Editing Open Problems:86
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{{Header | {{Header | ||
| + | |title=Equivalence testing lower bound via communication complexity | ||
|source=focs17 | |source=focs17 | ||
|who=Clément Canonne | |who=Clément Canonne | ||
}} | }} | ||
| − | Blais, Canonne, and Gur {{cite|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 | + | Blais, Canonne, and Gur {{cite|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 {{cite|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 [[Instance-specific Hellinger testing|this other open problem]]). |
Recall that the ''equivalence'' testing problem (also known as closeness testing) is the generalization of identity testing where, instead of having a known reference distribution $q$ and access to samples from an unknown $p$, now both distributions are unknown and only available via samples. | Recall that the ''equivalence'' testing problem (also known as closeness testing) is the generalization of identity testing where, instead of having a known reference distribution $q$ and access to samples from an unknown $p$, now both distributions are unknown and only available via samples. | ||
| − | Can one use the {{cite|BlaisCG-17}} framework to re-establish an $\tilde{\Omega}(n^{2/3})$ sample complexity lower bound for equivalence testing? If so, can it yield some sort of ''instance-specific'' equivalence testing lower bound, for some meaningful notion of | + | Can one use the {{cite|BlaisCG-17}} framework to re-establish an $\tilde{\Omega}(n^{2/3})$ sample complexity lower bound for equivalence testing? If so, can it yield some sort of ''instance-specific'' equivalence testing lower bound, for some meaningful notion of "instance-specific"? |
