ODIn/paper-BagRelationalPDBsAreHard
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Some minor changes.

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Aaron Huber 2020-04-29 15:57:52 -04:00
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@ -236,7 +236,7 @@ Functions $\surj:[\prodsize]\mapsto [\dist], \surj':[\prodsize]\mapsto [\dist']$
\AH{\^---next on the agenda!} \AH{\^---next on the agenda!}
\begin{Lemma}\label{lem:sig-j-survive} \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
\[ \[
\term_1(\dw_{\surj(1)},\dots, \dw_{\surj(\prodsize)}, \dw_{\surj'(1)},\dots, \dw_{\surj'(\prodsize')}) = \frac{1}{\sketchCols^\dist}% \sum_{\substack{\dw_{1} \prec \cdots \prec \dw_{_\dist}\\ \in \wSet}}\prod_{i = 1}^{\prodsize}\vect_i(\dw_{\surj(i)})\vect_i(\dw_{\surj'(i)}) \term_1(\dw_{\surj(1)},\dots, \dw_{\surj(\prodsize)}, \dw_{\surj'(1)},\dots, \dw_{\surj'(\prodsize')}) = \frac{1}{\sketchCols^\dist}% \sum_{\substack{\dw_{1} \prec \cdots \prec \dw_{_\dist}\\ \in \wSet}}\prod_{i = 1}^{\prodsize}\vect_i(\dw_{\surj(i)})\vect_i(\dw_{\surj'(i)})
\] \]
@ -281,10 +281,10 @@ Note \cref{eq:term-1}, and consider the "generic term"--
Let's rewrite the term based on its exact definition: Let's rewrite the term based on its exact definition:
\begin{align*} \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{\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*} \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 $\dw_i$. 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 $\dist \neq \dist'$. %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, 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} \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) \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}\ind{\hfunc(\dw'_\ell) = j}\right) \right.\nonumber\\ \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) &\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} \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} \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 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*} \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, 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}\ind{\hfunc(\dw'_\ell) = 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, 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} \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*} \end{align*}
@ -361,7 +361,7 @@ have the condition $\dist \neq \dist'$, we still have that for any $\dist \in [\
\end{proof} \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} \begin{enumerate}
%\item $\dist = \dist'$, i.e., the output size of both functions is the same, %\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. \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, Recalling that we require $\dw_i = \dw'_i$ this means,
\AH{This also covers the case when m = 1, i.e. $|\surj^{-1}(i)| = \prodsize$.} \AH{This also covers the case when m = 1, i.e. $|\surj^{-1}(i)| = \prodsize$.}
\begin{align} \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}\\ = &\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 = &\frac{1}{\sketchCols^{\dist}}\nonumber
\end{align} \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}, 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} \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\\ \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}, \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}\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}, \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} = &\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}\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} %= &\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} \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. 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)}) - \[\sigsq = \sum_{\dist = 2}^{\prodsize}\frac{1}{\sketchCols^{\dist}}\sum_{\substack{\surj, \surj'\\\match{\surj}{\surj'}}}\sum_{\substack{\dw_{_1}, \ldots, \dw_{_\dist}\\ \in W}}\prod_{i = 1}^{\prodsize}\vect_i(\dw_{\surj(i)})\vect_i(\dw_{\surj'(i)}) -
\frac{1}{B^2}\sum_{\wElem \in W}\prod_{i = 1}^{\prodsize}v_i^2(\wElem)\label{eq:cvar-bound}.\] \frac{1}{B^2}\sum_{\wElem \in W}\prod_{i = 1}^{\prodsize}v_i^2(\wElem).\]
\AH{After the lemma for eq 95, next on the agenda, type up the expectation calculations, then start on SOP.} \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.} \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.}