From e4676fedc819043f86f91fe3a08be77430212588 Mon Sep 17 00:00:00 2001
From: Aaron Huber <ahuber@buffalo.edu>
Date: Tue, 3 Sep 2019 15:16:29 -0400
Subject: [PATCH] Minor corrections

---
 combining.tex | 13 ++++++++-----
 1 file changed, 8 insertions(+), 5 deletions(-)

diff --git a/combining.tex b/combining.tex
index a6c9b5d..dc52504 100644
--- a/combining.tex
+++ b/combining.tex
@@ -79,10 +79,11 @@ Note that with an odd number of sketches being multiplied, such as 3, we get the
 For $\est{3}$, multiplying an even number of sketches yields
 \begin{align*}
 &\expect{\sum_{j \in \sketchCols}\sCom{1}{j} \cdot \sCom{2}{j}}\\
-=&\expect{\sum_{j \in \sketchCols}\left(\sum_{\substack{\wVec \in \pw \st\\\hashP{\wVec} = j}}\gVP{1}{\wVec}\polP{\wVec}\cdot \sum_{\substack{\wVecPrime \in \pw \st\\\hashP{\wVecPrime} = j}}\gVP{2}{\wVecPrime}\polP{\wVecPrime}\right)}\\
-=&\expect{\sum_{j \in \sketchCols}\sum_{\substack{\wVec, \wVecPrime \in \pw \st\\\hashP{\wVec} = j\\\wVec = \wVecPrime}}\gVP{1}{\wVec}\gVP{2}{\wVec}\polP{\wVec}\polP{\wVec}\sum_{\substack{\wVec, \wVecPrime \in \pw \st\\\hashP{\wVec} = j\\\wVec \neq \wVecPrime}}\gVP{1}{\wVec}\gVP{2}{\wVecPrime}\polP{\wVec}\polP{\wVecPrime}}\\
-=&\expect{\sum_{\wVec \in \pw}\gVP{1}{\wVec}\gVP{2}{\wVec}}\\
-=&\gVP{1}{\wVec}\gVP{2}{\wVec}
+=&\expect{\sum_{j \in \sketchCols}\left(\sum_{\substack{\wOne \in \pw \st\\\hashP{\wOne} = j}}\gVP{1}{\wOne}\polP{\wOne}\cdot \sum_{\substack{\wTwo \in \pw \st\\\hashP{\wTwo} = j}}\gVP{2}{\wTwo}\polP{\wTwo}\right)}\\
+=&\mathbb{E}\big[\sum_{j \in \sketchCols}\sum_{\substack{\wOne, \wTwo \in \pw \st\\\hashP{\wOne} = j\\\wOne = \wTwo}}\gVP{1}{\wOne}\gVP{2}{\wOne}\polP{\wOne}\polP{\wOne} +\\
+&\qquad \gVP{1}{\wOne}\polP{\wOne}\sum_{\substack{\wOne, \wTwo \in \pw \st\\\hashP{\wOne} = j\\\wOne \neq \wTwo}}\gVP{2}{\wTwo}\polP{\wTwo}\big]\\
+=&\expect{\sum_{\wOne \in \pw}\gVP{1}{\wOne}\gVP{2}{\wOne}}\\
+=&\sum_{\wOne \in \pw}\gVP{1}{\wOne}\gVP{2}{\wOne}
 \end{align*}
 Following the reversal of the pattern of $\est{2}$, an odd number of sketches would produce an expectation of $0$, since each product in the sum has an operand whose expectation evaluates to $0$, as seen in the following, 
 \begin{align*}
@@ -112,7 +113,9 @@ The case for an odd number of sketches can be reduced to the even case by includ
 							\wTwo \neq \wOne}}\gVP{2}{\wTwo}\gVP{3}{\wTwo}.
 \end{align*}
 
-
+One potential work around would be to store additional sketches with independent $\pol$ functions.  For $\est{2}$, this would result in
+\begin{align*}
+\end{align*}
 For the case of multiplication, when assumming independent variables, it is a known result that
 \[
 \varParam{X \cdot Y} = \expect{X^2}\expect{Y^2} - (\expect{X})^2 (\expect{Y})^2.