Comments from 050120 meeting.

sop.tex

@@ -84,6 +84,8 @@ For notational convenience set
 \term_2\left(\wElem_1,\ldots, \wElem_{\prodsize}, \wElem'_1,\ldots, \wElem'_{\prodsize}\right) = &\ex{\prod_{i = 1}^{\prodsize}\sine(\wElem_i)\ind{\hfunc(\wElem_i) = j}} \cdot \ex{\prod_{i = 1}^{\prodsize}\conj{\sine(\wElem'_i)}\ind{\hfunc(\wElem'_i) = j'}} \label{eq:term-2}
 \end{align}
 Focusing on $\term_1$, observe that $\term_1 = 1$ if and only if all the $\wElem_i$'s are equal, all the $\wElem'_i$'s are equal, and the two groups of variables do not equal each other, 
+\AH{Need to back up the above statement with lemma 1, indicator vars, 2kwise independence of s, h}
+\AH{Both of these terms need to be fixed, we forgot the indicator variables.}
 \begin{equation*}
 \term_1\left(\wElem_1,\ldots, \wElem_{\prodsize}\right) =
  		\begin{cases}
@@ -388,7 +390,7 @@ When $\match{\surj}{\surj'}$, with $\dw_i = \dw'_i$ for all $i \in [\dist]$, \cr
 \AH{Should we mention this weird case for $\dist = \dist' = 1$, where for all $i \in [\prodsize]$ $\dw_i = \wElem$ and for all $i' \in [\prodsize]$ $\dw'_{i'} = \wElem'$ \textit{but} $\wElem \neq \wElem'$?  This case makes $\term_1 - \term_2 = 0$, and doesn't change the results.}
 
 By the fact that the expectations cancel when $\forall i, i', j, j'\in [\prodsize], \wElem_i = \wElem_j = \wElem, \wElem_{i'}' = \wElem_{j'}' = \wElem'$, for both $\wElem = \wElem'$ and $\wElem \neq \wElem'$, we can rid ourselves of $\term_2$, (\cref{eq:term-2}), the case when there exists only one distinct world value.  This is precisely why we have not needed to account for the last two expectations in \cref{eq:sig-j-last}.  We then need to sum up all the $\dist$ distinct world value possibilities for $\dist \in [2, \prodsize]$.  Starting with \cref{eq:sig-j-distinct},
-
+\AH{redo bringing in the $\term_2$.  Advised to separate both (in the $\sigsq_j$ case) $\term_1, \term_2$, and do separate analysis for each.}
 \begin{align}
 \sigsq_j = &\sum_{\dist  = 1}^{\prodsize}\sum_{\dist'  = 1}^{\prodsize}\sum_{\surj, \surj'}\sum_{\substack{\dw_{_1} \prec \cdots \prec \dw_{_\dist},\\\dw'_{1},\ldots,\dw'_{\dist'}\\ \in W}}\prod_{i = 1}^{\prodsize}\vect_i(\dw_{\surj(i)})\vect_i(\dw'_{\surj'(i)})\cdot \left(\term_1\left(\dw_{\surj(1)},\ldots,\dw_{\surj(\prodsize)}, \dw'_{\surj'(1)},\ldots, \dw'_{\surj'(\prodsize)}\right) - \term_2\left(\dw_{\surj(1)},\ldots,\dw_{\surj(\prodsize)}, \dw'_{\surj'(1)},\ldots, \dw'_{\surj'(\prodsize)}\right)\right)\nonumber\\
 = &\sum_{\dist  = 2}^{\prodsize}\sum_{\surj, \surj'}\sum_{\substack{\dw_{_1} \prec  \cdots \prec \dw_{_\dist},\\\dw'_{1} \prec \cdots \prec\dw'_{\dist}\\ \in W}}\prod_{i = 1}^{\prodsize}\vect_i(\dw_{\surj(i)})\vect_i(\dw'_{\surj'(i)})\cdot \term_1\left(\dw_{\surj(1)},\ldots,\dw_{\surj(\prodsize)}, \dw'_{\surj'(1)},\ldots, \dw'_{\surj'(\prodsize)}\right)\label{eq:sig-j-bnd-1}\\
@@ -400,11 +402,18 @@ By the fact that the expectations cancel when $\forall i, i', j, j'\in [\prodsiz
 We obtain \cref{eq:sig-j-bnd-1} by the fact that $\dist = \dist'$ and the removal of $\term_2$.  We conclude with \cref{eq:sig-j-bnd-2} by \cref{lem:sig-j-survive}.% as well as bringing out the indicator variables of $\term_1$.  Equation \ref{eq:sig-j-bnd-3} is derived from the fact that $\forall i \in [\dist], \dw_i = \dw'_i$.  We arrive at \cref{eq:sig-j-bnd-4}, since with $\dist$ distinct variables, the product of indicator variables will result in multiplying the uniform distribution probability distribution $\dist$ times. 
 
 Using \cref{eq:cvar-bound} and \cref{eq:sig-j-bnd-2}, we state the general bounds for $\sigsq$,
+\AH{Needs to be redone.  Missing the outer $\sum_j$ and $\sum_{j \neq j'}$.  Use another letter than h for the iteration, h is the hash function.}
 \[\sigsq = \sum_{\dist  = 2}^{\prodsize}\frac{1}{\sketchCols^{\dist}}\sum_{\substack{\surj, \surj'\\\match{\surj}{\surj'}}}\sum_{\substack{\dw_{_1}, \ldots,			\dw_{_\dist}\\ \in W}}\prod_{i = 1}^{\prodsize}\vect_i(\dw_{\surj(i)})\vect_i(\dw_{\surj'(i)}) - 
 		\frac{1}{B^2}\sum_{\wElem \in W}\prod_{i = 1}^{\prodsize}v_i^2(\wElem).\]
 \AH{Can start on SOP.  Another thing I could work on would be revising lemma 1.}
 \AR{Remaining TODOs: (1) Give expression for general $\sigma^2$, i.e. deal with the general $\lambda(j,j')$ term. (2) Show how to use the analysis for general $k$-product to handle generic  SoP expressions-- the expectation arguments would just follow from the above and linearity of expectation but the variance bounds might need a bit of extra work.}
 
+\AH{make h a subscript.  Avoid superscripts generally unless you already have a subscript; make another file with notation for a quick easy table. Make every product term to have the same k product terms.
+Also, the game plan is:
+use lemma 6 for when $\ell = \ell'$
+write lemma 6' for when $\ell \neq \ell'$.
+ 
+}
 
 \section{SOP}
 \subsection{Notation}