Started description and analysis of David's scheme.

davidscheme.tex

@@ -0,0 +1,11 @@
+%root: main.tex
+\section{David's Scheme}
+Here we define a sampling scheme for which to compare to our scheme.  Given two vectors, $\sone = \vecform{v}_1$ and $\stwo = \vecform{v}_2 \had \cdots \had \vecform{v}_\numTup$, we want to estimate $\sone \cdot \stwo$ using sampling. Recall that $\had$ is defined as the pointwise product of all vector elements.  Specifically, a vector $\wVec = \vecform{v}_1 \had \vecform{v}_2$ iff for all $i \in [\veclen], \vecform{\wElem}[i] = \vecform{v}_1[i] \cdot \vecform{v}_2[i]$.
+
+It can be shown that with $O(\frac{1}{\epsilon^2})$ words of space, with constant probability we can achieve an additive error of $O(\epsilon \norm{\sone}_2 \hp(\vecform{v}_2,\ldots, \vecform{v}_\numTup))$.
+
+\subsection{POS queries}
+Further, define $\sone = \vecform{v}_{1} + \cdots + \vecform{v}_{\numTup}$ and $\stwo = \stwo_1 \had \cdots \had \stwo_{\prodsize}$, where for each $i \in [\prodsize]$, $\stwo_i$ is a summation of at most $\numTup$ vectors.  Since we are measuring the error of estimating $\prodsize$ products, set $\sone = \wVec[v]_1$ and $\stwo = \wVec[v]_2 \had \cdots \had \wVec[v]_\prodsize$.  We want to estimate the hadamard product $\sone \had \stwo$ through sampling.
+
+\AH{This part I am not sure on...David speaks of taking a random sample from a stream, but in our situation, where does the stream originate?}
+\AH{Also, why is it a fair comparison to only sample one POS term, and not from all POS terms?  Sorry if this is a trivial question, but I don't understand, that if we are comparing with sampling, why is it that we assume we have all of $\stwo$?  Shouldn't a fair comparison require that we sample from each of the $\prodsize$ terms?}

macros.tex

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 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 % NOTATION
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%David's Scheme
+\newcommand{\vecform}[1]{\textbf{#1}}%auxiliary cmd to use only with macros, so that we can change the format easily if need be
+\newcommand{\sone}{\vecform{x}}
+\newcommand{\stwo}{\vecform{y}}
+\newcommand{\veclen}{n}%vector length
+\newcommand{\hp}{HP}%hadamard product notation
 %
 %Sum of Products
 %

main.tex

@@ -137,6 +137,7 @@
 \input{var_estj}
 \input{pos}
 \input{sop}
+\input{davidscheme}
 %\input{}