Sampling, M_sketch, B_sketch bounds

est_bounds.tex

@@ -3,6 +3,9 @@
 \section{Bounding the Estimates}
 
 \newcommand{\bMu}{\epsilon\mu_{\sketchCols_{sum}}}
+\newcommand{\bBnd}{\sketchCols_{sketch}}
+\newcommand{\mBnd}{\sketchRows_{sketch}}
+\newcommand{\sBnd}{m_{sketch}}
 
 For a $\sketchCols$ estimate, denoted $\sketchCols_{est}$, and given the following:
 
@@ -27,8 +30,31 @@ Pr[|X - \mu| \leq (1 - \epsilon)\mu] \leq e^{-\frac{\epsilon^2}{2 + \epsilon}\mu
 \end{equation*}
 Solving for $\delta$,
 \begin{align*}
-\delta \geq e^{-\frac{(\frac{1}{3})^2}{2 + \frac{1}{3}}\frac{2}{3}\sketchRows}\\
-\delta \geq e^{-\frac{63}{2}\sketchRows}\\
-e^{\frac{63}{2}\sketchRows} \geq \frac{1}{\delta}\\
-\sketchRows \geq \frac{63}{2}ln(\frac{1}{\delta})
-\end{align*}
+\delta \geq e^{-\frac{(\frac{1}{2})^2}{2 + \frac{1}{2}}\frac{2}{3}\sketchRows}\\
+\delta \geq e^{-\frac{1}{15}\sketchRows}\\
+e^{\frac{1}{15}\sketchRows} \geq \frac{1}{\delta}\\
+\sketchRows \geq \frac{15}{1}ln(\frac{1}{\delta})
+\end{align*}
+
+We are now ready to combine the bounds we have derived for both $\sketchCols$ and $\sketchRows$ to which we will refer to as $\bBnd$ and $\mBnd$ respectively.
+\begin{align*}
+&\mBnd \cdot \bBnd \\
+= &  \frac{15}{1}ln(\frac{1}{\delta}) \cdot \frac{3\left(\norm{\genV}_2^2\left(|\pw|\right) + \norm{\genV}_1^2\right)}{\epsilon^2 p^2}\\ 
+= & \frac{45\left(\norm{\genV}_2^2\left(|\pw|\right) + \norm{\genV}_1^2\right)}{\epsilon^2 p^2}ln(\frac{1}{\delta})
+\end{align*}
+Sampling bounds, $\sBnd$, are obtained via Chernoff Bounds.  Given,
+\begin{align*}
+&X = \sum_{i = 1}^{m}X_i\\
+&X_i \text{is i.i.d. r.v.} \in [0, 1] \\
+&p = \frac{\norm{\genV}_1}{|W|}\\
+&\bar{X} = \frac{X}{m}\\
+&P[|\bar{X} - p| \geq \epsilon p] \leq 2e^{-\frac{\epsilon^2}{2 + \epsilon}pm} \rightarrow\\
+&\delta \geq  2e^{-\frac{\epsilon^2}{2 + \epsilon}pm} \\
+&e^{\frac{\epsilon^2}{2 + \epsilon}pm} \geq \frac{2}{\delta} \\
+&\frac{\epsilon^2}{2 + \epsilon}pm \geq ln(\frac{2}{\delta})\\
+&m \geq \frac{2 + \epsilon}{\epsilon^2 p}ln(\frac{2}{\delta})
+\end{align*}
+  We are particularly interested when the former are a lower bound to the latter.  We want to know when the following is true.
+\begin{equation*}
+\frac{2 + \epsilon}{\epsilon^2 p}ln(\frac{2}{\delta}) >  \frac{45\left(\norm{\genV}_2^2\left(|\pw|\right) + \norm{\genV}_1^2\right)}{\epsilon^2 p^2}ln(\frac{1}{\delta})
+\end{equation*}