Some minor changes.

sop.tex

@@ -236,7 +236,7 @@ Functions $\surj:[\prodsize]\mapsto [\dist], \surj':[\prodsize]\mapsto [\dist']$
 \AH{\^---next on the agenda!}
 
 \begin{Lemma}\label{lem:sig-j-survive}
-When $\surj, \surj'$are matching, where for every $j \in[\dist], \dw_{j} = \dw'_{j}$, %\cref{eq:sig-j-distinct} is exactly 
+When $\surj, \surj'$are matching, where for every $i \in[\dist], \dw_{i} = \dw'_{i}$, %\cref{eq:sig-j-distinct} is exactly 
 \[
 \term_1(\dw_{\surj(1)},\dots, \dw_{\surj(\prodsize)}, \dw_{\surj'(1)},\dots, \dw_{\surj'(\prodsize')}) = \frac{1}{\sketchCols^\dist}% \sum_{\substack{\dw_{1} \prec \cdots \prec \dw_{_\dist}\\ \in \wSet}}\prod_{i = 1}^{\prodsize}\vect_i(\dw_{\surj(i)})\vect_i(\dw_{\surj'(i)})
 \]
@@ -281,10 +281,10 @@ Note \cref{eq:term-1}, and consider the "generic term"--
 Let's rewrite the term based on its exact definition:
 \begin{align*}
 = &\ex{\prod_{i = 1}^{\prodsize}\sine(\dw_{\surj(i)})\cdot\conj{\sine(\dw'_{\surj'(i)})}\ind{\hfunc(\dw_i) = j}\ind{\hfunc(\dw'_i) = j}}\\
-= &\ex{\left(\prod_{i = 1}^{\dist}\sine(\dw_{i})^{|\surj^{-1}(i)|}\right) \cdot \left(\prod_{\ell = 1}^{\dist'}\conj{\sine(\dw'_{\ell})}^{|\surj'^{-1}(\ell)|}\right)\ind{\hfunc(\dw_i) = j}\ind{\hfunc(\dw'_i) = j}}
+= &\ex{\left(\prod_{i = 1}^{\dist}\sine(\dw_{i})^{|\surj^{-1}(i)|}\right) \cdot \left(\prod_{\ell = 1}^{\dist}\conj{\sine(\dw'_{\ell})}^{|\surj'^{-1}(\ell)|}\right)\ind{\hfunc(\dw_i) = j}}
 \end{align*}
 Notice that each $i \in [\prodsize]$ has its own mapping to an element in $[\dist]$.  We can thus rearrange all the elements of the product such that the preimage of function $\surj(i)$, i.e., $\surj^{-1}(i)$ yields the number of terms that will be mapped to a distinct variable $\dw_i$.
-Further see how the requirement that $\dw_i = \dw'_i$ gives us the precise combinations we are looking for, where each random $\sine$ output value has its own matching complex conjugate.
+Further see how the requirement that $\dw_i = \dw'_i$ gives us the precise combinations we are looking for, where each random $\sine$ output value has its own matching complex conjugate.  This condition also cancels one of the indicator variables as well.
 
 %To prove that \cref{lem:sig-j-survive} is true, consider what the expectation looks like when $\surj, \surj'$ are not matching.  The first condition for $\surj, \surj'$ to be matching is violated when $\dist \neq \dist'$.  
 
@@ -310,15 +310,15 @@ Such cross terms exist since
 Let $n = \{i ~|~ |\surj^{-1}(i)| \neq |\surj'^{-1}(i)|\}$.  Further, let $\dist_* = [\dist] - n$,  $\surj^{*-1}(i) = min\left(\surj^{-1}(i), \surj'^{-1}(i)\right)$, and $\hat{f}^{-1}(i) = \biggm| |\surj^{-1}(i)| - |\surj'^{-1}(i)| \biggm|$.  Then,
 
