Finished Lemma 2.
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main.tex
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main.tex
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@ -7,7 +7,8 @@
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\usepackage{amsmath}
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\usepackage{amssymb}
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%\let\proof\relax
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%\let\endproof\relax
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%
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\let\endproof\relax
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\usepackage{amsthm}
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\usepackage{mathtools}
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\usepackage{etoolbox}
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sop.tex
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sop.tex
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@ -178,37 +178,55 @@ To complete the proof, we now approach the case where $\dist = \dist'$, but ther
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\begin{align*}
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&\exists i \in [\dist], |f^{-1}(i)| \neq |f'^{-1}(i)|\\
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\implies &\exists j \in [m] ~|~i \neq j, |f^{-1}(j)| \neq |f'^{-1}(j)|\\
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\implies &\exists i, j \in [\dist], i \neq j ~|~ |\dw_i| \neq |\dw_j|\\
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\implies &\exists i, j \in [\dist], i \neq j ~|~ |\dw_i| \neq |\dw_i|, |\dw_j| \neq |\dw'_j|, \\
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%\implies &\exists \wElem_i \in \wSet ~|~ \nexists \wElem_i' \in \wSet ~|~ \wElem_i = \wElem_i'
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\end{align*}
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The above means that we will have at least two world values that don't match. Put another way, after the optimal number of matching world value pairs have been assigned, there will be at least one world value whose matching conjugate product is not the conjugate of the sine of the same world value. i.e. for $i \neq j$, there will exist at least one product of $\sine(\dw_i) \conj{\sine(\dw_{j}')}$.
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The above means that we will have at least two world values that don't match. Put another way, after the optimal number of matching world value pairs have been assigned, there will be at least one world value whose matching conjugate product is not the conjugate of the sine of the same world value, i.e. for $i \neq j$, there will exist at least one product of $\sine(\dw_i) \conj{\sine(\dw_{j}')}$.
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Such cross terms exist since
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\[\left(\sum_{\substack{i \in [\dist],\\|f^{-1}(i)| \neq |f'^{-1}(i)|}}|f^{-1}(i)|\right) = \left(\sum_{\substack{i' \in [\dist],\\|f^{-1}(i')| \neq |f'^{-1}(i')|}}|f'^{-1}(i')|\right)\]
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Let $n = \{i ~|~ |f^{-1}(i)| \neq |f'^{-1}(i)|\}$. Further, let $\dist_* = [\dist] - n$ and $f^{*-1}(i) = min\left(f^{-1}(i), f'^{-1}(i)\right)$. Then,
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\begin{align*}
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\term_1 = &\ex{\left(\prod_{i \in [\dist_*]} \sine(\dw_i)^{|f^{-1}(i)|} \conj{\sine(\dw_i)}^{|f'^{-1}(i)|}\right)
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\left(\prod_{j \in n}\sine(\dw_i)^{|f^{*-1}|} \conj{\sine(\dw_i)^{|f^{*-1}|}}\right)
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\left(\prod_{\substack{i' \in n ~|~\\f^{-1}(i') > f'{-1}(i')}} \sine(\dw_{i'})^{|f^{-1}(i')| - |f'^{-1}(i')|} \prod_{\substack{j' \in n ~|~\\ f'^{-1}('j) > f^{-1}(j')}} \conj{\sine(\dw_{j'})}^{|f'^{-1}(j')| - |f^{-1}(j')|}\right)}\\
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= &\ex{\left(\prod_{i \in [\dist_*]} \sine(\dw_i)^{|f^{-1}(i)|} \conj{\sine(\dw_i)}^{|f'^{-1}(i)|}\right)
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\left(\prod_{j \in n}\sine(\dw_i)^{|f^{*-1}|} \conj{\sine(\dw_i)^{|f^{*-1}|}}\right)} \cdot
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\ex{\left(\prod_{\substack{i' \in n ~|~\\f^{-1}(i') > f'{-1}(i')}} \sine(\dw_{i'})^{|f^{-1}(i')| - |f'^{-1}(i')|} \prod_{\substack{j' \in n ~|~\\ f'^{-1}('j) > f^{-1}(j')}} \conj{\sine(\dw_{j'})}^{|f'^{-1}(j')| - |f^{-1}(j')|}\right)}\\
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= &\ex{\left(\prod_{i \in [\dist_*]} \sine(\dw_i)^{|f^{-1}(i)|} \conj{\sine(\dw_i)}^{|f'^{-1}(i)|}\right)
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\left(\prod_{j \in n}\sine(\dw_i)^{|f^{*-1}|} \conj{\sine(\dw_i)^{|f^{*-1}|}}\right)} \cdot
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\ex{\prod_{\substack{i' \in n ~|~\\f^{-1}(i') > f'{-1}(i')}} \sine(\dw_{i'})^{|f^{-1}(i')| - |f'^{-1}(i')|}} \cdot \ex{\prod_{\substack{j' \in n ~|~\\ f'^{-1}('j) > f^{-1}(j')}} \conj{\sine(\dw_{j'})}^{|f'^{-1}(j')| - |f^{-1}(j')|}}\\
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= &\ex{\left(\prod_{i \in [\dist_*]} \sine(\dw_i)^{|f^{-1}(i)|} \conj{\sine(\dw_i)}^{|f'^{-1}(i)|}\right)
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\left(\prod_{j \in n}\sine(\dw_i)^{|f^{*-1}|} \conj{\sine(\dw_i)^{|f^{*-1}|}}\right)} \cdot
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\prod_{\substack{i' \in n ~|~\\f^{-1}(i') > f'{-1}(i')}} \ex{\sine(\dw_{i'})^{|f^{-1}(i')| - |f'^{-1}(i')|}} \cdot \prod_{\substack{j' \in n ~|~\\ f'^{-1}('j) > f^{-1}(j')}} \ex{\conj{\sine(\dw_{j'})}^{|f'^{-1}(j')| - |f^{-1}(j')|}}\\
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& = 0.
