}}\gVP{3}{\wVecPrime'}\polP{\wVecPrime'}\right)\big]\\
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= &\mathbb{E}\big[\sum_{\w \in \pw}\gVP{1}{\w}\polP{\w}\gVP{2}{\w}\polP{\w}\gVP{3}{\w}\polP{\w} + \\
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&\qquad \gVP{1}{\w}\polP{\w}\gVP{2}{\w}\polP{\w}\sum_{\substack{\w'' \in \pw\st\\\w''\neq\w}}\gVP{3}{\w''}\polP{\w''} + \\
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&\qquad\gVP{1}{\w}\polP{\w}\gVP{3}{\w}\polP{\w}\sum_{\substack{\wVecPrime \in \pw\st\\\wVecPrime\neq\w\\}}\gVP{2}{\wVecPrime}\polP{\wVecPrime} + \\
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&\qquad \gVP{1}{\w}\polP{\w}\sum_{\substack{\wVecPrime \in \pw\st\\\wVecPrime\neq\w\\}}\gVP{2}{\wVecPrime}\polP{\wVecPrime}\sum_{\substack{\w'' \in \pw\st\\\w''\neq\w}}\gVP{3}{\w''}\polP{\w''}\big] \\
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= & 0.
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\end{align*}
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The case for an odd number of sketches can likewise be reduced to the even case as in $\est{2}$. The expectation, albeit, does not yield the ground truth,
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\begin{align*}
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&\expect{\sum_{j \in \sketchCols}\sCom{1}{j}\sCom{2}{j}\sCom{3}{j}\gIJ}\\
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&= \sum_{j \in \sketchCols}\sum_{\substack{\wOne \in \pw \st\\
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\hashP{\wOne} = j}}\gVP{1}{\wOne}\gVP{2}{\wOne}\gVP{3}{\wOne} \\
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&\qquad + \gVP{1}{\wOne}\gVP{3}{\wOne}\sum_{\substack{\wTwo \in \pw \st \\
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\hashP{\wTwo} = j,\\
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\wTwo \neq \wOne}}\gVP{2}{\wTwo}\\
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&\qquad + \gVP{1}{\wOne}\gVP{2}{\wOne}\sum_{\substack{\wThree \in \pw \st\\
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\hashP{\wThree} = j,\\
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\wThree \neq \wOne}}\gVP{3}{\wThree} \\
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&\qquad + \gVP{1}{\wOne}\sum_{\substack{\wTwo \in \pw \st\\
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\hashP{\wTwo} = j,\\
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\wTwo \neq \wOne}}\gVP{2}{\wTwo}\gVP{3}{\wTwo}.
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\end{align*}
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We desire an expectation which yields the ground truth. Thus we seek to find sketch products whose expectation computes to the extraneous terms above in order to cancel them out.
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One potential work around would be to store additional sketches with independent $\pol$ functions. For the first unwanted term, assuming independent $\pol$ functions between the $\mathcal{S}_1, \mathcal{S}_2$ and $\mathcal{S}_3, \mathcal{S}_4$ pairs allows us to use linearity of expectations resulting in
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\begin{align*}
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&\expect{\sum_{j \in \sketchCols}\sCom{1}{j}\sCom{2}{j}\sCom{3}{j}\sCom{4}{j}}\\
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%&= \expect{\sum_{j \in \sketchCols}\sCom{1}{j}\sCom{2}{j}}\expect{\sum_{j \in \sketchCols}\sCom{3}{j}\sCom{4}{j}}\\
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&= \sum_{j \in \sketchCols}\expect{\sum_{\substack{\wOne, \wTwo,\\ \wThree, \wFour \in \pw \st\\\hashP{\wOne} =\hashP{\wTwo}\\=\hashP{\wThree} = \hashP{\wFour}}}\gVP{1}{\wOne}\polI{1}{\wOne}\gVP{2}{\wTwo}\polI{1}{\wTwo}\gVP{3}{\wThree}\polI{2}{\wThree}\gVP{4}{\wFour}\polI{2}{\wFour}}\\
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&=\sum_{j \in \sketchCols}\expect{\sum_{\substack{\wOne, \wTwo \in \pw \st \\ \hashP{\wOne} = \hashP{\wTwo}}}\gVP{1}{\wOne}\polI{1}{\wOne}\gVP{2}{\wTwo}\polI{1}{\wTwo}}\\
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&\qquad\cdot\expect{\sum_{\substack{\wThree, \wFour \in \pw \st \\ \hashP{\wThree} = \hashP{\wFour}}}\gVP{3}{\wThree}\polI{2}{\wThree}\gVP{4}{\wFour}\polI{2}{\wFour}}
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\end{align*}
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which reduces by \eqref{eq:two-sk-prod} to
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\begin{equation*}
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\sum_{\wOne, \wTwo \in \pw}\gVP{1}{\wOne}\gVP{2}{\wOne}\cdot \sum_{\wThree, \wFour \in \pw}\gVP{3}{\wThree}\gVP{4}{\wFour}.
