Tree Density Estimation

We study the problem of estimating the density
$f({\mathbf {x}})$
of a random vector
${ {\mathbf {X}}}$
in
${\mathbb R}^{d}$
. For a spanning tree
$T$
defined on the vertex set
$\{1, {\dots },d\}$
, the tree density
$f_{T}$
is a product of bivariate conditional densities. An optimal spanning tree minimizes the Kullback-Leibler divergence between
$f$
and
$f_{T}$
. From i.i.d. data we identify an optimal tree
$T^{*}$
and efficiently construct a tree density estimate
$f_{n}$
such that, without any regularity conditions on the density
$f$
, one has
$\lim _{n\to \infty } \int | f_{n}({\mathbf {x}})-f_{T^{*}}({\mathbf {x}})|d {\mathbf {x}}=0$
a.s. For Lipschitz
$f$
with bounded support,
${\mathbb E}\left \{{ \int | f_{n}({\mathbf {x}})-f_{T^{*}}({\mathbf {x}})|d {\mathbf {x}}}\right \}=O\big (n^{-1/4}\big)$
, a dimension-free rate.