Corrections.
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sop.tex
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sop.tex
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@ -1,7 +1,7 @@
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%root--main.tex
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\section{Analysis of a $\prodsize$-way join}
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There are several steps involved to obtaining bounds on the Sum of Products (SOP) query. We start by analyzing a $\prodsize$ product. Define the $j_{th}$ bucket of a sketch $\sk$ for a vector $\vect$ as
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There are several steps involved to obtaining bounds on the Sum of Products (SOP) query. We start by analyzing a $\prodsize$ product. Define the $j^{\text{th}}$ bucket of a sketch $\sk$ for a vector $\vect$ as
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\[\sk^\vect[j] = \sum_{\substack{\wElem \in \wSet,\\ \hfunc(\wElem) = j}}\vect(\wElem)\sine(\wElem)\].
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Define the estimate of the $j_{th}$ bucket to be
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@ -13,7 +13,11 @@ For notational convenience define
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&\term_j = \sum_{\wElem \in \wSet_j} \prod_{i = 1}^{\prodsize}\vect_i(\wElem)
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\end{align*}
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Let us show first that the expectation of the estimate does in fact yield the value we are estimating, $\term_j$.
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\AR{You should convert the above statement into a formal lemma. Otherwise it is weird to see a proof without any formal statement of what it is proving.}
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\begin{Lemma}
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The expectation of an estimate $\est_j$ is the sum of its pointwise vector products.
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\end{Lemma}
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\begin{proof}
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\begin{align*}
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@ -22,19 +26,20 @@ Let us show first that the expectation of the estimate does in fact yield the va
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= &\ex{\sum_{\substack{\wElem_1,\ldots, \wElem_{\prodsize}\\ \in \wSet_j}} \prod_{i = 1}^{\prodsize}\vect_i(\wElem_i)\prod_{i = 1}^{\prodsize}\sine(\wElem_i)}\\
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= &\sum_{\substack{\wElem_1,\ldots, \wElem_{\prodsize}\\ \in \wSet_j}} \prod_{i = 1}^{\prodsize}\vect_i(\wElem_i)\ex{\prod_{i = 1}^{\prodsize}\sine(\wElem_i)}
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\end{align*}
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Fix the variables $\wElem_1,\ldots, \wElem_{\prodsize}$. Define $\dist$ to be the number of distinct worlds in $\wElem_1,\ldots, \wElem_{\prodsize}$ and $e_l$ to be the number of repetitions for the $l_{th}$ \AR{General typesetting comments. (1) You should always use $\ell$ instead of $l$. (2) Typeset $l_{th}$ as $\ell^{\text{th}}$-- note that ``th" is in superscript and not in math mode.} distinct world value. For $\term_1^{\est_j} = \ex{\prod_{i = 1}^{\prodsize} \sine(\wElem_i)}$, \AR{Why are you defining the new notation $\term_1^{\est_j}$. You should always be wary of introducing new notation since it makes things hard to read.} we get
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Fix the variables $\wElem_1,\ldots, \wElem_{\prodsize}$. Define $\dist$ to be the number of distinct worlds in $\wElem_1,\ldots, \wElem_{\prodsize}$ and $e_{\ell}$ to be the number of repetitions for the $\ell^{\text{th}}$ distinct world value. Focusing on the expectation factor, $\ex{\prod_{i = 1}^{\prodsize} \sine(\wElem_i)}$, we get
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\begin{align*}
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\term_1^{\est_j} = &\ex{\prod_{i = 1}^{\prodsize}\sine(\wElem_i)}\\
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= &\ex{\prod_{l = 1}^{\dist} \sine(\wElem_l)^{e_l}}\\
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&\ex{\prod_{i = 1}^{\prodsize}\sine(\wElem_i)}\\
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= &\ex{\prod_{\ell = 1}^{\dist} \sine(\wElem_{\ell})^{e_{\ell}}}\\
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= & \begin{cases}
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0 &1 <\dist < \prodsize\\
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1 & \dist = 1.
