Problem 86: Equivalence Testing Lower Bound via Communication Complexity

Suggested by	Clément Canonne
Source	FOCS 2017
Short link	https://sublinear.info/86

Blais, Canonne, and Gur [BlaisCG-17] recently described a reduction technique to obtain distribution testing lower bounds from communication complexity (specifically, even from the simultaneous message-passing (SMP) setting in communication complexity). Using this technique, which is the analogue of the "usual property testing" framework of Blais, Brody, and Matulef [BlaisBM-12], they prove and (re)-derive lower bounds for many of the standard distribution testing questions, including identity testing (and instance-specific identity testing — see this other open problem).

Can one use this framework to re-establish an $\tilde{\Omega}(n^{2/3})$ sample complexity lower bound for equivalence testing (a.k.a. closeness testing; where now both distributions are unknown, instead of one known and one unknown, available through sample access)? If so, can it yield some sort of instance-specific equivalence testing lower bound?