Working on generalizing v_t

analysis.tex

@@ -3,10 +3,10 @@
 \label{sec:analysis}
 We begin the analysis by showing that with high probability an estimate is approximately $\numWorldsP$, where $p$ is a tuple's probability measure for a given TIPD.  Note that 
 \begin{equation}
-\numWorldsP = \numWorldsSum\label{eq:mu}.
+\gVt{k\cdot}\numWorldsP = \numWorldsSum\label{eq:mu}.
 \end{equation}
 
-We begin by making the claim that the expectation of the estimate of a tuple t's membership across all worlds is $\numWorldsSum$, formally
+We begin by making the claim that the expectation of the estimate of a tuple t's membership across all worlds is $\sum\limits_{\wVec \in \pw}\kMapParam{\wVec}$, formally
 \begin{equation}
 \expect{\sum_{\wVec \in \pw} \sketchJParam{\sketchHashParam{\wVec}} \cdot \sketchPolarParam{\wVec}} = \sum_{\wVec \in \pw}\kMapParam{\wVec}\label{eq:allWorlds-est}.
 \end{equation}
@@ -115,10 +115,9 @@ Note that four-wise independence is assumed across all four random variables of
 \begin{equation}
 \sketchPolarParam{\wa}\cdot\sketchPolarParam{\wb}\cdot\sketchPolarParam{\wc}\cdot\sketchPolarParam{\wVecD} \label{eq:polar-product}
 \end{equation}
-we make some key observations.%it can be seen that for $\wOne, \wOneP \in \pw$ and $\wTwo, \wTwoP \in \pw'$, all four random variables in \eqref{eq:polar-product} take their values from $\pw$, although we have iteration over two separate sets $\pw$.
+we see that %it can be seen that for $\wOne, \wOneP \in \pw$ and $\wTwo, \wTwoP \in \pw'$, all four random variables in \eqref{eq:polar-product} take their values from $\pw$, although we have iteration over two separate sets $\pw$.
 %\AR{I do not know what you mean by ``iteration"} \AH{I don't know how to word what I am saying any better...by iteration I mean if you pictured the summation as nested for loops, one could have one level of nesting, where the outer loop would be iterating over the set $\pw$ and the inner loop would be iterating over a separate set of $\pw$.  However, maybe this is unnecessary to point out, and for now I have commented this out.} 
-
-Thus, there are five possible sets of $\wVec$ variable combinations, namely for $a, b, c, d \in \{1, 1', 2, 2'\} \st a \neq b \neq c \neq d$:
+there are five possible sets of $\wVec$ variable combinations, namely for $a, b, c, d \in \{1, 1', 2, 2'\} \st a \neq b \neq c \neq d$:
 \begin{align*}
 &\distPattern{1}:&\forElems{\cOne}\\
 &\distPattern{2}:&\forElems{\cTwo}\\

hash_const.tex

@@ -48,6 +48,13 @@ Examining the former term of equation \eqref{eq:exact-results}, we fix $\kMap{t}
 					0,		&\text{otherwise}.
 				  \end{cases}
 \end{equation*}
+\gVt{(generalizing)$\cdot$
+\begin{equation*}
+\kMapParam{\wVec} = \begin{cases}
+					k,&\text{if } w_t = 1\\
+					0, &\text{otherwise}.
+				\end{cases}
+\end{equation*}}
 %Therefore, by definition we have
 %\begin{equation*}
 %\sum_{\wVec \in \pw}\sketchJParam{\sketchHashParam{\wVec}} = \sum_{\wVec \in \pw}\kMapParam{\wVec}\sketchPolarParam{\wVec},
@@ -58,10 +65,10 @@ Using the same argument as in $\gIJ$ yields
 \end{equation*}
 Setting $T_3 = \sum\limits_{\wVec \in \pw \st \sketchPolarParam{\wVec} = 0}\kMapParam{\wVec}$, $T_4 = \sum\limits_{\wVec \in \pw \st \sketchPolarParam{\wVec} = 1}\kMapParam{\wVec}$ gives an exact calculation for each term given a fixed $\buck$:
 \begin{equation*}
-T_3 = | \{\wVec \st \matrixH \cdot \wVec = \buck^{(0)}, \kMapParam{\wVec} = 1\}\rightarrow T_3 \in [0, 2^{\numTup - rank(\matrixH')}]
+T_3 = \gVt{(k \cdot)} | \{\wVec \st \matrixH \cdot \wVec = \buck^{(0)}, \kMapParam{\wVec} =  \gVt{(k) }1\}\rightarrow T_3 \in [0, 2^{\numTup - rank(\matrixH')}]
 \end{equation*}
 \begin{equation*}
-T_4 = | \{\wVec \st \matrixH \cdot \wVec = \buck^{(1)}, \kMapParam{\wVec} = 1\}\rightarrow T_4 \in [0, 2^{\numTup - rank(\matrixH')}]
+T_4 = \gVt{(k \cdot)} | \{\wVec \st \matrixH \cdot \wVec = \buck^{(1)}, \kMapParam{\wVec} = \gVt{(k) 1}\}\rightarrow T_4 \in [0, 2^{\numTup - rank(\matrixH')}]
 \end{equation*}
 
 

macros.tex

@@ -48,6 +48,7 @@
 %\newcommand{\kMap}{v_t}
 \newcommand{\kMap}[1]{v_{#1}}
 \newcommand{\kMapParam}[1]{\kMap{t}\paramBox{#1}}
+\newcommand{\gVt}[1]{\textcolor{blue}{#1}}
 \newcommand{\wVec}[1][w]{\textbf{#1}}
 \newcommand{\wVecPrime}{\wVec[w']}
 %%%%%%%%%%%%%%%%