From 2e0190e4f3ef31446e78e8c94e60093d6e705930 Mon Sep 17 00:00:00 2001 From: Aaron Huber Date: Mon, 16 Sep 2019 09:32:20 -0400 Subject: [PATCH] Expectation of 4 Products with Independent Polarity --- combining.tex | 36 ++++++++++++++++++++++++++---------- 1 file changed, 26 insertions(+), 10 deletions(-) diff --git a/combining.tex b/combining.tex index d987acc..2d9aa78 100644 --- a/combining.tex +++ b/combining.tex @@ -81,15 +81,15 @@ The even case can be reduced to the odd case by including the one's vector as an &\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} + \\ &\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}. \end{align*} -For $\est{3}$, multiplying an even number of sketches yields -\begin{align*} -&\expect{\sum_{j \in \sketchCols}\sCom{1}{j} \cdot \sCom{2}{j}}\\ -=&\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)}\\ -=&\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} +\\ -&\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]\\ -=&\expect{\sum_{\wOne \in \pw}\gVP{1}{\wOne}\gVP{2}{\wOne}}\\ -=&\sum_{\wOne \in \pw}\gVP{1}{\wOne}\gVP{2}{\wOne} -\end{align*} +For $\est{3}$, multiplying two sketches yields +\begin{align} +&\expect{\sum_{j \in \sketchCols}\sCom{1}{j} \cdot \sCom{2}{j}} \nonumber\\ +=&\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\\ +=&\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\\ +&\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\\ +=&\expect{\sum_{\wOne \in \pw}\gVP{1}{\wOne}\gVP{2}{\wOne}} \nonumber \\ +=&\sum_{\wOne \in \pw}\gVP{1}{\wOne}\gVP{2}{\wOne}\label{eq:two-sk-prod} +\end{align} 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, \begin{align*} &\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''}}\\ @@ -117,8 +117,23 @@ The case for an odd number of sketches can likewise be reduced to the even case \hashP{\wTwo} = j,\\ \wTwo \neq \wOne}}\gVP{2}{\wTwo}\gVP{3}{\wTwo}. \end{align*} +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. -One potential work around would be to store additional sketches with independent $\pol$ functions. For $\est{2}$, this would result in +One potential work around would be to store additional sketches with independent $\pol$ functions. 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 + +\begin{align*} +&\expect{\sum_{j \in \sketchCols}\sCom{1}{j}\sCom{2}{j}\sCom{3}{j}\sCom{4}{j}}\\ +%&= \expect{\sum_{j \in \sketchCols}\sCom{1}{j}\sCom{2}{j}}\expect{\sum_{j \in \sketchCols}\sCom{3}{j}\sCom{4}{j}}\\ +&= \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}}\\ +&=\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}}\\ +&\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}} +\end{align*} +which reduces by \eqref{eq:two-sk-prod} to +\begin{equation*} +\sum_{\wOne, \wTwo \in \pw}\gVP{1}{\wOne}\gVP{2}{\wOne}\cdot \sum_{\wThree, \wFour \in \pw}\gVP{3}{\wThree}\gVP{4}{\wFour}. +\end{equation*} +The remaining additional terms can be analogously found. +\newline For $\est{2}$, this would result in \begin{align*} &\expect{\sum_{\wVec \in \pw}\polI{1}{\wVec}\sum_{\substack{\wOne, \wTwo, \wThree \in \pw \st\\ \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}}\\ @@ -129,6 +144,7 @@ One potential work around would be to store additional sketches with independent \hashP{\wTwo} = \hashP{\wVec}}}\gVP{2}{\wTwo}\gVP{3}{\wTwo}\polI{2}{\wTwo}^2\right)\big]\\ &= \sum_{\wVec \in \pw}\gVP{1}{\wVec}\sum_{\wTwo \in \pw}\gVP{2}{\wTwo}\gVP{3}{\wTwo} \end{align*} + \startOld{Old Content} For the case of multiplication, when assumming independent variables, it is a known result that \[