  Graphs are playing a crucial role in different fields since they are powerful
tools to unveil intrinsic relationships among signals. In many scenarios, an
accurate graph structure representing signals is not available at all and that
motivates people to learn a reliable graph structure directly from observed
signals. However, in real life, it is inevitable that there exists uncertainty
in the observed signals due to noise measurements or limited observability,
which causes a reduction in reliability of the learned graph. To this end, we
propose a graph learning framework using Wasserstein distributionally robust
optimization (WDRO) which handles uncertainty in data by defining an
uncertainty set on distributions of the observed data. Specifically, two models
are developed, one of which assumes all distributions in uncertainty set are
Gaussian distributions and the other one has no prior distributional
assumption. Instead of using interior point method directly, we propose two
algorithms to solve the corresponding models and show that our algorithms are
more time-saving. In addition, we also reformulate both two models into
Semi-Definite Programming (SDP), and illustrate that they are intractable in
the scenario of large-scale graph. Experiments on both synthetic and real world
data are carried out to validate the proposed framework, which show that our
scheme can learn a reliable graph in the context of uncertainty.