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\end{equation*}
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To compute variance of the above product, the independence of $\pol$ functions can be exploited as follows:
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\begin{align}
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&\varParam{\sum_{j \in \sketchCols}\sCom{1}{j}\sCom{2}{j}\sCom{3}{j}\sCom{4}{j}}\nonumber\\
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&= \varParam{\sum_{j \in \sketchCols}\sum_{\substack{\wOne, \wTwo,\\ \wThree, \wFour \in \pw \st\\\hashP{\wOne} =\hashP{\wTwo}\\=\hashP{\wThree} = \hashP{\wFour}}}\gVP{1}{\wOne}\polI{1}{\wOne}\gVP{2}{\wTwo}\polI{1}{\wTwo}\gVP{3}{\wThree}\polI{2}{\wThree}\gVP{4}{\wFour}\polI{2}{\wFour}}\nonumber \\
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&= \sum_{j \in \sketchCols}\expect{\sum_{\substack{\wOne, \wTwo,\\ \wThree, \wFour \in \pw \st\\\hashP{\wOne} =\hashP{\wTwo}\\=\hashP{\wThree} = \hashP{\wFour}}}\gVP{1}{\wOne}\polI{1}{\wOne}\gVP{2}{\wTwo}\polI{1}{\wTwo}\gVP{3}{\wThree}\polI{2}{\wThree}\gVP{4}{\wFour}\polI{2}{\wFour}}^2 \nonumber \\
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&\qquad - \left(\sum_{\wOne, \wTwo \in \pw}\gVP{1}{\wOne}\gVP{2}{\wOne}\cdot \sum_{\wThree, \wFour \in \pw}\gVP{3}{\wThree}\gVP{4}{\wFour}\right)^2\nonumber \\
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&=\sum_{j \in \sketchCols}\expect{\sum_{\substack{\wOne, \wTwo \in \pw \st \\ \hashP{\wOne} = \hashP{\wTwo}}}\gVP{1}{\wOne}\polI{1}{\wOne}\gVP{2}{\wTwo}\polI{1}{\wTwo}^2}\nonumber\\
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&\qquad\cdot\expect{\sum_{\substack{\wThree, \wFour \in \pw \st \\ \hashP{\wThree} = \hashP{\wFour}}}\gVP{3}{\wThree}\polI{2}{\wThree}\gVP{4}{\wFour}\polI{2}{\wFour}^2} \nonumber \\
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&~\qquad - \left(\sum_{\wOne, \wTwo \in \pw}\gVP{1}{\wOne}\gVP{2}{\wOne}\cdot \sum_{\wThree, \wFour \in \pw}\gVP{3}{\wThree}\gVP{4}{\wFour}\right)^2\nonumber.
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\end{align}
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Expanding the first term, we have
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\begin{align}
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&\sum_{j \in \sketchCols}\expect{\sum_{\substack{\wOne, \wTwo \in \pw \st \\ \hashP{\wOne} = \hashP{\wTwo}}}\gVP{1}{\wOne}\polI{1}{\wOne}\gVP{1}{\wOne'}\polI{1}{\wOne'}\gVP{2}{\wTwo}\polI{1}{\wTwo}\gVP{2}{\wTwo'}\polI{1}{\wTwo'}}\nonumber \\
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&= \sum_{\wOne \in \pw}\gVP{1}{\wOne}^2\gVP{2}{\wOne}^2 +\sum_{\substack{\wOne, \wTwo \in \pw \st\\ \wOne \neq \wTwo}}\gVP{1}{\wOne}^2\gVP{2}{\wTwo}^2 + \nonumber \\
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&\qquad 2 \cdot \left(\sum_{\substack{\wOne, \wTwo \in \pw \st\\ \wOne \neq \wTwo}}\gVP{1}{\wOne}\gVP{2}{\wOne}\gVP{1}{\wTwo}\gVP{2}{\wTwo}\right). \nonumber
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\end{align}
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The second term expands analogously, leaving the product of the two expansions minus the expectation squared term.
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The expectation and variance calculations for the remaining additional terms can be analogously found.