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\end{cases}
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\end{align*}
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\AR{Why is the last equality true? You need to justify it by explicitly showing how you are using Lemma~\ref{lem:exp-sine} to prove it.}
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Notice, that the above leaves us with the condition that $\forall i, j \in [\prodsize], \wElem_i = \wElem_j$,
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We obtain the final equality by \cref{lem:exp-sine}, which states that the only way in expectation that $\sine(\wElem_{\ell})^{e_{\ell}}$ can be something other than $0$ is when $e_{\ell} = \prodsize$. It can further be seen that the only way this can happen is when $\dist = \prodsize$.
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Notice, that the above leaves us with the only remaining condition that $\forall i, j \in [\prodsize], \wElem_i = \wElem_j$,
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\begin{align*}
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= &\sum_{\wElem \in \wSet_j}\prod_{i = 1}^{\prodsize} \vect_i(w) \cdot \term_1^{\est_j} = \term_j.
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= &\sum_{\wElem \in \wSet_j}\prod_{i = 1}^{\prodsize} \vect_i(w) \cdot \ex{\prod_{i = 1}^{\prodsize} \sine(\wElem_i)} = \term_j.
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\end{align*}
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\end{proof}
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@ -44,40 +49,39 @@ We need to compute the variance of the $\prodsize$-way product $\est$. We wish
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\begin{equation}
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\sigsq \leq \sum_j \sigsq_j \label{eq:var-to-prove}.
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\end{equation}
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Therefore, substituting in the definition of variance for complex numbers,
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For notational convenience let $\cvar{j, j'} = \ex{\est_j \cdot \overline{\est_j'}} - \ex{\est_j}\ex{\conj{\est_{j'}}}$.
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Substituting in the definition of variance for complex numbers,
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\begin{align}
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\sigsq &= \ex{\sum_j \est_j \cdot \conj{\sum_{j'} \est_j'}} - \ex{\sum_j \est_j}\cdot\ex{\conj{\sum_{j'} \est_{j'}}}\nonumber\\
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&= \ex{\sum_j \est_j \cdot \sum_{j'} \conj{\est_j'}} - \ex{\sum_j \est_j}\cdot\ex{\sum_{j'} \conj{\est_{j'}}}\nonumber\\
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&= \sum_{j, j'}\left(\ex{\est_j \cdot \overline{\est_j'}} - \ex{\est_j}\ex{\overline{\est_{j'}}} = \cvar{j, j'}\right)\nonumber\\
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&= \sum_{j, j'}\left(\ex{\est_j \cdot \overline{\est_j'}} - \ex{\est_j}\ex{\overline{\est_{j'}}}\right)\nonumber\\
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&= \sum_j\ex{\est_j \cdot \overline{\est_j'}} - \ex{\est_j}\ex{\overline{\est_j}} + \sum_{j \neq j'}\cvar{j, j'}\nonumber\\
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&= \sum_j \sigsq_j + \sum_{j \neq j'}\cvar{j, j'} \label{eq:sigsq-jneqj}
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&= \sum_j \sigsq_j + \sum_{j \neq j'}\cvar{j, j'} \label{eq:sigsq}
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\end{align}
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\AR{The above is a terrible way to define $\lambda(j,j')$. Pretty much any reader will miss the fact that you defined it here. Define $\lambda(j,j')$ ideally outside the align statement especially since this definition will be used later on as well.}
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Notice that assuming independence of $\sigsq_j ~\forall j \in \sketchCols$, we can push the variance through the sum and obtain the result
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\begin{align*}
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&\sigsq - \sum_j \sigsq_j = \cvar{j, j'}\\
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&\implies \cvar{j, j'} \leq 0.
