  We develop Square Root Graphical Models (SQR), a novel class of parametric
graphical models that provides multivariate generalizations of univariate
exponential family distributions. Previous multivariate graphical models [Yang
et al. 2015] did not allow positive dependencies for the exponential and
Poisson generalizations. However, in many real-world datasets, variables
clearly have positive dependencies. For example, the airport delay time in New
York---modeled as an exponential distribution---is positively related to the
delay time in Boston. With this motivation, we give an example of our model
class derived from the univariate exponential distribution that allows for
almost arbitrary positive and negative dependencies with only a mild condition
on the parameter matrix---a condition akin to the positive definiteness of the
Gaussian covariance matrix. Our Poisson generalization allows for both positive
and negative dependencies without any constraints on the parameter values. We
also develop parameter estimation methods using node-wise regressions with
$\ell_1$ regularization and likelihood approximation methods using sampling.
Finally, we demonstrate our exponential generalization on a synthetic dataset
and a real-world dataset of airport delay times.