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\startOld{Evaluating Estimate 2}
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\newline For $\est{2}$, this would result in
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\begin{align*}
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&\expect{\sum_{\wVec \in \pw}\polI{1}{\wVec}\sum_{\substack{\wOne, \wTwo, \wThree \in \pw \st\\
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\hashP{\wVec} = \hashP{\wOne} =\\ \hashP{\wTwo} = \hashP{\wThree}}}\gVP{1}{\wOne}\polI{1}{\wOne}\gVP{2}{\wTwo}\polI{2}{\wTwo}\gVP{3}{\wThree}\polI{2}{\wThree}}\\
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&= \mathbb{E}\big[\sum_{\wVec \in \pw}\polP{\wVec}\left(\gVP{1}{\wVec}\polI{1}{\wVec} + \sum_{\substack{\wOne \in \pw \st\\
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\hashP{\wOne} = \hashP{\wVec},\\
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\wOne \neq \wVec}}\gVP{1}{\wOne}\polI{1}{\wOne}\right)\\
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&\qquad\left(\sum_{\substack{\wTwo \in \pw \st \\
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\hashP{\wTwo} = \hashP{\wVec}}}\gVP{2}{\wTwo}\gVP{3}{\wTwo}\polI{2}{\wTwo}^2\right)\big]\\
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&= \sum_{\wVec \in \pw}\gVP{1}{\wVec}\sum_{\wTwo \in \pw}\gVP{2}{\wTwo}\gVP{3}{\wTwo}
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\end{align*}
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\finOld
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\startOld{Old Content}
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For the case of multiplication, when assumming independent variables, it is a known result that
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\[
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\varParam{X \cdot Y} = \expect{X^2}\expect{Y^2} - (\expect{X})^2 (\expect{Y})^2.
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\]
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It is necessary then to calculate the expectation of the square of the sum of estimates. Assuming discreet variables the expectation of the square of a random variable is simply the sum of its weighted squares. This yields
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\begin{align}
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&\expect{\left(\sum_{\wVec \in \pw}\sketchJParam{\hashP{\wVec}}\cdot \polP{\wVec}\right)^2}\label{eq:rand-sq}\\
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=& \sum_{\wVec \in \pw}\expect{\left(\sketchJParam{\hashP{\wVec}}\cdot\polP{\wVec}\right)^2}\label{eq:rand-sq-ex-push}\\
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=& \sum_{\wVec \in \pw}\expect{\left(\sum_{\substack{\wVecPrime \in \pw \st \\
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\hashP{\wVecPrime} = \hashP{\wVec}}} \genVParam{\wVecPrime}\polP{\wVecPrime}\polP{\wVec}\right)^2}\label{eq:rand-sq-equiv}\\
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=& \sum_{\wVec \in \pw}\expect{\left(\genVParam{\wVec}^2\polP{\wVec}^2 + \sum_{\substack{\wVecPrime \in \pw \st \\
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\hashP{\wVecPrime} = \hashP{\wVec},
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\wVecPrime \neq \wVec}} \genVParam{\wVecPrime}\polP{\wVecPrime}\polP{\wVec}\right)^2}\label{eq:rand-sq-assoc}\\
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=& \sum_{\wVec \in \pw}\expect{\genVParam{\wVec}^2}\label{eq:rand-sq-reduce}\\
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=& \sum_{\wVec \in \pw}\genVParam{\wVec}^2\label{eq:rand-sq-final}.
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\end{align}
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\begin{Justification}
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\hfill
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\begin{itemize}
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\item Starting out with \eqref{eq:rand-sq} since we need to know the expectation of the square of the sum of estimates.
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\item \eqref{eq:rand-sq-ex-push} is the sum of weighted squares, or alternatively, pushes the expectation inside the summation by linearity of expectation.
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\item \eqref{eq:rand-sq-equiv} substitutes the definition of a sketch bucket.
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\item \eqref{eq:rand-sq-assoc} uses associativity to rearrange the operands of the sum.
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\item \eqref{eq:rand-sq-reduce} reduces the second term of \eqref{eq:rand-sq-assoc} to $0$ by the property of uniform distribution of $\pol$.
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\item \eqref{eq:rand-sq-final} is obtained by the fact that the expectation of $\genVParam{\wVec}$ is simply itself.
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\end{itemize}
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\end{Justification}
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\begin{Assumption}
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\hfill
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\begin{itemize}\item Uniform distribution of both $\hash$ and $\pol$.\end{itemize}
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\end{Assumption}
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It then follows that the variance corresponding to the muliplication of two base sketches is
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\begin{align}
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&\sum_{\wVec \in \pw}\genV_1\paramBox{\wVec}^2\sum_{\wVec \in \pw}\genV_2\paramBox{\wVec}^2 - \left(\sum_{\wVec \in \pw} \genV_1\paramBox{\wVec}\right)^2\left(\sum_{\wVec \in \pw} \genV_2\paramBox{\wVec}\right)^2\\
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=&\norm{\genV_1}_2^2\cdot\norm{\genV_2}_2^2 - \norm{\genV_1}_1^2\cdot\norm{\genV_2}_1^2.
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\end{align}
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\AH{I don't think this equation makes sense. Where am I missing it?}
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The subscript notation for $\genV$ is used to denote sketch identity. Substituting upper bounds obtained for the L1 norm squared from \eqref{eq:norm1-sq-cauchy} results in
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\[
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\norm{\genV_1}_2^2\cdot\norm{\genV_2}_2^2 - \left(|\pw|\right)\norm{\genV_1}_2^2 \cdot \left(|\pw|\right)\norm{\genV_2}_2^2.
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\]
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\finOld |