Semi-Supervised Learning via Sparse Label Propagation

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.