From 95be054863b38620f3983854634981e02035562d Mon Sep 17 00:00:00 2001 From: Aaron Huber Date: Thu, 25 Jul 2019 09:52:02 -0400 Subject: [PATCH] Working on generalizing v_t --- analysis.tex | 9 ++++----- hash_const.tex | 11 +++++++++-- macros.tex | 1 + 3 files changed, 14 insertions(+), 7 deletions(-) diff --git a/analysis.tex b/analysis.tex index 33d5e7b..6b43e3c 100644 --- a/analysis.tex +++ b/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}\\ diff --git a/hash_const.tex b/hash_const.tex index 0717edd..5a2b023 100644 --- a/hash_const.tex +++ b/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*} diff --git a/macros.tex b/macros.tex index 9115216..07a8c6f 100644 --- a/macros.tex +++ b/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']} %%%%%%%%%%%%%%%%