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\end{align*}
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Recall that we started this section out by seeking to prove \cref{eq:var-to-prove}. Should this be true, the use of $\leq$ in the above implication results from the fact that $\sigsq \leq \sum_j \sigsq_j \implies \cvar{j, j'} \leq 0$.
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\AH{I'm really not so sure about the above results. This was from a conversation we had months ago, but we're basing an implication on something we haven't proved. That doesn't seem right to me.}
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\AR{Yeah, the para above does not make sense.}
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\AH{The implication above was discussed months ago, but I don't see how it's true. Is it true?}
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One can see that \cref{eq:sigsq-jneqj} is composed of two addends. We now bound each of them separately.
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\subsection{Bounding $\sum_{j \neq j'}\cvar{j, j'}$}
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One can see that \cref{eq:sigsq} is composed of two addends. We now bound each of them separately.
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\subsection{Bounding $\cvar{j, j'}$}
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\AR{You need to re-write the stuff below. First in the 2nd equality suddenly the sum on $j\ne j'$ has vanished. Also I think you should first analyze $\lambda(j,j')$ for both $j=j'$ and $j\ne j'$ for as long as you can. Only when it is needed should you divide into the two cases-- do not do the division up front.}
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\begin{align*}
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\sum_{j \neq j'}\cvar{j, j'} &= \sum_{j \neq j'} \ex{\est_j \cdot \conj{\est_{j'}}} - \ex{\est_j}\cdot\ex{\conj{\est_{j'}}}\\
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&=\ex{\prod_{i = 1}^{\prodsize}\sum_{\wElem \in W}v_i(\wElem)s(\wElem)\ind{h(\wElem) = j}\cdot \prod_{i = 1}^{\prodsize}\sum_{\wElem' \in W}v_i(\wElem')\conj{s(\wElem')}\ind{h(\wElem') = j'}} - \ex{\prod_{i = 1}^{\prodsize}\sum_{\wElem \in W}v_i(\wElem)s(\wElem)\ind{h(\wElem) = j}}\cdot \ex{\prod_{i = 1}^{\prodsize}\sum_{\wElem' \in W}v_i(\wElem')\conj{s(\wElem')}\ind{h(\wElem') = j'}}\\
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&=\ex{\sum_{\substack{\wElem_1,\cdots,\wElem_\prodsize,\\\wElem'_1,\cdots,\wElem'_\prodsize\\\in W}}\prod_{i = 1}^{\prodsize}v_i(\wElem_i)s(\wElem_i)v_i(\wElem'_i)\conj{s(\wElem'_i)} \ind{h(\wElem_i) = j} \ind{h(\wElem'_i) = j'}} - \ex{\sum_{\substack{\wElem_1,\cdots, \wElem_\prodsize\\\in W}}\prod_{i = 1}^{\prodsize}v_i(\wElem_i)s(\wElem_i) \ind{h(\wElem_i) = j}}\cdot\ex{\sum_{\substack{\wElem'_1,\cdots, \wElem'_\prodsize\\\in W}}\prod_{i = 1}^{\prodsize}v_i(\wElem'_i)\conj{s(\wElem'_i)} \ind{h(\wElem'_i) = j'}}\\
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Notice we have two cases of $\cvar{j, j'}$, the first is when $j = j'$, i.e. $(\sigsq_j)$, and the second when $j \neq j'$.
