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@@ -60,4 +60,6 @@ One thing we can try is to tighten the second line.
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Recall that for each dimension $i$ a random subset $S_i\subset X$ is selected and the value of $f_i(x)$ is $\min_{s\in S_i} d(x,s)$.
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We want to show that for any fixed $x,y\in X$ and any dimension $i$ the event that distance $|f_i(x)-f_i(y)|^2$ is much smaller than $d(x,y)^2$ happends with high probability.
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Now consider a subset $S_i$ by sampling each node in $X$ iid with probability $2^{-i}$.
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\end{document}
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