11 lines
1.6 KiB
TeX
11 lines
1.6 KiB
TeX
%root: main.tex
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\section{David's Scheme}
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Here we define a sampling scheme for which to compare to our scheme. Given two vectors, $\sone = \vecform{v}_1$ and $\stwo = \vecform{v}_2 \had \cdots \had \vecform{v}_\numTup$, we want to estimate $\sone \cdot \stwo$ using sampling. Recall that $\had$ is defined as the pointwise product of all vector elements. Specifically, a vector $\wVec = \vecform{v}_1 \had \vecform{v}_2$ iff for all $i \in [\veclen], \vecform{\wElem}[i] = \vecform{v}_1[i] \cdot \vecform{v}_2[i]$.
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It can be shown that with $O(\frac{1}{\epsilon^2})$ words of space, with constant probability we can achieve an additive error of $O(\epsilon \norm{\sone}_2 \hp(\vecform{v}_2,\ldots, \vecform{v}_\numTup))$.
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\subsection{POS queries}
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Further, define $\sone = \vecform{v}_{1} + \cdots + \vecform{v}_{\numTup}$ and $\stwo = \stwo_1 \had \cdots \had \stwo_{\prodsize}$, where for each $i \in [\prodsize]$, $\stwo_i$ is a summation of at most $\numTup$ vectors. Since we are measuring the error of estimating $\prodsize$ products, set $\sone = \wVec[v]_1$ and $\stwo = \wVec[v]_2 \had \cdots \had \wVec[v]_\prodsize$. We want to estimate the hadamard product $\sone \had \stwo$ through sampling.
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\AH{This part I am not sure on...David speaks of taking a random sample from a stream, but in our situation, where does the stream originate?}
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\AH{Also, why is it a fair comparison to only sample one POS term, and not from all POS terms? Sorry if this is a trivial question, but I don't understand, that if we are comparing with sampling, why is it that we assume we have all of $\stwo$? Shouldn't a fair comparison require that we sample from each of the $\prodsize$ terms?} |