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\begin{align}
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\cvar{j, j'} &= \ex{\est_j \cdot \conj{\est_{j'}}} - \ex{\est_j}\cdot\ex{\conj{\est_{j'}}}\nonumber\\
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&=\ex{\prod_{i = 1}^{\prodsize}\sum_{\wElem \in W}v_i(\wElem)s(\wElem)\ind{h(\wElem) = j}\cdot \prod_{i = 1}^{\prodsize}\sum_{\wElem' \in W}v_i(\wElem')\conj{s(\wElem')}\ind{h(\wElem') = j'}} - \ex{\prod_{i = 1}^{\prodsize}\sum_{\wElem \in W}v_i(\wElem)s(\wElem)\ind{h(\wElem) = j}}\cdot \ex{\prod_{i = 1}^{\prodsize}\sum_{\wElem' \in W}v_i(\wElem')\conj{s(\wElem')}\ind{h(\wElem') = j'}}\nonumber\\
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&=\ex{\sum_{\substack{\wElem_1,\cdots,\wElem_\prodsize,\\\wElem'_1,\cdots,\wElem'_\prodsize\\\in W}}\prod_{i = 1}^{\prodsize}v_i(\wElem_i)s(\wElem_i)v_i(\wElem'_i)\conj{s(\wElem'_i)} \ind{h(\wElem_i) = j} \ind{h(\wElem'_i) = j'}} - \ex{\sum_{\substack{\wElem_1,\cdots, \wElem_\prodsize\\\in W}}\prod_{i = 1}^{\prodsize}v_i(\wElem_i)s(\wElem_i) \ind{h(\wElem_i) = j}}\cdot\ex{\sum_{\substack{\wElem'_1,\cdots, \wElem'_\prodsize\\\in W}}\prod_{i = 1}^{\prodsize}v_i(\wElem'_i)\conj{s(\wElem'_i)} \ind{h(\wElem'_i) = j'}}\nonumber\\
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&=\sum_{\substack{\wElem_1,\cdots,\wElem_\prodsize,\\\wElem'_1,\cdots,\wElem'_\prodsize\\\in W}}\ex{\prod_{i = 1}^{\prodsize}v_i(\wElem_i)s(\wElem_i)v_i(\wElem'_i)\conj{s(\wElem'_i)}\ind{h(\wElem_i) = j}\ind{h(\wElem'_i) = j'}} - \ex{\prod_{i = 1}^{\prodsize} v_i(\wElem_i)s(\wElem_i) \ind{h(\wElem_i) = j}} \cdot \ex{\prod_{i = 1}^{\prodsize}v_i(\wElem'_i)\conj{s(\wElem'_i)}\ind{h(\wElem'_i) = j'}}\\
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&= \sum_{\substack{\wElem_1,\cdots,\wElem_\prodsize,\\\wElem'_1,\cdots,\wElem'_\prodsize\\\in W}}\prod_{i = 1}^{\prodsize}v_i(\wElem_i)v_i(\wElem'_i)\ex{\prod_{i = 1}^{\prodsize}s(\wElem_i)\conj{s(\wElem'_i)}\ind{h(\wElem_i) = j}\ind{h(\wElem'_i) = j'}} - \prod_{i = 1}^{\prodsize}v_i(\wElem_i)\ex{\prod_{i = 1}^{\prodsize}s(\wElem_i)\ind{h(\wElem_i) = j}}\cdot \prod_{i = 1}^{\prodsize}v_i(\wElem'_i)\ex{\prod_{i = 1}^{\prodsize}\conj{s(\wElem'_i)}\ind{h(\wElem_i') = j'}}\\
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&= \sum_{\substack{\wElem_1,\cdots,\wElem_\prodsize,\\\wElem'_1,\cdots,\wElem'_\prodsize\\\in W}}\prod_{i = 1}^{\prodsize}v_i(\wElem_i)v_i(\wElem'_i)\left(\ex{\prod_{i = 1}^{\prodsize}s(\wElem_i)\conj{s(\wElem'_i)}\ind{h(\wElem_i) = j}\ind{h(\wElem'_i) = j'}} - \ex{\prod_{i = 1}^{\prodsize}s(\wElem_i)\ind{h(\wElem_i) = j}}\cdot\ex{\prod_{i = 1}^{\prodsize}\conj{s(\wElem'_i)}\ind{h(\wElem_i') = j'}} \right).
