Problem 85: Sample Stretching

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Suggested by Ryan O'Donnell
Source FOCS 2017
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Let $p$ be an unknown (discrete) probability distribution over a discrete domain $\Omega$ (e.g., $\Omega=[n]$) and $k\in\mathbb{N}$. As usual, $\operatorname{d}_{\rm TV}$ denotes the total variation distance.

What is the minimum value of $\varepsilon\in(0,1]$ such that there exists an algorithm that, on input $k$ i.i.d. samples from $p$, outputs $k+1$ i.i.d. samples from some $p'$ such that $\operatorname{d}_{\rm TV}(p,p')\leq \varepsilon$?

Note: this is of a similar spirit as Open Problem 69 and the setting of sampling correctors and improvers [CanonneGR-16].