Problem 69: Correcting Independence of Distributions

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Suggested by Clément Canonne
Source Baltimore 2016
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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 $$ d_{\rm TV}(p,q) = \max_{S\subseteq [n]\times[n]} \left( p(S) - q(S)\right) \leq \varepsilon. $$

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):

  • $d_{\rm TV}(p,\tilde{p}) = O(\varepsilon)$ (i.e., the corrected distribution is faithful to the original one),
  • $\tilde{p} = \tilde{p}_1\times \tilde{p}_2$ (i.e., the corrected distribution is a product distribution),

with a rate as good as possible, where the rate is the number of samples from $p$ required to provide a single sample from $\tilde{p}$.

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 $p$ [BatuFFKRW-01].

Question 1: Is a rate $r < 2$ achievable? What about an amortized rate (to provide $q=o(n^2)$ samples from the same distribution $\tilde{p}$[1])?

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?

Note: This question fits within the framework of “sampling improvers,” introduced in [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.

  1. 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.