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| {{Header | | {{Header |
| + | |title=Title of the problem |
| |source=baltimore16 | | |source=baltimore16 |
| |who=Clément Canonne | | |who=Clément Canonne |
| }} | | }} |
− | | + | The open problem will appear here. |
− | Let $p$ be an unknown (discrete) probability distribution over product space $[n]\times [n]$, and $\varepsilon\in(0,1]$. Suppose $p$ is ''$\varepsilon$-close to independent'' (in total variation distance), i.e., there exists a product distribution $q=q_1\times q_2$ on $[n]\times [n]$ such that
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− | $$
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− | d_{\rm TV}(p,q) = \max_{S\subseteq [n]\times[n]} \left( p(S) - q(S)\right) \leq \varepsilon.
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− | $$
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− | Given access to independent samples drawn from $p$, the goal is to ''correct'' $p$, that is, to provide access to independent samples from a distribution $\tilde{p}$ satisfying (with high probability):
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− | * $d_{\rm TV}(p,\tilde{p}) = O(\varepsilon)$ (i.e., the corrected distribution is faithful to the original one),
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− | * $\tilde{p} = \tilde{p}_1\times \tilde{p}_2$ (i.e., the corrected distribution is a product distribution),
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− | with a ''rate'' as good as possible, where the rate is the number of samples from $p$ required to provide a single sample from $\tilde{p}$.
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− | Achieving a rate of $2$ is simple: drawing $(x_1,y_1)$ and $(x_2,y_2)$ from $p$ and outputting $(x_1,y_2)$ provides sample access to $\tilde{p} = p_1\times p_2$, which can be shown to be $3\varepsilon$-close to $p$ {{cite|BatuFFKRW-01}}.
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− | '''Question 1:''' Is a rate $r < 2$ achievable? What about an amortized rate (to provide $q=o(n^2)$ samples from the same distribution $\tilde{p}$<ref>The restriction $o(n^2)$ comes from the fact that, after $n^2$ samples, one can actually learn the distribution $p$, and then compute a good corrected definition $\tilde{p}$ offline. Hence, the range of interest is when having to provide a number of samples negligible in front of what learning $p$ would require.</ref>)?
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− | '''Question 2:''' What about the same question, when relaxing the second item to only ask that $p$ be ''improved'': that is, to provide sample access to a distribution $\tilde{p}$ guaranteed to be $\frac{\varepsilon}{2}$-close to a product distribution?
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− | '''Note:''' This question fits within the framework of “sampling improvers,” introduced in {{cite|CanonneGR-16}}. In this framework, given access to a probability distribution only close to having a desired property, one aims at providing access to corrected samples from a nearby distribution that exhibits this property.
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− | <references />
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