Title: Which Distribution Distances are Sublinearly Testable?,
Abstract: Given samples from an unknown distribution $p$ and a description of a
distribution $q$, are $p$ and $q$ close or far? This question of "identity
testing" has received significant attention in the case of testing whether $p$
and $q$ are equal or far in total variation distance. However, in recent work,
the following questions have been been critical to solving problems at the
frontiers of distribution testing:
-Alternative Distances: Can we test whether $p$ and $q$ are far in other
distances, say Hellinger?
-Tolerance: Can we test when $p$ and $q$ are close, rather than equal? And if
so, close in which distances?
Motivated by these questions, we characterize the complexity of distribution
testing under a variety of distances, including total variation, $\ell_2$,
Hellinger, Kullback-Leibler, and $\chi^2$. For each pair of distances $d_1$ and
$d_2$, we study the complexity of testing if $p$ and $q$ are close in $d_1$
versus far in $d_2$, with a focus on identifying which problems allow strongly
sublinear testers (i.e., those with complexity $O(n^{1 - \gamma})$ for some
$\gamma > 0$ where $n$ is the size of the support of the distributions $p$ and
$q$). We provide matching upper and lower bounds for each case. We also study
these questions in the case where we only have samples from $q$ (equivalence
testing), showing qualitative differences from identity testing in terms of
when tolerance can be achieved. Our algorithms fall into the classical paradigm
of $\chi^2$-statistics, but require crucial changes to handle the challenges
introduced by each distance we consider. Finally, we survey other recent
results in an attempt to serve as a reference for the complexity of various
distribution testing problems.