More fixes per Atri's suggestions
This commit is contained in:
parent
179a794592
commit
0af40b80a1
23
analysis.tex
23
analysis.tex
|
@ -47,8 +47,31 @@ For a given $\wVec \in \pw$, substituting definitions we have
|
||||||
\expect{\sum\limits_{\substack{\wVecPrime, \wVec \in \pw \st \\
|
\expect{\sum\limits_{\substack{\wVecPrime, \wVec \in \pw \st \\
|
||||||
\sketchHashParam{\wVecPrime} = \sketchHashParam{\wVec} \\
|
\sketchHashParam{\wVecPrime} = \sketchHashParam{\wVec} \\
|
||||||
\wVecPrime \neq \wVec}}\kMapParam{\wVecPrime}\sketchPolarParam{\wVecPrime}\sketchPolarParam{\wVec}}\label{eq:step-three}\\
|
\wVecPrime \neq \wVec}}\kMapParam{\wVecPrime}\sketchPolarParam{\wVecPrime}\sketchPolarParam{\wVec}}\label{eq:step-three}\\
|
||||||
|
=&~\expect{\sum\limits_{\substack{\wVecPrime \in \pw \st \\
|
||||||
|
\sketchHashParam{\wVecPrime} = \sketchHashParam{\wVec},\\
|
||||||
|
\wVecPrime = \wVec}}\kMapParam{\wVecPrime}} \cdot \expect{\sum\limits_{\substack{\wVecPrime \in \pw \st \\
|
||||||
|
\sketchHashParam{\wVecPrime} = \sketchHashParam{\wVec},\\
|
||||||
|
\wVecPrime = \wVec}}\sketchPolarParam{\wVecPrime}^2} + \nonumber\\
|
||||||
|
&\qquad\expect{\sum\limits_{\substack{\wVecPrime, \wVec \in \pw \st \\
|
||||||
|
\sketchHashParam{\wVecPrime} = \sketchHashParam{\wVec} \\
|
||||||
|
\wVecPrime \neq \wVec}}\kMapParam{\wVecPrime}}\expect{\sum\limits_{\substack{\wVecPrime, \wVec \in \pw \st \\
|
||||||
|
\sketchHashParam{\wVecPrime} = \sketchHashParam{\wVec} \\
|
||||||
|
\wVecPrime \neq \wVec}}\sketchPolarParam{\wVecPrime}\sketchPolarParam{\wVec}}\label{eq:step-three-a}\\
|
||||||
|
=&~{\sum\limits_{\substack{\wVecPrime \in \pw \st \\
|
||||||
|
\sketchHashParam{\wVecPrime} = \sketchHashParam{\wVec},\\
|
||||||
|
\wVecPrime = \wVec}}\kMapParam{\wVecPrime}} \cdot \expect{\sum\limits_{\substack{\wVecPrime \in \pw \st \\
|
||||||
|
\sketchHashParam{\wVecPrime} = \sketchHashParam{\wVec},\\
|
||||||
|
\wVecPrime = \wVec}}\sketchPolarParam{\wVecPrime}^2} + \nonumber\\
|
||||||
|
&\qquad\sum\limits_{\substack{\wVecPrime, \wVec \in \pw \st \\
|
||||||
|
\sketchHashParam{\wVecPrime} = \sketchHashParam{\wVec} \\
|
||||||
|
\wVecPrime \neq \wVec}}\kMapParam{\wVecPrime}\expect{\sum\limits_{\substack{\wVecPrime, \wVec \in \pw \st \\
|
||||||
|
\sketchHashParam{\wVecPrime} = \sketchHashParam{\wVec} \\
|
||||||
|
\wVecPrime \neq \wVec}}\sketchPolarParam{\wVecPrime}\sketchPolarParam{\wVec}}\label{eq:step-three-b}\\
|
||||||
%\] from which the last term evaluates to $0$ and we have
|
%\] from which the last term evaluates to $0$ and we have
|
||||||
%\[
|
%\[
|
||||||
|
=&~\sum\limits_{\substack{\wVecPrime \in \pw \st \\
|
||||||
|
\sketchHashParam{\wVecPrime} = \sketchHashParam{\wVec},\\
|
||||||
|
\wVecPrime = \wVec}}\kMapParam{\wVecPrime} \label{eq:step-three-c}\\
|
||||||
=&~\expect{\sum\limits_{\substack{\wVecPrime\in \pw \st \\
|
=&~\expect{\sum\limits_{\substack{\wVecPrime\in \pw \st \\
|
||||||
\sketchHashParam{\wVecPrime} = \sketchHashParam{\wVec},\\
|
\sketchHashParam{\wVecPrime} = \sketchHashParam{\wVec},\\
|
||||||
\wVecPrime = \wVec}}\kMapParam{\wVecPrime}\sketchPolarParam{\wVecPrime}^2}\label{eq:step-four}\\
|
\wVecPrime = \wVec}}\kMapParam{\wVecPrime}\sketchPolarParam{\wVecPrime}^2}\label{eq:step-four}\\
|
||||||
|
|
BIN
main.synctex(busy)
Normal file
BIN
main.synctex(busy)
Normal file
Binary file not shown.
Loading…
Reference in a new issue