How to learn a graph from smooth signals

We propose a framework that learns the graph structure underlying a set of smooth signals. Given a real m by n matrix X whose rows reside on the vertices of an unknown graph, we learn the edge weights w under the smoothness assumption that trace(X^TLX) is small. We show that the problem is a weighted l-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.