# Semi-supervised Learning in Network-Structured Data via Total Variation Minimization

## TL;DR

This paper introduces a scalable semi-supervised learning method for network-structured data using total variation minimization, leveraging graph signal recovery and network flow optimization for effective label propagation.

## Contribution

It presents a novel scalable algorithm based on TV minimization and primal-dual methods, with theoretical guarantees for cluster recovery in large network datasets.

## Key findings

- Algorithm is highly scalable for big data applications.
- Theoretical conditions guarantee cluster recovery via TV minimization.
- Numerical experiments confirm effectiveness and scalability.

## Abstract

We propose and analyze a method for semi-supervised learning from partially-labeled network-structured data. Our approach is based on a graph signal recovery interpretation under a clustering hypothesis that labels of data points belonging to the same well-connected subset (cluster) are similar valued. This lends naturally to learning the labels by total variation (TV) minimization, which we solve by applying a recently proposed primal-dual method for non-smooth convex optimization. The resulting algorithm allows for a highly scalable implementation using message passing over the underlying empirical graph, which renders the algorithm suitable for big data applications. By applying tools of compressed sensing, we derive a sufficient condition on the underlying network structure such that TV minimization recovers clusters in the empirical graph of the data. In particular, we show that the proposed primal-dual method amounts to maximizing network flows over the empirical graph of the dataset. Moreover, the learning accuracy of the proposed algorithm is linked to the set of network flows between data points having known labels. The effectiveness and scalability of our approach is verified by numerical experiments.