  We present a novel k-way high-dimensional graphical model called the
Generalized Root Model (GRM) that explicitly models dependencies between
variable sets of size k > 2---where k = 2 is the standard pairwise graphical
model. This model is based on taking the k-th root of the original sufficient
statistics of any univariate exponential family with positive sufficient
statistics, including the Poisson and exponential distributions. As in the
recent work with square root graphical (SQR) models [Inouye et al.
2016]---which was restricted to pairwise dependencies---we give the conditions
of the parameters that are needed for normalization using the radial
conditionals similar to the pairwise case [Inouye et al. 2016]. In particular,
we show that the Poisson GRM has no restrictions on the parameters and the
exponential GRM only has a restriction akin to negative definiteness. We
develop a simple but general learning algorithm based on L1-regularized
node-wise regressions. We also present a general way of numerically
approximating the log partition function and associated derivatives of the GRM
univariate node conditionals---in contrast to [Inouye et al. 2016], which only
provided algorithm for estimating the exponential SQR. To illustrate GRM, we
model word counts with a Poisson GRM and show the associated k-sized variable
sets. We finish by discussing methods for reducing the parameter space in
various situations.
