  We investigate a generic problem of learning pairwise exponential family
graphical models with pairwise sufficient statistics defined by a global
mapping function, e.g., Mercer kernels. This subclass of pairwise graphical
models allow us to flexibly capture complex interactions among variables beyond
pairwise product. We propose two $\ell_1$-norm penalized maximum likelihood
estimators to learn the model parameters from i.i.d. samples. The first one is
a joint estimator which estimates all the parameters simultaneously. The second
one is a node-wise conditional estimator which estimates the parameters
individually for each node. For both estimators, we show that under proper
conditions the extra flexibility gained in our model comes at almost no cost of
statistical and computational efficiency. We demonstrate the advantages of our
model over state-of-the-art methods on synthetic and real datasets.