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\end{align*}
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\AH{Perhaps a formal proof is necessary below.}
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\AR{Most definitely.}
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For $\term_1^{\cvar{j, j'}} = \ex{\prod_{i = 1}^{\prodsize}s(\wElem_i)s(\wElem'_i)\ind{h(\wElem_i) = j}\ind{h(\wElem'_i) = j'}}$, because hash function $h$ cannot bucket the same world to two different buckets, the only instance $\term_1^{\cvar{j, j'}} = 1$ occurs when there is no overlap between the $\wElem_i$ and $\wElem'_i$ variables. Given the condition of no overlap, $\term_1^{\cvar{j, j'}} = 1$ only with the further condition that $\forall i \in [\prodsize], \wElem_i = \wElem, \wElem'_i = \wElem', \wElem \neq \wElem'$. Notice, however, given the conditions, the product of the remaining expectations will cancel this out. Looking at the remaining two expectations $\term_2^{\cvar{j, j'}} = \ex{\prod_{i = 1}^{\prodsize}\sine(\wElem_i) \ind{\hfunc(\wElem_i) = j}} \cdot \ex{\prod_{i = 1}^{\prodsize}\conj{\sine(\wElem'_i)} \ind{\hfunc(\wElem'_i) = j'}}$, that $\term_2^{\cvar{j, j'}} = 1$ only when $\forall i \in [\prodsize], \wElem_i = \wElem, \wElem'_i = \wElem'$. Taken together, the constraints leave us with only one possible case for $\term_1^{\cvar{j, j'}} - \term_2^{\cvar{j, j'}} \neq 0$, when all variables are the same world. Thus,
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&= \sum_{\substack{\wElem_1,\cdots,\wElem_\prodsize,\\\wElem'_1,\cdots,\wElem'_\prodsize\\\in W}}\prod_{i = 1}^{\prodsize}v_i(\wElem_i)v_i(\wElem'_i)\ex{\prod_{i = 1}^{\prodsize}s(\wElem_i)\conj{s(\wElem'_i)}\ind{h(\wElem_i) = j}\ind{h(\wElem'_i) = j'}} - \prod_{i = 1}^{\prodsize}v_i(\wElem_i)\ex{\prod_{i = 1}^{\prodsize}s(\wElem_i)\ind{h(\wElem_i) = j}}\cdot \prod_{i = 1}^{\prodsize}v_i(\wElem'_i)\ex{\prod_{i = 1}^{\prodsize}\conj{s(\wElem'_i)}\ind{h(\wElem_i') = j'}}\nonumber\\
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&= \sum_{\substack{\wElem_1,\cdots,\wElem_\prodsize,\\\wElem'_1,\cdots,\wElem'_\prodsize\\\in W}}\prod_{i = 1}^{\prodsize}v_i(\wElem_i)v_i(\wElem'_i)\left(\ex{\prod_{i = 1}^{\prodsize}s(\wElem_i)\conj{s(\wElem'_i)}\ind{h(\wElem_i) = j}\ind{h(\wElem'_i) = j'}} - \ex{\prod_{i = 1}^{\prodsize}s(\wElem_i)\ind{h(\wElem_i) = j}}\cdot\ex{\prod_{i = 1}^{\prodsize}\conj{s(\wElem'_i)}\ind{h(\wElem_i') = j'}} \right).\label{eq:var-lambda-j-j'}
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\end{align}
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\AH{How can I present the derivation of the bounds below in a \textit{better} way?}
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Equation ~\eqref{eq:var-lambda-j-j'} for $j \neq j'$ bounds to the rightmost sum of \cref{eq:sigsq}. For $\term_1^{\cvar{j, j'}} = \ex{\prod_{i = 1}^{\prodsize}s(\wElem_i)s(\wElem'_i)\ind{h(\wElem_i) = j}\ind{h(\wElem'_i) = j'}}$, because hash function $h$ cannot bucket the same world to two different buckets, the only instance $\term_1^{\cvar{j, j'}} = 1$ occurs when there is no overlap between the $\wElem_i$ and $\wElem'_i$ variables. Given the condition of no