  Sum-Product Networks (SPNs) can be regarded as a form of deep graphical
models that compactly represent deeply factored and mixed distributions. An SPN
is a rooted directed acyclic graph (DAG) consisting of a set of leaves
(corresponding to base distributions), a set of sum nodes (which represent
mixtures of their children distributions) and a set of product nodes
(representing the products of its children distributions).
  In this work, we initiate the study of the sample complexity of PAC-learning
the set of distributions that correspond to SPNs. We show that the sample
complexity of learning tree structured SPNs with the usual type of leaves
(i.e., Gaussian or discrete) grows at most linearly (up to logarithmic factors)
with the number of parameters of the SPN. More specifically, we show that the
class of distributions that corresponds to tree structured Gaussian SPNs with
$k$ mixing weights and $e$ ($d$-dimensional Gaussian) leaves can be learned
within Total Variation error $\epsilon$ using at most
$\widetilde{O}(\frac{ed^2+k}{\epsilon^2})$ samples. A similar result holds for
tree structured SPNs with discrete leaves.
  We obtain the upper bounds based on the recently proposed notion of
distribution compression schemes. More specifically, we show that if a (base)
class of distributions $\mathcal{F}$ admits an "efficient" compression, then
the class of tree structured SPNs with leaves from $\mathcal{F}$ also admits an
efficient compression.