 \begin{align}
-\term_1 = &\mathbb{E}\left[\left(\prod_{i \in [\dist_*]} \sine(\dw_i)^{|\surj^{-1}(i)|} \conj{\sine(\dw'_i)}^{|\surj'^{-1}(i)|}\ind{\hfunc(\dw_i) = j}\ind{\hfunc(\dw'_i) = j}\right) 
-	\left(\prod_{\ell \in [n]}\sine(\dw_\ell)^{|\surj^{*-1}(\ell)|} \conj{\sine(\dw'_\ell)}^{|\surj^{*-1}(\ell)|}\ind{\hfunc(\dw_\ell) = j}\ind{\hfunc(\dw'_\ell) = j}\right) \right.\nonumber\\
+\term_1 = &\mathbb{E}\left[\left(\prod_{i \in [\dist_*]} \sine(\dw_i)^{|\surj^{-1}(i)|} \conj{\sine(\dw'_i)}^{|\surj'^{-1}(i)|}\ind{\hfunc(\dw_i) = j}\right) 
+	\left(\prod_{\ell \in [n]}\sine(\dw_\ell)^{|\surj^{*-1}(\ell)|} \conj{\sine(\dw'_\ell)}^{|\surj^{*-1}(\ell)|}\ind{\hfunc(\dw_\ell) = j}\right) \right.\nonumber\\
 		&\qquad\qquad\qquad\qquad\qquad\qquad\left.\left(\prod_{\substack{i' \in [n],\\\surj^{-1}(i') > \surj'^{-1}(i')}} \sine(\dw_{i'})^{|\surj^{-1}(i')| - |\surj'^{-1}(i')|}\ind{\hfunc(\dw_{i'}) = j}\right)
  					\left(\prod_{\substack{\ell' \in n,\\ \surj'^{-1}(\ell') > \surj^{-1}(\ell')}} \conj{\sine(\dw'_{\ell'})}^{|\surj'^{-1}(\ell')| - |\surj^{-1}(\ell')|}\ind{\hfunc(\dw_{\ell'}) = j}\right)\right] \label{eq:lem-match-pt2-line1}
 \end{align}
 Notice that the two rightmost factors in the product are distinct values with no matching conjugates, and by the independence of $\sine$, we can push the expectation through the product.  If we label the four factors as 
 \begin{align*}
-\term_{1, 1} =&\prod_{i \in [\dist_*]} \sine(\dw_i)^{|\surj^{-1}(i)|} \conj{\sine(\dw'_i)}^{|\surj'^{-1}(i)|}\ind{\hfunc(\dw_i) = j}\ind{\hfunc(\dw'_i) = j}\\
-\term_{1, 2} =& \prod_{\ell \in [n]}\sine(\dw_\ell)^{|\surj^{*-1}(\ell)|} \conj{\sine(\dw'_\ell)}^{|\surj^{*-1}(\ell)|}\ind{\hfunc(\dw_\ell) = j}\ind{\hfunc(\dw'_\ell) = j}\\
+\term_{1, 1} =&\prod_{i \in [\dist_*]} \sine(\dw_i)^{|\surj^{-1}(i)|} \conj{\sine(\dw'_i)}^{|\surj'^{-1}(i)|}\ind{\hfunc(\dw_i) = j}\\
+\term_{1, 2} =& \prod_{\ell \in [n]}\sine(\dw_\ell)^{|\surj^{*-1}(\ell)|} \conj{\sine(\dw'_\ell)}^{|\surj^{*-1}(\ell)|}\ind{\hfunc(\dw_\ell) = j}\\
 \term_{1, 3} =& \prod_{\substack{i' \in [n],\\\surj^{-1}(i') > \surj'^{-1}(i')}} \sine(\dw_{i'})^{|\surj^{-1}(i')| - |\surj'^{-1}(i')|}\ind{\hfunc(\dw_{i'}) = j}\\
 \term_{1, 4} =&\prod_{\substack{\ell' \in n,\\ \surj'^{-1}(\ell') > \surj^{-1}(\ell')}} \conj{\sine(\dw'_{\ell'})}^{|\surj'^{-1}(\ell')| - |\surj^{-1}(\ell')|}\ind{\hfunc(\dw_{\ell'}) = j}
 \end{align*}
@@ -361,7 +361,7 @@ have the condition $\dist \neq \dist'$, we still have that for any $\dist \in [\
 \end{proof}
 