overlap, $\term_1^{\cvar{j, j'}} = 1$ only with the further condition that $\forall i \in [\prodsize], \wElem_i = \wElem, \wElem'_i = \wElem', \wElem \neq \wElem'$. Notice, however, given the conditions, the product of the remaining expectations will cancel this out. Looking at the remaining two expectations $\term_2^{\cvar{j, j'}} = \ex{\prod_{i = 1}^{\prodsize}\sine(\wElem_i) \ind{\hfunc(\wElem_i) = j}} \cdot \ex{\prod_{i = 1}^{\prodsize}\conj{\sine(\wElem'_i)} \ind{\hfunc(\wElem'_i) = j'}}$, that $\term_2^{\cvar{j, j'}} = 1$ only when $\forall i \in [\prodsize], \wElem_i = \wElem, \wElem'_i = \wElem'$. Taken together, the constraints leave us with only one possible case for $\term_1^{\cvar{j, j'}} - \term_2^{\cvar{j, j'}} \neq 0$, when all variables are the same world. Thus,
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\begin{align}
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&\sum_{j \neq j'}\cvar{j, j'} = - \frac{1}{B^2}\sum_{\wElem \in W}\prod_{i = 1}^{\prodsize}v_i^2(\wElem)\label{eq:cvar-bound}.
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\end{align}
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@ -85,22 +89,23 @@ For $\term_1^{\cvar{j, j'}} = \ex{\prod_{i = 1}^{\prodsize}s(\wElem_i)s(\wElem'_
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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We now move on to bound the variance of a $\prodsize$-way join.
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Taking a look at the leftmost term of \cref{eq:sigsq}, we establish bounds the variance of the $j^{\text{th}}$ bucket of a $\prodsize$-way join. Note in this case that \cref{eq:var-lambda-j-j'} has that $j = j'$, and can be written in the following way,
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\begin{align}
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&\sigsq_j = \ex{\est_j \cdot \overline{\est_j}} - \ex{\est_j} \cdot \ex{\overline{\est_j}} \nonumber\\
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&= \ex{\prod_{i = 1}^{\prodsize}\sum_{w \in W_j}v_i(w)s(w) \cdot \prod_{i = 1}^\prodsize\sum_{w' \in W_j}v_i(w')\overline{s(w')}} -
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\ex{\prod_{i = 1}^{\prodsize}\sum_{w \in W_j}v_i(w)s(w)}\cdot \ex{\prod_{i = 1}^\prodsize\sum_{w' \in W_j}v_i(w')\overline{s(w')}}\nonumber\\
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&= \ex{\sum_{\substack{w_1...w_\prodsize\\w'_1...w'_\prodsize\\ \in W}}\prod_{i = 1}^\prodsize v_i(w_i)v(w'_i)s(w_i)\overline{s(w'_i)}\ind{h(w_i) = j}\ind{h(w'_i) = j}} -
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\ex{\sum_{w_1...w_\prodsize \in W} \prod_{i = 1}^\prodsize v_i(w_i)s(w_i)\ind{h(w_i) = j}} \cdot
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\ex{\sum_{w'_1...w'_\prodsize \in W} \prod_{i = 1}^\prodsize v_i(w'_i)\overline{s(w'_i)}\ind{h(w'_i) = j}}\nonumber\\
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=&\sum_{\substack{w_1...w_\prodsize\\w'_1...w'_\prodsize\\ \in W}}\ex{\prod_{i = 1}^\prodsize v_i(w_i)v_i(w'_i)s(w_i)\overline{s(w'_i)}\ind{h(w_i) = j}\ind{h(w'_i) = j}} -
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\ex{\prod_{i = 1}^kv_i(w_i)s(w_i)\ind{h(w_i) = j}} \cdot \ex{\prod_{i = 1}^\prodsize v_i(w'_i)\overline{s(w'_i)}\ind{h(w'_i) = j}}\nonumber\\