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\end{align*}
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Note that each $\sine$ function in the first expectation has its matching complex conjugate in the product terms. However, for each of the expectations in the rightmost products, each $\sine$ function has a distinct input. Therefore, by \cref{lem:exp-sine}, each of those inner expectations computes to $0$. This computation zeroes out the whole product.
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\begin{align}
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\term_1 = &\ex{\left(\prod_{i \in [\dist_*]} \sine(\dw_i)^{|f^{-1}(i)|} \conj{\sine(\dw'_i)}^{|f'^{-1}(i)|}\right)
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\left(\prod_{j \in [n]}\sine(\dw_i)^{|f^{*-1}|} \conj{\sine(\dw'_i)}^{|f^{*-1}|}\right)
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\left(\prod_{\substack{i' \in [n],\\f^{-1}(i') > f'^{-1}(i')}} \sine(\dw_{i'})^{|f^{-1}(i')| - |f'^{-1}(i')|} \prod_{\substack{j' \in n ~|~\\ f'^{-1}('j) > f^{-1}(j')}} \conj{\sine(\dw'_{j'})}^{|f'^{-1}(j')| - |f^{-1}(j')|}\right)} \label{eq:lem-match-pt2-line1}\\
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= &\ex{\left(\prod_{i \in [\dist_*]} \sine(\dw_i)^{|f^{-1}(i)|} \conj{\sine(\dw'_i)}^{|f'^{-1}(i)|}\right)
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\left(\prod_{j \in [n]}\sine(\dw_i)^{|f^{*-1}|} \conj{\sine(\dw'_i)}^{|f^{*-1}|}\right)} \cdot
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\ex{\left(\prod_{\substack{i' \in n ~|~\\f^{-1}(i') > f'^{-1}(i')}} \sine(\dw_{i'})^{|f^{-1}(i')| - |f'^{-1}(i')|} \prod_{\substack{j' \in n ~|~\\ f'^{-1}('j) > f^{-1}(j')}} \conj{\sine(\dw'_{j'})}^{|f'^{-1}(j')| - |f^{-1}(j')|}\right)}\nonumber\\
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= &\ex{\left(\prod_{i \in [\dist_*]} \sine(\dw_i)^{|f^{-1}(i)|} \conj{\sine(\dw'_i)}^{|f'^{-1}(i)|}\right)
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\left(\prod_{j \in [n]}\sine(\dw_i)^{|f^{*-1}|} \conj{\sine(\dw'_i)}^{|f^{*-1}|}\right)} \cdot
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\ex{\prod_{\substack{i' \in n ~|~\\f^{-1}(i') > f'^{-1}(i')}} \sine(\dw_{i'})^{|f^{-1}(i')| - |f'^{-1}(i')|}} \cdot \ex{\prod_{\substack{j' \in n ~|~\\ f'^{-1}('j) > f^{-1}(j')}} \conj{\sine(\dw'_{j'})}^{|f'^{-1}(j')| - |f^{-1}(j')|}}\nonumber\\
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= &\ex{\left(\prod_{i \in [\dist_*]} \sine(\dw_i)^{|f^{-1}(i)|} \conj{\sine(\dw'_i)}^{|f'^{-1}(i)|}\right)
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\left(\prod_{j \in [n]}\sine(\dw_i)^{|f^{*-1}|} \conj{\sine(\dw'_i)}^{|f^{*-1}|}\right)} \cdot
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\prod_{\substack{i' \in n ~|~\\f^{-1}(i') > f'^{-1}(i')}} \ex{\sine(\dw_{i'})^{|f^{-1}(i')| - |f'^{-1}(i')|}} \cdot \prod_{\substack{j' \in n ~|~\\ f'^{-1}('j) > f^{-1}(j')}} \ex{\conj{\sine(\dw'_{j'})}^{|f'^{-1}(j')| - |f^{-1}(j')|}}\label{eq:lem-match-pt2-last}\\
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& = 0.\nonumber
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\end{align}
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Looking at \cref{eq:lem-match-pt2-line1}, each $\sine$ function in the first two products has its matching complex conjugate in the product terms. However, the rightmost products of the expectation are all distinct world value inputs, i.e. random $\sine$ values with no matching conjugate counterparts. Since we have distinct, non-matching world value inputs for the rightmost products, we can push the expectation through the products until we arrive at \cref{eq:lem-match-pt2-last}, where finally, by \cref{lem:exp-sine}, each of those inner expectations computes to $0$. This in turn zeroes out the whole product.
