From 14dc360dd1e67dcd8263f2d1314be8c4c3bf0754 Mon Sep 17 00:00:00 2001
From: Aaron Huber <ahuber@buffalo.edu>
Date: Fri, 16 Aug 2019 11:32:16 -0400
Subject: [PATCH] New Chebyshev bounds

---
 analysis.tex | 35 ++++++++++++++++++++++++++++++++++-
 1 file changed, 34 insertions(+), 1 deletion(-)

diff --git a/analysis.tex b/analysis.tex
index 7b09740..7f0c08c 100644
--- a/analysis.tex
+++ b/analysis.tex
@@ -327,9 +327,41 @@ Substituting $\Delta = k\sigma \rightarrow k = \frac{\Delta}{\sigma} \rightarrow
 \begin{equation*}
 Pr\left[~|X - \mu|~> \Delta~\right] < \frac{\sigma^2}{\Delta^2}
 \end{equation*}
-\AH{This next bit needs to be redone.}
 
 For the case when $\Delta = \mu\epsilon$, taking both Chebyshev bounds, setting them equal to each other, simplifying and solving for $\sketchCols$ results in  
+
+\begin{align*}
+\frac{\sigma^2}{\Delta^2} &= \frac{1}{3}\\
+\frac{\norm{\genV}_2^2 \cdot \left(|\pw|\right) + \norm{\genV}_1^2}{\sketchCols \norm{\genV}_1^2 \cdot \epsilon^2} &= \frac{1}{3}\\
+\frac{3\norm{\genV}_2^2\left(|\pw|\right) + \norm{\genV}_1^2}{\norm{\genV}_1^2 \cdot \epsilon^2} &= \sketchCols
+\end{align*}
+
+A brief digression is desirable for the purpose of simply the above bounds.  Recall the Cauchy Schwarts inequality which states:
+\[\sum_i a_i \cdot b_i \leq \norm{a}_2 \cdot \norm{b}_2\].
+The L1 norm can be expanded to the following expression, 
+\[\norm{\genV}_1 = \sum_{\wVec \in \pw} 1 \cdot \genVParam{\wVec}.\]
+Notice that the constant term can be viewed as a vector of $1$'s with size $n$ (the size of $\genV$).  Calling this vector $x$ and taking the L2 norm gives\begin{align}
+\norm{x} &= \sqrt{1_1^2 + 1_2^2 + \cdots + 1_n^2}\nonumber\\
+&= \sqrt{n * 1} \nonumber\\
+&= \sqrt{n}\nonumber\\
+&= \sqrt{|\pw|}\label{eq:w-card}
+\end{align}
+By \eqref{eq:w-card} and Cauchy Swarts, we then have
+\[
+\norm{\genV}_1 \leq \sqrt{|\pw|} \cdot \norm{\genV}_2,
+\]
+which squared yields
+\[
+\norm{\genV}_1^2 \leq |\pw| \cdot \norm{\genV}_2^2.
+\]
+
+Substituting the Cauchy Schwarts bounds into the Chebyshev calculations gives 
+\begin{align}
+&\sketchCols \leq \frac{3\norm{\genV}_2^2\left(|\pw|\right) + \norm{\genV}_2^2\left(|\pw|\right)}{\norm{\genV}_2\sqrt{|\pw|}}\nonumber\\
+&\sketchCols \leq \frac{4\norm{\genV}_2^2\left(|\pw|\right)}{\norm{\genV}_2\sqrt{|\pw|}}\nonumber\\
+&\sketchCols \leq 4\norm{\genV}_2\sqrt{|\pw|}
+\end{align}
+\AH{\textbf{BEGIN}: Old  Bound calculations}
 \begin{align*}
 \frac{\sigma^2}{\Delta^2} &= \frac{1}{3}\\
 \frac{ 2^{2N}\big(\frac{2\prob}{\sketchCols}\big)}{\mu^2\epsilon^2} &= \frac{1}{3}\\
@@ -348,6 +380,7 @@ Setting $\Delta = \epsilon\numWorlds$ gives
 \end{align*}
 
 Other cases for $\Delta$ can be solved similarly.
+\AH{\textbf{END}: Old Bound calculations}