paper-BagRelationalPDBsAreHard/combining.tex

% -*- root: main.tex -*-
\section{Combining Sketches}
\label{sec:combining}
\subsection{Adding Sketches}
When assuming that the variables are independent, as in the TIDB model, it is a known result that 
\[
\varParam{X + Y} = \varParam{X} + \varParam{Y}.
\]
It then immediately follows that adding $n$ base sketches results in the following variance:
\[
n \cdot 4\norm{\genV}_2 |\pw|^{1/2}.
\]

\subsection{Multiplying Sketches}
For the case of multiplication, when assumming independent variables, it is a known result that
\[
\varParam{X \cdot Y} = \expect{X^2}\expect{Y^2} - (\expect{X})^2 (\expect{Y})^2.
\]
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
\begin{align}
&\expect{\left(\sum_{\wVec \in \pw}\sketchJParam{\sketchHashParam{\wVec}}\cdot \sketchPolarParam{\wVec}\right)^2}\label{eq:rand-sq}\\
=& \sum_{\wVec \in \pw}\expect{\left(\sketchJParam{\sketchHashParam{\wVec}}\cdot\sketchPolarParam{\wVec}\right)^2}\label{eq:rand-sq-ex-push}\\
=& \sum_{\wVec \in \pw}\expect{\left(\sum_{\substack{\wVecPrime \in \pw \st \\
									\sketchHashParam{\wVecPrime} = \sketchHashParam{\wVec}}} \genVParam{\wVecPrime}\sketchPolarParam{\wVecPrime}\sketchPolarParam{\wVec}\right)^2}\label{eq:rand-sq-equiv}\\
=& \sum_{\wVec \in \pw}\expect{\left(\genVParam{\wVec}^2\sketchPolarParam{\wVec}^2 + \sum_{\substack{\wVecPrime \in \pw \st \\
									\sketchHashParam{\wVecPrime} = \sketchHashParam{\wVec},
									\wVecPrime \neq \wVec}} \genVParam{\wVecPrime}\sketchPolarParam{\wVecPrime}\sketchPolarParam{\wVec}\right)^2}\label{eq:rand-sq-assoc}\\
=& \sum_{\wVec \in \pw}\expect{\genVParam{\wVec}^2}\label{eq:rand-sq-reduce}\\
=& \sum_{\wVec \in \pw}\genVParam{\wVec}^2\label{eq:rand-sq-final}.
\end{align}
\begin{Justification}
\hfill
	\begin{itemize}
		\item Starting out with \eqref{eq:rand-sq} since we need to know the expectation of the square of the sum of estimates.
		\item \eqref{eq:rand-sq-ex-push} pushes the expectation inside the summation by linearity of expectation.
		\item \eqref{eq:rand-sq-equiv} substitutes the definition of a sketch bucket.
		\item \eqref{eq:rand-sq-assoc} uses associativity to rearrange the operands of the sum.
		\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 $\sketchPolar$.
		\item \eqref{eq:rand-sq-final} is obtained by the fact that the expectation of $\genVParam{\wVec}$ is simply itself.
	\end{itemize}
\end{Justification}
It then follows that the variance corresponding to the muliplication of two base sketches is
\begin{align}
&\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\\
=&\norm{\genV_1}_2^2\cdot\norm{\genV_2}_2^2 - \norm{\genV_1}_1^2\cdot\norm{\genV_2}_1^2.
\end{align}
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 
\[
\norm{\genV_1}_2^2\cdot\norm{\genV_2}_2^2 - \left(|\pw|\right)\norm{\genV_1}_2^2 \cdot \left(|\pw|\right)\norm{\genV_2}_2^2.
\]