How to learn a graph from smooth signals

TL;DR

This paper introduces a scalable framework for learning graph structures from smooth signals, utilizing weighted l1 minimization to produce sparse, accurate graphs that outperform existing methods in various settings.

Contribution

It presents a novel graph learning model based on smoothness assumptions, with efficient algorithms, that improves upon state-of-the-art techniques.

Findings

The proposed model achieves better accuracy than existing methods.

Efficient primal-dual algorithms enable scalability to large datasets.

The framework unifies and extends known graph learning techniques.

Abstract

We propose a framework that learns the graph structure underlying a set of smooth signals. Given $X\in\mathbb{R}^{m\times n}$ whose rows reside on the vertices of an unknown graph, we learn the edge weights $w\in\mathbb{R}_+^{m(m-1)/2}$ under the smoothness assumption that $\text{tr}{X^\top LX}$ is small. We show that the problem is a weighted $\ell$-1 minimization that leads to naturally sparse solutions. We point out how known graph learning or construction techniques fall within our framework and propose a new model that performs better than the state of the art in many settings. We present efficient, scalable primal-dual based algorithms for both our model and the previous state of the art, and evaluate their performance on artificial and real data.