How to Learn a Graph from Smooth Signals

We propose a framework to learn the graph structure underlying a set of smooth signals. Given X∈\mathbbR^m\times n whose rows reside on the vertices of an unknown graph, we learn the edge weights w∈\mathbbR_+^m(m-1)/2 under the smoothness assumption that \rm trX^⊤LX is small, where L is the graph Laplacian. We show that the problem is a weighted \ell-1 minimization that leads to naturally sparse solutions. We prove that the standard graph construction with Gaussian weights w_ij = \exp(-\frac1σ^2\|x_i-x_j\|^2) and the previous state of the art are special cases of our framework. We propose a new model and present efficient, scalable primal-dual based algorithms both for this and the previous state of the art, to evaluate their performance on artificial and real data. The new model performs best in most settings.