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%&\sigsq_j = \ex{\est_j \cdot \overline{\est_j}} - \ex{\est_j} \cdot \ex{\overline{\est_j}} \nonumber\\
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%&= \ex{\prod_{i = 1}^{\prodsize}\sum_{w \in W_j}v_i(w)s(w) \cdot \prod_{i = 1}^\prodsize\sum_{w' \in W_j}v_i(w')\overline{s(w')}} -
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%\ex{\prod_{i = 1}^{\prodsize}\sum_{w \in W_j}v_i(w)s(w)}\cdot \ex{\prod_{i = 1}^\prodsize\sum_{w' \in W_j}v_i(w')\overline{s(w')}}\nonumber\\
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%&= \ex{\sum_{\substack{w_1...w_\prodsize\\w'_1...w'_\prodsize\\ \in W}}\prod_{i = 1}^\prodsize v_i(w_i)v(w'_i)s(w_i)\overline{s(w'_i)}\ind{h(w_i) = j}\ind{h(w'_i) = j}} -
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%\ex{\sum_{w_1...w_\prodsize \in W} \prod_{i = 1}^\prodsize v_i(w_i)s(w_i)\ind{h(w_i) = j}} \cdot
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%\ex{\sum_{w'_1...w'_\prodsize \in W} \prod_{i = 1}^\prodsize v_i(w'_i)\overline{s(w'_i)}\ind{h(w'_i) = j}}\nonumber\\
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%=&\sum_{\substack{w_1...w_\prodsize\\w'_1...w'_\prodsize\\ \in W}}\ex{\prod_{i = 1}^\prodsize v_i(w_i)v_i(w'_i)s(w_i)\overline{s(w'_i)}\ind{h(w_i) = j}\ind{h(w'_i) = j}} -
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%\ex{\prod_{i = 1}^kv_i(w_i)s(w_i)\ind{h(w_i) = j}} \cdot \ex{\prod_{i = 1}^\prodsize v_i(w'_i)\overline{s(w'_i)}\ind{h(w'_i) = j}}\nonumber\\
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&= \sum_{\substack{w_1...w_\prodsize\\w'_1...w'_\prodsize\\ \in W}}\prod_{i = 1}^\prodsize v_i(w_i)v_i(w'_i)\cdot\left( \ex{\prod_{i = 1}^\prodsize s(w_i)\overline{s(w'_i)}\ind{h(w_i) = j}\ind{h(w'_i) = j}} -
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\ex{\prod_{i = 1}^ks(w_i)\ind{h(w_i) = j}}\cdot \ex{\prod_{i = 1}^\prodsize\overline{s(w'_i)}\ind{h(w'_i) = j}} \right)\label{eq:sig-j-last}.
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\end{align}
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Before proceeding, we introduce some notation and terminology that will aid in communicating the bounds we are about to establish. We refer to the leftmost expectation of \cref{eq:sig-j-last} in the following way:
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\AR{dangling eq ref}
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\AR{dangling eq ref}\AH{I don't see one}
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\[\term_1\left(\wElem_1,\ldots,\wElem_\prodsize, \wElem_1',\ldots, \wElem_\prodsize'\right) = \ex{\prod_{i = 1}^\prodsize s(w_i)\overline{s(w'_i)}\ind{h(w_i) = j}\ind{h(w'_i) = j}}.%\text{, and}
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\]
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%\[\term_2\left(\wElem_1,\ldots,\wElem_\prodsize, \wElem_1',\ldots, \wElem_\prodsize'\right) = \ex{\prod_{i = 1}^ks(w_i)\ind{h(w_i) = j}}\cdot \ex{\prod_{i = 1}^\prodsize\overline{s(w'_i)}\ind{h(w'_i) = j}}. \]
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Reference in a new issue