Problem 82: Beyond Identity Testing

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Suggested by Clément Canonne
Source FOCS 2017
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Given access to i.i.d. samples from two unknown probability distributions $p$ and $q$ over a discrete domain (say $[n]=\{1,\dots,n\}$) and distance parameter $\varepsilon\in(0,1]$, the equivalence (also known as closeness) testing problem asks to distinguish w.h.p. between (i) $p=q$ and (ii) $\operatorname{d}_{\rm TV}(p,q)>\varepsilon$. (The identity problem is the analogue when $q$ is fixed and explicitly known beforehand.)

Equivalence up to a permutation would then be the variant where one must test whether $p$ and $q$ are equal up to relabeling of the elements: (i) $\exists \pi\in\mathcal{S}_n$ s.t. $p=q\circ\pi$ vs. (ii) $\forall \pi\in\mathcal{S}_n$, $\operatorname{d}_{\rm TV}(p,q\circ \pi)>\varepsilon$. Results of Valiant and Valiant [ValiantV-11] imply that this question has sample complexity $\Theta\left(\frac{n}{\varepsilon^2\log n}\right)$.

The most general question then is:

Given a fixed class $\mathcal{F}$ of functions from $[n]$ to $[m]$, distinguish between (i) $\exists \pi\in\mathcal{F}$ s.t. $p=q\circ\pi$ vs. (ii) $\forall \pi\in\mathcal{F}$, $\operatorname{d}_{\rm TV}(p,q\circ \pi)>\varepsilon$.

In particular, one can study how the sample complexity depends on $\mathcal{F}$, or what it is for some classes of interest (e.g., $n=m$ for $\mathcal{F}$ a subgroup of the symmetric group $\mathcal{S}_n$; or $m\ll n$ and $\mathcal{F}$ being a class of “coarsenings,” capturing whether $p$ and $q$ are the same distribution but with a different discretization/binning).