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We now seek to show that when $f, f'$ are matching, that $\term_1$ will always equal 1. Recall that when $\match{f}{f'}$, that
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\begin{enumerate}
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\item $\dist = \dist'$, i.e., the output size of both functions is the same,
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\item $\forall i \in [\dist],| f^{-1}(i)| = |f'^{-1}(i)|$, i.e. each $\dw_i$ has the same number of variables assigned to it as its $\dw'_i$ counterpart.
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\end{enumerate}
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This means,
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\begin{align}
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\term_1 = &\ex{\prod_{i = 1}^{\dist}\sine(\dw_i)^{|f^{-1}(i)|}\conj{\sine(\dw'_i)}^{|f'^{-1}(i)|}}\nonumber\\
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= &\ex{\prod_{i = 1}^{\dist}\left(\sine(\dw_i) \cdot \conj{\sine(\dw'_i)}\right)^{|f^{-1}(i)|}}\label{eq:lem-match-pt3-2}\\
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= &1\nonumber
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\end{align}
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We arrive at \cref{eq:lem-match-pt3-2} since $\forall i \in [\dist], |f^{-1}(i)| = |f'^{-1}(i)|$ and we can use the distributive law of exponents
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over multiplication. This then implies that each individiual $\sine(\dw_i)$ has its own matching conjugate $\conj{\sine(\dw'_i)}$,and by the property of roots of unity in complex numbers, each $\sine(\dw_i)\cdot \conj{\sine(\dw'_i)} = 1$, yielding an overall product of $1$.
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\AH{Next on the agenda v------}
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We now seek to show that when $f, f'$ are matching, that $\term_1 - \term_2$ will always equal 1.
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\end{proof}
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\AH{Next on the agenda, }
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Using the above definitions, we can now present the variance bounds for $\sigsq_j$ based on \eqref{eq:sig-j-distinct}.
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By the fact that the expectations cancel when $\forall i, i', j, j'\in [\prodsize], \wElem_i = \wElem_j = \wElem_{i'}' = \wElem_{j'}' = \wElem$, we can rid ourselves of the case when there exists only one distinct world value.\AR{This is where you are subtracting off the $T_2$ terms, which is why you do not need to consider them in the arguments above.} We then need to sum up all the $\dist$ distinct world value possibilities for $\dist \in [2, \prodsize]$. Note that the number of distinct values $\dist$ affects the randomness of the hash function $\hfunc$. E.g. only $\dist = 2$ distinct values will yield $\frac{1}{\sketchCols} \cdot \frac{1}{\sketchCols} = \frac{1}{\sketchCols^2} = \frac{1}{\sketchCols^\dist}$. By lemma \ref{lem:sig-j-survive} and equation \eqref{eq:sig-j-distinct} we get
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By the fact that the expectations cancel when $\forall i, i', j, j'\in [\prodsize], \wElem_i = \wElem_j = \wElem_{i'}' = \wElem_{j'}' = \wElem$, we can rid ourselves of 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 $\dist \in [2, \prodsize]$. Note that the number of distinct values $\dist$ affects the randomness of the hash function $\hfunc$. E.g. only $\dist = 2$ distinct values will yield $\frac{1}{\sketchCols} \cdot \frac{1}{\sketchCols} = \frac{1}{\sketchCols^2} = \frac{1}{\sketchCols^\dist}$. By lemma \ref{lem:sig-j-survive} and equation \eqref{eq:sig-j-distinct} we get
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\AR{Um, no this is not an argument for general situation. You need to explicitly argue how each term in Lemma~\eqref{eq:sig-j-distinct} fares under Lemma~\ref{lem:sig-j-survive} and then how those give you the expression below. In particular, you should explicitly argue what happens to the indicator variable terms. Also doing this for $\lambda(j,j')$ would mean you'll have to deal with those explicitly in any case.}
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%
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%\begin{equation*}
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