  We consider the problem of inferring graph topology from smooth graph signals
in a novel but practical scenario where data are located in distributed clients
and prohibited from leaving local clients due to factors such as privacy
concerns. The main difficulty in this task is how to exploit the potentially
heterogeneous data of all clients under data silos. To this end, we first
propose an auto-weighted multiple graph learning model to jointly learn a
personalized graph for each local client and a single consensus graph for all
clients. The personalized graphs match local data distributions, thereby
mitigating data heterogeneity, while the consensus graph captures the global
information. Moreover, the model can automatically assign appropriate
contribution weights to local graphs based on their similarity to the consensus
graph. We next devise a tailored algorithm to solve the induced problem, where
all raw data are processed locally without leaving clients. Theoretically, we
establish a provable estimation error bound and convergence analysis for the
proposed model and algorithm. Finally, extensive experiments on synthetic and
real data are carried out, and the results illustrate that our approach can
learn graphs effectively in the target scenario.
