Title: Generalized Uniformity Testing,
Abstract: In this work, we revisit the problem of uniformity testing of discrete
probability distributions. A fundamental problem in distribution testing,
testing uniformity over a known domain has been addressed over a significant
line of works, and is by now fully understood.
The complexity of deciding whether an unknown distribution is uniform over
its unknown (and arbitrary) support, however, is much less clear. Yet, this
task arises as soon as no prior knowledge on the domain is available, or
whenever the samples originate from an unknown and unstructured universe. In
this work, we introduce and study this generalized uniformity testing question,
and establish nearly tight upper and lower bound showing that -- quite
surprisingly -- its sample complexity significantly differs from the
known-domain case. Moreover, our algorithm is intrinsically adaptive, in
contrast to the overwhelming majority of known distribution testing algorithms.