  This work proposes a novel method for semi-supervised learning from partially
labeled massive network-structured datasets, i.e., big data over networks. We
model the underlying hypothesis, which relates data points to labels, as a
graph signal, defined over some graph (network) structure intrinsic to the
dataset. Following the key principle of supervised learning, i.e., similar
inputs yield similar outputs, we require the graph signals induced by labels to
have small total variation. Accordingly, we formulate the problem of learning
the labels of data points as a non-smooth convex optimization problem which
amounts to balancing between the empirical loss, i.e., the discrepancy with
some partially available label information, and the smoothness quantified by
the total variation of the learned graph signal. We solve this optimization
problem by appealing to a recently proposed preconditioned variant of the
popular primal-dual method by Pock and Chambolle, which results in a sparse
label propagation algorithm. This learning algorithm allows for a highly
scalable implementation as message passing over the underlying data graph. By
applying concepts of compressed sensing to the learning problem, we are also
able to provide a transparent sufficient condition on the underlying network
structure such that accurate learning of the labels is possible. We also
present an implementation of the message passing formulation allows for a
highly scalable implementation in big data frameworks.
