  Graph-based representations play a key role in machine learning. The
fundamental step in these representations is the association of a graph
structure to a dataset. In this paper, we propose a method that aims at finding
a block sparse representation of the graph signal leading to a modular graph
whose Laplacian matrix admits the found dictionary as its eigenvectors. The
role of sparsity here is to induce a band-limited representation or,
equivalently, a modular structure of the graph. The proposed strategy is
composed of two optimization steps: i) learning an orthonormal sparsifying
transform from the data; ii) recovering the Laplacian, and then topology, from
the transform. The first step is achieved through an iterative algorithm whose
alternating intermediate solutions are expressed in closed form. The second
step recovers the Laplacian matrix from the sparsifying transform through a
convex optimization method. Numerical results corroborate the effectiveness of
the proposed methods over both synthetic data and real brain data, used for
inferring the brain functionality network through experiments conducted over
patients affected by epilepsy.