 
-We now seek to show that when $\surj, \surj'$ are matching, that $\term_1$ will always equal 1.  Recall that when $\match{\surj}{\surj'}$, that 
+We now seek to show that when $\surj, \surj'$ are matching, that $\term_1$ will always equal $\frac{1}{\sketchCols^\dist}$.  Recall that when $\match{\surj}{\surj'}$, that 
 \begin{enumerate}
 	%\item $\dist = \dist'$, i.e., the output size of both functions is the same,
 	\item $\forall i \in [\dist],| \surj^{-1}(i)| = |\surj'^{-1}(i)|$, i.e. each $\dw_i$ has the same number of variables assigned to it as its $\dw'_i$ counterpart.
@@ -369,7 +369,7 @@ We now seek to show that when $\surj, \surj'$ are matching, that $\term_1$ will
 Recalling that we require $\dw_i = \dw'_i$ this means,
 \AH{This also covers the case when m = 1, i.e. $|\surj^{-1}(i)| = \prodsize$.}
 \begin{align}
-\term_1 = &\ex{\prod_{i = 1}^{\dist}\sine(\dw_i)^{|\surj^{-1}(i)|}\conj{\sine(\dw'_i)}^{|\surj'^{-1}(i)|}\ind{\hfunc(\dw_i) = j}\ind{\hfunc(\dw'_i) = j}}\nonumber\\
+\term_1 = &\ex{\prod_{i = 1}^{\dist}\sine(\dw_i)^{|\surj^{-1}(i)|}\conj{\sine(\dw'_i)}^{|\surj'^{-1}(i)|}\ind{\hfunc(\dw_i) = j}}\nonumber\\
 = &\ex{\prod_{i = 1}^{\dist}\left(\sine(\dw_i) \cdot \conj{\sine(\dw'_i)}\right)^{|\surj^{-1}(i)|}\ind{\hfunc(\dw_i) = j}}\label{eq:lem-match-pt3-2}\\
 = &\frac{1}{\sketchCols^{\dist}}\nonumber
 \end{align}
@@ -389,17 +389,17 @@ When $\match{\surj}{\surj'}$, with $\dw_i = \dw'_i$ for all $i \in [\dist]$, \cr
 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},
 
 \begin{align}
-\sigsq_j = &\sum_{\dist  = 1}^{\prodsize}\sum_{\dist'  = 1}^{\prodsize}\sum_{\surj, \surj'}\sum_{\substack{\dw_{_1}, \ldots,\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}, \ldots,\dw_{_\dist},\\\dw'_{1},\ldots,\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}\\
-= &\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)})\label{eq:sig-j-bnd-2}
+\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}\\
+= &\sum_{\dist  = 2}^{\prodsize}\frac{1}{\sketchCols^{\dist}}\sum_{\substack{\surj, \surj'\\\match{\surj}{\surj'}}}\sum_{\substack{\dw_{_1} \prec \cdots \prec \dw_{_\dist},\\ \in W}}\prod_{i = 1}^{\prodsize}\vect_i(\dw_{\surj(i)})\vect_i(\dw'_{\surj'(i)})\label{eq:sig-j-bnd-2}
 %= &\sum_{\dist  = 2}^{\prodsize}\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)})\cdot \prod_{i = 1}^{\dist}\ind{\hfunc(\dw_i) = j}\label{eq:sig-j-bnd-3}\\
 %= &\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)})\label{eq:sig-j-bnd-4}
 \end{align}
 
 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-4}, we state the general bounds for $\sigsq$,
+Using \cref{eq:cvar-bound} and \cref{eq:sig-j-bnd-2}, we state the general bounds for $\sigsq$,
 \[\sigsq = \sum_{\dist  = 2}^{\prodsize}\frac{1}{\sketchCols^{\dist}}\sum_{\substack{\surj, \surj'\\\match{\surj}{\surj'}}}\sum_{\substack{\dw_{_1}, \ldots,			\dw_{_\dist}\\ \in W}}\prod_{i = 1}^{\prodsize}\vect_i(\dw_{\surj(i)})\vect_i(\dw_{\surj'(i)}) - 
-		\frac{1}{B^2}\sum_{\wElem \in W}\prod_{i = 1}^{\prodsize}v_i^2(\wElem)\label{eq:cvar-bound}.\]
-\AH{After the lemma for eq 95, next on the agenda, type up the expectation calculations, then start on SOP.}
+		\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.}