ODIn/paper-BagRelationalPDBsAreHard
1
0
Fork 0
Code Issues Pull requests Projects Releases Wiki Activity

Most recent version

Browse source
This commit is contained in:
Aaron Huber 2019-09-05 11:49:27 -04:00
parent 018d7438d5
commit 9b9ca0ef32
2 changed files with 10 additions and 3 deletions
Show stats Download patch file Download diff file Expand all files Collapse all files

11
combining.tex
View file

@ -76,6 +76,11 @@ Note that with an odd number of sketches being multiplied, such as 3, we get the
&\qquad + \gVP{3}{\wVec}\sum_{\substack{\wOne'\in \pw \st \\ \hashP{\wOne} = \hashP{\wVec}\\
\wOne \neq \wVec}} \gVP{1}{\wOne}\gVP{2}{\wOne}.
\end{align*}
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
\begin{align*}
&\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}}\\
@ -97,7 +102,7 @@ Following the reversal of the pattern of $\est{2}$, an odd number of sketches wo
= & 0.
\end{align*}
The case for an odd number of sketches can be reduced to the even 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, albeit, does not yield the ground truth,
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,
\begin{align*}
&\expect{\sum_{j \in \sketchCols}\sCom{1}{j}\sCom{2}{j}\sCom{3}{j}\gIJ}\\
&= \sum_{j \in \sketchCols}\sum_{\substack{\wOne \in \pw \st\\
@ -124,6 +129,7 @@ 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
\[
\varParam{X \cdot Y} = \expect{X^2}\expect{Y^2} - (\expect{X})^2 (\expect{Y})^2.
@ -165,4 +171,5 @@ It then follows that the variance corresponding to the muliplication of two base
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
\[
\norm{\genV_1}_2^2\cdot\norm{\genV_2}_2^2 - \left(|\pw|\right)\norm{\genV_1}_2^2 \cdot \left(|\pw|\right)\norm{\genV_2}_2^2.
\]
\]
\finOld

2
macros.tex
View file

@ -141,7 +141,7 @@
\newcommand{\AH}[1]{\todo[inline, backgroundcolor=blue]{\textbf{Aaron says:$\,$} #1}}
\newcommand{\SR}[1]{\todo[inline, backgroundcolor=white]{\textbf{Note to self:$\,$} #1}}
\newcommand{\AR}[1]{\todo[inline, color=green]{\textbf{Atri says:$\,$} #1}}
\newcommand{\startOld}[1]{\textcolor{purple}{\newline-------------------------\newline\textbf{Old Content:\newline-------------------------\newline} #1}}
\newcommand{\startOld}[1]{\textcolor{purple}{\newline-------------------------\newline\textbf{Old Content:\newline-------------------------\newline} #1}\newline}
\newcommand{\finOld}{\newline\textcolor{purple}{------------------------------\newline\textbf{END} Old Content\newline ------------------------------\newline}}
%\newcommand{\comment}[1]{}