  Graphical models are useful tools for describing structured high-dimensional
probability distributions. Development of efficient algorithms for learning
graphical models with least amount of data remains an active research topic.
Reconstruction of graphical models that describe the statistics of discrete
variables is a particularly challenging problem, for which the maximum
likelihood approach is intractable. In this work, we provide the first
sample-efficient method based on the Interaction Screening framework that
allows one to provably learn fully general discrete factor models with
node-specific discrete alphabets and multi-body interactions, specified in an
arbitrary basis. We identify a single condition related to model
parametrization that leads to rigorous guarantees on the recovery of model
structure and parameters in any error norm, and is readily verifiable for a
large class of models. Importantly, our bounds make explicit distinction
between parameters that are proper to the model and priors used as an input to
the algorithm. Finally, we show that the Interaction Screening framework
includes all models previously considered in the literature as special cases,
and for which our analysis shows a systematic improvement in sample complexity.
