  We investigate the problem of identity testing for multidimensional histogram
distributions. A distribution $p: D \rightarrow \mathbb{R}_+$, where $D
\subseteq \mathbb{R}^d$, is called a $k$-histogram if there exists a partition
of the domain into $k$ axis-aligned rectangles such that $p$ is constant within
each such rectangle. Histograms are one of the most fundamental nonparametric
families of distributions and have been extensively studied in computer science
and statistics. We give the first identity tester for this problem with {\em
sub-learning} sample complexity in any fixed dimension and a nearly-matching
sample complexity lower bound.
  In more detail, let $q$ be an unknown $d$-dimensional $k$-histogram
distribution in fixed dimension $d$, and $p$ be an explicitly given
$d$-dimensional $k$-histogram. We want to correctly distinguish, with
probability at least $2/3$, between the case that $p = q$ versus $\|p-q\|_1
\geq \epsilon$. We design an algorithm for this hypothesis testing problem with
sample complexity $O((\sqrt{k}/\epsilon^2) 2^{d/2} \log^{2.5 d}(k/\epsilon))$
that runs in sample-polynomial time. Our algorithm is robust to model
misspecification, i.e., succeeds even if $q$ is only promised to be {\em close}
to a $k$-histogram. Moreover, for $k = 2^{\Omega(d)}$, we show a sample
complexity lower bound of $(\sqrt{k}/\epsilon^2) \cdot \Omega(\log(k)/d)^{d-1}$
when $d\geq 2$. That is, for any fixed dimension $d$, our upper and lower
bounds are nearly matching. Prior to our work, the sample complexity of the
$d=1$ case was well-understood, but no algorithm with sub-learning sample
complexity was known, even for $d=2$. Our new upper and lower bounds have
interesting conceptual implications regarding the relation between learning and
testing in this setting.
