  We present a convex cone program to infer the latent probability matrix of a
random dot product graph (RDPG). The optimization problem maximizes the
Bernoulli maximum likelihood function with an added nuclear norm regularization
term. The dual problem has a particularly nice form, related to the well-known
semidefinite program relaxation of the MaxCut problem. Using the primal-dual
optimality conditions, we bound the entries and rank of the primal and dual
solutions. Furthermore, we bound the optimal objective value and prove
asymptotic consistency of the probability estimates of a slightly modified
model under mild technical assumptions. Our experiments on synthetic RDPGs not
only recover natural clusters, but also reveal the underlying low-dimensional
geometry of the original data. We also demonstrate that the method recovers
latent structure in the Karate Club Graph and synthetic U.S. Senate vote graphs
and is scalable to graphs with up to a few hundred nodes.
